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Letters to the Editor
The Mathematical Intelligencer
encourages comments about the
material in this issue. Letters
to the editor should be sent to the
editor-in-chief, Chandler Davis.
Girls and Boys in Moscow
If a wild goose came across Konrad
Lorenz's wonderful books on ethology,
it would read with great interest and
probably would like to add something.
I have a similar feeling reading about a
"country from which ... reliable data
is not obtainable" in "Impoverishment,
Feminization, and Glass Ceilings:
Women in Mathematics in Russia" by
Karin Johnsgard (Mathematical Intelligencer, vol. 22 (2000), no. 4, 20-32).
Let me first thank her for her sincere
interest and sympathy for Russia's
(certainly difficult) situation; but let
me add a few comments.
I am a teacher in a specialized math
school which selects students from the
whole Moscow region by running a se
ries of problem-solving sessions. (Oc
casionally physics problems are in
cluded.) Usually 100-200 students aged
13 and 14 participate in these sessions
(each student comes 2-4 times), and
the 20-25 students with the best results
are selected and invited to the school.
Typically most students that come
to the problem session are boys. Writ
ing this, I have looked in our files. In
1996 there were about 60 girls among
270 applicants; the disproportion is
similar among the students with the
best results, with 6 girls among the 25
students selected. In some years the
disproportion was even greater, and
we decided to lower the threshold
somewhat for girls (which has evident
drawbacks). Similarly in departments
of mathematics, most applicants are
male and most students are male.
Karin Johnsgard writes, "American
professors know that their female stu
dents are as good as and often better
than their male students; why isn't this
obvious to our Russian counterparts?
[emphasis hers]" What is "this"? That
some female students are better than
most male students? This is indeed ob
vious. (Nor have I seen any indication
that girls have special difficulties in
"time-critical competitions," as Johns
gard suggests. Several girls from the
class mentioned above were winners in
the Moscow Mathematics Olympiad. I
was sorry, by the way, that one of these
told us later that she does not want to
continue mathematics studies.) On the
other hand, we do find that more boys
than girls are interested in mathemat
ics and perform well. Thus the graph
in the accompanying figure shows re
sults in a mathematics contest where
simple math problems were sent to
schools with an open invitation to stu
dents to write down their solutions and
send them in by mail.
I am not sure that profound insights
can be gained by measuring correla
tions between gender (or race) and
scientific achievements. But I believe
that, whatever statistics are gathered,
one should set aside one's preconcep
tions and deal with the facts as one
finds them.
Alexander Shen
Institute for Problems of Information
Transmission
Ermolovoi 19
K-51 Moscow GSP-4, 1 01 447
Russia
e-mail: shen@landau.ac.ru
.1' ..... , •' •,
I '
" .... .
0 120 357 boys and 191 girls ages 10-14 years have sent their papers with solutions of 20 prob
lems. Grades are in the range 0 to 120. Solid line is a histogram for girls; dotted line is for
boys.
© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 1, 2002 3
GERALD T. CARGO, JACK E. GRAVER, AND JOHN L. TROUTMAN
Designing a Mirror that Inverts in a Circe
Dedicated to our mentors, George Piranian, Ernst Snapper, and Max Schiffer
• f Cf6 is a circle with center 0 and P is a point distinct from 0 in the plane of Cf6, the � inverse (image) of P under inversion in Cf6 is the unique point Q on the ray from 0
through P for which the product of the lengths of the segments OQ and OP equals
the square of the radius of Cf6. As with reflection in a line, inversion in a circle can
easily be carried out pointwise with a straightedge and a pair of compasses.
Introduction
During the early part of the Industrial Revolution, engineers and mathematicians tried to design linkages to carry out these transformations. Linkages for reflection in a line were easy to produce. The interest in the more difficult problem of designing a linkage for inversion in a circle 'i6 is based on the well-known fact that, under inversion in 'i6, circles through 0 become lines not through 0, and lines not through 0 become circles through 0. In 1864 the French military engineer Peaucellier designed a linkage that converts circular motion to mathematically perfect linear motion. Cf. [1; Ch. 4] and [2].
Because reflection in a line can be effected with a flat mirror, while controlled optical distortions can be produced through reflection (in the optical sense) in curved mirrors, it is natural to wonder whether inversion in a circle can be achieved through reflection in a suitable mirrored surface. In this note we give some positive answers to this question, including equations for constructing such mirrors. Specifically, we show how to design a mirror in which the viewer sees the exterior of a disk as though it had been geometrically inverted to the interior of the disk.
The Mirror
If such a mirror exists, it is a surface of revolution somewhat similar in shape to a cone. (In fact, it more closely resembles a bell.) Its exact shape depends upon the point E
4 THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
where the observer's eye is located on the axis of revolution, which we take to be the y-axis of a standard euclidean coordinate system in R3. We further suppose that E is above the xz-plane, which meets the mirror in a circle of radius r0 s 1 centered at the origin.
Under simple optical inversion with respect to the unit circle C(6 in the xz-plane, a dot at a point D* in the plane outside C(6 would be seen by the observer at E as if it were located inside C(6 at the point D on the segment between the origin 0 and D* for which IOD*I · IODI = 1. To achieve this, our mirror must reflect a ray from D* to E at an intermediate point M in such a way that the reflected ray appears to come from D, as indicated in Figure 1. (From geometric optics, the tangent line to the mirror surface at M in the plane containing the incident ray and the reflected ray makes equal angles with these rays.) The mirror images of lines outside C(6 would then appear as circles inside 'i6.
It will suffice to restrict our attention to a tangent line to the cross section of the mirror in the xy-plane, as depicted in Figure 2. In this figure, Y is the y-coordinate of the point E (the observer's eye), w* is the x-coordinate of the point on the x-axis whose reflection is being viewed by the observer, and w is is the x-coordinate of its virtual image.
The Differential Equation
Let y = f(x) be the equation of the cross section of the hypothesized mirror for x 2: 0. If (x,y) represents a point on the mirror, let a denote the angle that the tangent line to the graph off at (x,y) makes with the line of sight from the
y-aXIS E (Eye)
x-axis D*
observer at (O,Y) to this point. Let u denote the angle the tangent line makes with the horizontal and y the angle it makes with the vertical. We note that rLr = -tan( y) and
dy conclude that
-1 tan(y) = -, .
y (1)
There are four other relations that we can easily see from Figure 2:
Y- y w* = --· x Y '
Y- y tan(a + u) = --; X
w* - x tan(a + y) = .
y
From (1) we get
(2)
(3)
(4)
(5)
. ( 7T) 1 1 - tan2( y) 1 - (y')2 u = -tan 2'Y- 2 = tan(2y) = 2 tan( y) = 2y' ;
y-axis (Eye) E (O,Y )
-1
l§tijil;ifW
also, by (2)-(5)
tan(2y- ;) = tan(y- u) =tan ((a+ y)- (a+ u)) w*-x
_ Y- y
so that
y .r
1 + (w*y-x) (Y�y)
= x(w* - x) - y(Y- y) xy + (w* - x)(Y- y)
_ (1 - Yy)(Y- y) - x2Y
- x x2Yy + (Y- y)(Y- y- x2Y)
(6)
The first expression for u gives the quadratic equation (y')Z + 2uy' - 1 = 0. Noting that y' is never positive, we see that
y' = -u - VUT+l; (7)
and when (6) is used to replace u, we get a first-order differential equation for the meridian curve. Note that y' =
-1 when x = 0. Before working with this general equation, we consider
the more tractable limiting case as the viewer moves toward positive infinity.
The View from Infinity
When Y---> x, we see from (6) that u---> xy/(1 - x2); and, when u = xy/(1 - x2), the right side of (7) has the partial derivative with respect to y given by
-(1 + �) uy = -(1 + �) 1 �x 2·
Since this partial derivative is bounded on each x-interval [O,b] where 0 < b < 1, it follows from a standard theorem (e.g., [3; p. 550]) that the limiting equation has a unique solution y = y(x) on [0,1) with prescribed y(O) = y0. We tum now to the solution of this equation.
When u = xy/(1 - x2), the quadratic equation for y' is
(y')Z + 2xy�?y'- 1 = 0, (0 s; x < 1).
1- :L- (8)
With the substitutions s = x2 and p = -y' lx (>0), equation (8) can be written
2y 1 -- =p--1- s sp ' where p = -2dy.
ds (9)
By differentiating with respect to s and eliminating y and dy, we get the first-order equation d.'
dp - p ds- s(s - 1)(sp2 + 1)
(0 < s < 1) (10)
which, although not standard, admits integration. Indeed, with the successive substitutions 1/s = 1 + pq,
p = v + q, and q = exp(w + v2/2), it reduces to the sepa-rable equation
VOLUME 24, NUMBER 1 , 2002 5
dw = e"'ev"l2_ dv
This leads to an implicit solution in the form
where
(1 - s) r eV212 dt = speV212 (for appropriate c) (11) v
s - 1 v = p + -- (= 2y + sp). sp
(12)
[In principle, equations (11) and (12) determine p in terms of s = x2, so that v and hence y = 112(v - sp) can be obtained as functions of x. ]
We can derive qualitative information about our implicitly determined solution. First, note that the integration constant c is given by
c = v(O) = 2y(O) = 2y0,
since ass\.. 0, sp = -xy'--> 0. Moreover, for s < 1, we have p(s) > 0 and dplds < 0 by (10) , so that as s )" 1 , p(s) decreases to a limit p1 2:: 0. In fact, p1 = 0 since otherwise v = p + (s - 1)/sp has the positive limit v1 = p1 which violates our integral relation (11). It follows that y' is negative and approaches zero as .x )" 1, while y(x) decreases to a finite limit y1 , say. (y1 is negative, since 2y/(l - s) = (p - 1)/sp--> -m ass/" 1.)
Proposition 1. Each solution curve y = y(x) has a unique inflection point, and that point lies on the graph of the equation
� y = x� � (0 :S X :S 1). (13)
Proof: Observe that y" = 2Vs( d/ds) (-Vsp) so that, for 0 < s < 1,
s gn y'' = - s gn(Vsdp + .1r p) = -s gn (-
1- + sp
2 + 1 ) ds 2vs s-1 2
= -s gn(-
1- + 1 + ysp ) = sgn(l- yp), s - 1 1 - s
where we have used (9) and (10), together with the positivity of p, s, sp2 + 1, and 1 - s.
We see that inflection occurs when p = 1/y or when 2y/(1 - s) = 1/y - y!s so that
s(l - s) ( 1 - x2 ) y2 = = xz __ _
1 + s 1 + x2
as claimed. (Inflection must occur because near s = 0 : 1 - yp < 0 which cannot hold when y becomes negative, since p > 0.) D
The value x0 where y(x0) = 0 is of practical interest because it locates the boundary of the physical mirror. Conversely, it is clearly desirable to have x0 as near 1 as possible and to know how large we must take y0 = y(O) to achieve this. However, when x = x0, we see that p = llx@p and v = x@p = x0. Then from our integral solution (with c = 2y0) we get the transcendental relation
(14)
6 THE MATHEMATICAL INTELLIGENCER
which implies that Yo--> +oo as x0 )" 1. If the integral in (14) is evaluated numerically, we find, for example, that when x0 = 0.999, then 2.0030 < y0 < 2.0031.
Equation (13) for the locus of inflection points can be obtained directly. If we differentiate (8) with respect to x, set y" = 0 and solve for y', we get
-x y'=-. y
Upon substituting this in (8), we recover (13). This approach also leads to an interesting geometrical fact. Consider the isocline associated with slope m < -1 obtained by replacing y' with min (8). We can put the resulting equation in the form:
and we see that the isocline is a hyperbola having as asymptotes they-axis and the line y = ("':,� 1) x. Moreover, the vertex of the relevant branch of the hyperbola has coordinates
� - 1 � x = � � ' y = -:;;: � � -But these coordinates satisfy (13), which characterizes an inflection point. Thus the locus of inflection points is the locus of the relevant vertices of the associated isoclines. In Figure 3 we exhibit the graphs of typical solutions and the locus of inflection points.
Solutions of the General Equation
For finite Y > 0, our differential equation (6) and (7) is considerably more complicated. However, it is straightforward to verify that y = Y( 1 - x) gives the only decreasing linear solution. Now, u = P!Q, where P = x[(Y - y)(yY - 1) + x2Y] and Q = (Y - y)Z(l - x2) + x2y2, which is positive, if 0 < x < 1 and y < Y. Consequently, for fixed Y > 1, u(x,y) is bounded on each set ((x,y) : 0::; x::; 1 - 8, y::; Y - 8) where 0 < 8 < 1, as is the partial derivative
+prijii;JIM
au Py Qy - = u = - - u-. ay y Q Q
y-axis
(0, f)
From the argument used at the beginning of the earlier section titled "The View From Infinity," we see that, for each y0 < Y, there is a unique decreasing solution y = y(x) of our equation on [0,1) with the initial value y(O) =Yo· Moreover, the associated solution curves for distinct Yo cannot intersect, nor can they meet the open segment L between the points (0, Y) and (1,0), because its defining function, y = Y(l - x), is also a solution of the equation. It follows that the solution must vanish at some x0 E (0, 1]; and conversely, for every x0 E (0,1), there is a unique solution y = y(x) on [0,1) with y(x0) = 0 and y(O) E (O,Y]. In particular, we can take x0 as near 1 as we please.
At an x0 E (0,1), we have, from (6), that u = -x0/Y and, from (7), that
y'(xo) = -(V(xo!Y)2 + 1 - xo!Y) > -1.
But if x0 = 1, the situation is less clear. In fact, when Y > 1, we note (see Fig. 4) that the point (1,0) ends the hyperbolic arc H defined by (Y - y )(y - l!Y) + x2 = 0 (0 :o:;
x < 1, 0 < y :o:; 1/Y) along which, by (6) and (7), u = 0 and y' = -1. On the other hand, it also ends the linear solution segment L. Since no other solution segment is admissible, we see geometrically that, when y0 E (1/Y,1], the solution either crosses H with an intervening inflection point or it avoids Hand L by having another inflection point. For y0 E (1,Y), the solution curve must cross the circular arc C, defined by x2 + y2 = 1, (0 :o:; x < X£, YL < y :o:; 1), where YL = -Y(xL - 1), as shown. At the crossing point, (xc, Yc), say, it can be easily verified from (6) and (7) that the solution curve has slope -ycl(l - xc) < -1. Again, the curve either crosses H with slope - 1 and thus has an inflection point, or it avoids H and L by tending (nonlinearly) toward (1,0) with an intervening inflection point. These arguments can be reinforced analytically, and they help establish our principal result:
Proposition 2. Suppose Y > 1. Then, if Yo E (l!Y, Y), the solution CUTVe has a unique inflection point; and, if y0 E (0, 1/Y], the solution curoe does not have an 1:njlection point.
(Of course, when Yo = Y the solution segment L has no inflection point.)
We only outline the arguments supporting the remaining assertions in this proposition. Note that along a solution curve y(x) of (7) we have
y" = -(1 + u(l + u2)-ll2)u' = y'u'(1 + u2)-ll2
where u'(x) = !,u(x,y(x)), so that u' = Ux + uyy' . Hence, in general, sgn y" = -sgn u'; and at an inflection point, u' = 0 with u.xUy 2: 0 (since y' < 0). Now, when (6) is used for fixed Y, then formally
u' = R(x, y, y'),
where R is a rational function of its variables that is linear in y' = -u - YT+U'2. By direct computation, we can show that u = xY and u'(x) * 0 at points on the horizontal open segment M of height m = (Y2 + 1)/2Y between L and the y-axis. Moreover, since u(O) = 0, it is easy to verify that sgn y"(O) = sgn(l!Y- y0) when Yo < Y. If we further differentiate and set y" = u' = 0, we find (eventually) that, with P and Q as before,
sgn y'"(x) = sgn ((y - m)[2x(Y - y - x2Y) + p- ylp2 + Q2]J,
where, for 0 < x < 1 < Y, the second factor is not positive and it is strictly negative unless y = Y(1 ± x). When y0 E (0, l!Y), y"(O) > 0 and it follows that y" cannot vanish at a "first" x value since there y111(x) > 0; the associated solution curves have no inflection points. We can extend this argument to the case Yo = l!Y where y"(O) = 0 but Y111(0) > 0, since then y"(x) > 0 for 0 < x :'S x1, with y(x1) < 1/Y.
When Yo E (1/Y, m], y"' will be positive at every inflection point, so that there cannot be more than one. Finally, if Yo E ( m, Y), then Yo > m and y"(O) < 0; hence, y" cannot vanish at a "first" x with y(x) > m since there y"'(x) < 0. It follows that all inflection points must occur below M, and again we conclude that there is at most one. D
By straightfmward extension of these arguments using L'Hospital's rule as needed, we can also prove:
y-axis
1
l@tijii;IIW
VOLUME 24, NUMBER 1, 2002 7
AUTHORS
GERALD T. CARGO
Department of Mathematics
Syracuse University
Syracuse, NY 13244-1150
USA
JACK E. GRAVER
Department of Mathematics
Syracuse University
Syracuse, NY 1 3244-1150
USA e-mail: jegraver@syr.edu
After earn ing a Master's degree in mathe- Jack Graver, whose doctorate is frorn lndi-
matical statistics from the University of Michi- ana University, has been on the faculty of
gan, Gerald Cargo served in the U.S. Army, Syracuse University for 35 years. His re-
where he worked with the world's first large- search has been on design theory, integer
scale computer, the ENIAC. He returned to and linear programming, and graph theory.
Mich igan and got a doctorate in 1959. Most Among his books is an undergraduate ex-
of his research publications have dealt with position of rig idity theory, MAA, 2001 . He
inequalities or the boundary behavior of an- gets particular satisfaction from teaching
alytic functions. He also worked with h igh- summer workshops for h igh-school teach-
school teachers who taught calculus courses ers, which he has done over the years in In-
for college credit. As Professor Emeritus he d iana, New York, the Virg in Islands, and Eng-
has had time to cultivate his many interests, land.
includ ing math, travel, and swimm ing .
JOHN L. TROUTMAN
Department of Mathematics
Syracuse University
Syracuse, NY 13244-1150
USA
John L. Troutman studied applied mathe
matics at Virginia Polytechnic Institute and at
Stanford University, where he received a
Ph.D. in 1964. During those years he also
worked on areoelastic problems at govern
ment laboratories that later became part of
NASA. He has taught mathematics at Stan
ford and Dartmouth, and has recently retired
after 30 years on the mathematics faculty at
Syracuse University. He has published arti
cles on real and complex analysis, and is the
author of textbooks on variational calculus
and boundary-value problems in applied
mathematics.
Corollary 1. L is the only solution curve that either originates at (0, Y) or terminates at (1, 0).
In particular, there cannot be a "perfect" mirror that inverts the entire unit disk. However, for specific Y, we can use standard methods to obtain numerical solutions to our equations; and in Figure 5 we present representative solution curves when Y = 10, for values of x0 = 0.8, 0.9, 0.95 with corresponding values of y0 = 0.887, 1.088, 1.245. In particular, the numerical solution with x0 = 0.95 (so
Yo = 1.245) gives the profile of a mirror that should faithfully invert the region exterior to the disk of 5-inch di-
ameter when viewed from a height of about 2 feet. It seems feasible to manufacture such a mirror on a computer-directed lathe1.
��- ����- ---------
1 Patent pending.
8 THE MATHEMATICAL INTELLIGENCER
REFERENCES 1 . Davis, P. J. The Thread: A Mathematical Yarn. The Harvester Press,
Birkhauser, Boston, 1983.
2. Kempe, A. B. How to Draw a Straight Line. National Council of
Teachers of Mathematics, Reston, VA, 1977.
3. Simmons, G. F. Differential Equations with Applications and Histor
ical Notes, Second Edition. McGraw-Hill, New York, 1991.
14@'1.i§,@ih£11§1§4@11,j,i§.id Michael Kleber and Ravi Vakil, Editors
This column is a place for those bits of
contagious mathematics that travel
from person to person in the
community, because they are so
elegant, suprising, or appealing that
one has an urge to pass them on.
Contributions are most welcome.
Please send all submissions to
Mathematical Entertainments Editor,
Ravi Vakil, Stanford University,
Department of Mathematics, Bldg. 380,
Stanford, CA 94305-21 25, USA
e-mail: vakil@math.stanford.edu
The Best Card Trick Michael Kleber
You, my friend, are about to witness the best card trick there is.
Here, take this ordinary deck of cards, and draw a hand of jive cards from it. Choose them deliberately or randomly, whichever you prefer-but do not show them to me! Show them instead to my lovely assistant, who will now give me four of them: the 7•, then the Q \?, the 8 "'· the 3 0. There is one card left in your hand, known only to you and my assistant. And the hidden card, my friend, is the K •.
Surely this is impossible. My lovely assistant passed me four cards, which means there are 48 cards left that could be the hidden one. I received the four cards in some specific order, and by
varying that order my assistant could pass me some information: one of 4! = 24 messages. It seems the bandwidth is off by a factor of two. Maybe we are passing one extra bit of information illicitly? No, I assure you: the only information I have is a sequence of four of the cards you chose, and I can name the fifth one.
The Story
If you haven't seen this trick before, the effect really is remarkable; reading it in print does not do it justice. (I am forever indebted to a graduate student in one audience who blurted out "No way!" just before I named the hidden card.) Please take a moment to ponder how the trick could work, while I relate some history and delay giving away the answer for a page or two. Fully appreciating the trick will involve
a little information theory and applications of the Birkhoff-von Neumann theorem as well as Hall's Marriage theorem. One caveat, though: fully appreciating this article involves taking
its title as a bit of showmanship, perhaps a personal opinion, but certainly not a pronouncement of fact!
The trick appeared in print in Wallace Lee's book Math Miracles, 1 in which he credits its invention to William Fitch Cheney, Jr., a.k.a. "Fitch." Fitch was born in San Francisco in 1894, son of a professor of medicine at Cooper Medical College, which later became the Stanford Medical School. After receiving his B.A and M.A. from the University of California in 1916 and 1917, Fitch spent eight years working for the First National Bank of San Francisco and then as statistician for the Bank of Italy. In 1927 he earned the first math Ph.D. ever awarded by MIT; it was supervised by C.L.E. Moore and titled "Infinitesimal deformation of surfaces in Riemannian space." Fitch was an instructor and assistant professor then at the University of Hartford (Hillyer College before 1957) until his retirement in 1971; he remained an aQjunct until his death in 1974.
For a look at his extra-mathematical activities, I am indebted to his son Bill Cheney, who writes:
My father, William Fitch Cheney, Jr., stage-name "Fitch the Magician," first became interested in the art of magic when attending vaudeville shows with his parents in San Francisco in the early 1900s. He devoted countless hours to learning sleightof-hand skills and other "pocket magic" effects with which to entertain friends and family. From the time of his initial teaching assignments at Tufts College in the 1920s, he enjoyed introducing magic effects into the classroom, both to il-
'Published by Seeman Printery, Durham, N.C., 1950: Wallace Lee's Magic Studio, Durham, N.C., 1960; Mickey
Hades International, Calgary, 1976.
© 2002 SPRINGER· VERLAG NEW YORK, VOLUME 24, NUMBER 1, 2002 9
lustrate points and to assure his students' attentiveness. He also trained himself to be ambidextrous (although naturally left-handed), and amazed his classes with his ability to write equations simultaneously with both hands, meeting in the center at the "equals" sign.
Each month the magazine M-U-M, official publication of the Society of American Magicians, includes a section of new effects created by society members, and "Fitch Cheney" was a regular by-line. A number of his contributions have a mathematical feel. His series of seven "Mental Dice Effects" (beginning Dec. 1963) will appeal to anyone who thinks it important to remember whether the numbers 1, 2, 3 are oriented clockwise or counterclockwise about their common vertex on a standard die. "Card Sense" (Oct. 1961) encodes the rank of a card (possibly a joker) using the fourteen equivalence classes of permutations of abed which remain distinct if you declare ac = ca and bd = db as substrings: the card is placed on a piece of paper whose four edges are folded over (by the magician) to cover it, and examining the creases gives precisely that much information about the order in which they were folded. 2
While Fitch was a mathematician, the five-card trick was passed down via Wallace Lee's book and the magic community (1 don't know whether it appeared earlier in M-U-M or not.) The trick seems to be making the rounds of the current math community and beyond, thanks to mathematician and magician Art Benjamin, who ran across a copy of Lee's book at a magic show, then taught the trick at the Hampshire College Summer Studies in Mathematics program3 in 1986. Since then it has turned up regularly in "brain teaser" puzzle-friendly fo-
rums; on the rec.puzzles newsgroup, I once heard that it was posed to a candidate at a job interview. It made a recent appearance in print in the "Problem Comer" section of the January 2001 Emissary, the newsletter of the Mathematical Sciences Research Institute. As a result of writing this column, I am learning about a slew of papers in prepa
ration that discuss it as well. It is a card trick whose time has come.
The Workings
Now to business. Our "proof' of impossibility ignored the other choice my lovely assistant gets to make: which of the five cards remains hidden. We can put that choice to good use. With five cards in your hand, there are certainly two of the same suit; we adopt the strategy that the first card my assistant shows me is of the same suit as the card that stays hidden. Once I see the first card, there are only twelve choices for the hidden card. But a bit more cleverness is required: by permuting the three remaining cards my assistant can send me one of only 3! = 6 messages, and again we are one bit short.
The remaining choice my assistant makes is which card from the samesuit pair is displayed and which is hidden. Consider the ranks of these cards to be two of the numbers from 1 to 13, arranged in a circle. It is always possible to add a number between 1 and 6 to one card (modulo 13) and obtain the other; this amounts to going around the circle "the short way." In summary, my assistant can show me one card and transmit a number from 1 to 6; I increment the rank of the card by the number, and leave the suit unchanged, to identify the hidden card.
It remains only for me and my assistant to pick a convention for representing the numbers from 1 to 6. First, totally order a deck of cards: say ini-
tially by rank, A23 ... JQK, and break ties by ordering the suits as in bridge (i.e., alphabetical) order, 4- 0 \? •· Then the three cards can be thought of as smallest, middle, and largest, and the six permutations can be ordered, e.g. , lexicographically. 4
Now go out and amaze (and illuminate5) your friends. But, please: just make sure that you and your assistant agree on conventions and can name the hidden card flawlessly, say 20 times in a row, before you try this in public. As we saw above, it's not hard to name the hidden card half the time-and it's tough to win back your audience if you happen to get the first one wrong. (I speak, sadly, from experience.)
The Big Time
Our scheme works beautifully with a standard deck, almost as if four suits of thirteen cards each were chosen just for this reason. While this satisfied Wallace Lee, we would like to know more. Can we do this with a larger deck of cards? And if we replace the hand size of five with n, what happens?
First we need a better analysis of the information-passing. My assistant is sending me a message consisting of an ordered set of four cards; there are 52 X 51 X 50 X 49 such messages. Since I see four of your cards and name the fifth, the information I ultimately extract is an unordered set of five cards, of which there are (5l), which for comparison we should write as 52 X 51 X 50 X 49 X 48/5!. So there is plenty of extra space: the set of messages is 1:� = 2.5 times as large as the set of situations. Indeed, we can see some of that slop space in our algorithm: some hands are encoded by more than one message (any hand with more than two cards of the same suit), and some messages never get used (any message which contains the card it encodes).
2This sort of "Purloined Letter" style hiding of information in plain sight is a cornerstone of magic. From that point of view, the "real" version of the five-card trick se
cretly communicates the missing bit of information; Persi Diaconis tells me there was a discussion of ways to do this in the late 1 950s. For our purposes we'll ignore
these clever but non-mathematical ruses.
3Unpaid advertisement: for more infomnation on this outstanding, intense, and enlightening introduction to mathematical thinking for talented high-school students, con·
tact David Kelly, Natural Science Department, Hampshire College, Amherst, MA 01 002, or dkelly@hampshire.edu.
4For some reason I personally find it easier to encode and decode by scanning for the position of a given card: place the smallest card in the left/middle/right position
to encode 1 2/34/56, respectively, placing medium before or after large to indicate the first or second number in each pair. The resulting order sm/, sfm, msf, Ism, mfs,
fms is just the lex order on the inverse of the permutation.
511 your goal is to confound instead, it is too transparent always to put the suit-indicating card first. Fitch recommended placing it (i mod 4)th for the ith performance
to the same audience.
10 THE MATHEMATICAL INTELLIGENCER
Generalize now to a deck with d cards, from which you draw a hand of n. Calculating as above, there are d(d - 1) · · · (d- n + 2) possible messages, and (�) possible hands. The trick really is impossible (without subterfuge) if there are more hands than messages, i. e. , unless d :::; n! + n - 1.
The remarkable theorem is that this upper bound on d is always attainable. While we calculated that there are enough messages to encode all the hands, it is far from obvious that we can match them up so each hand is encoded by a message using only the n cards available! But we can; the n = 5 trick, which we can do with 52 cards, can be done with a deck of 124. I will
give an algorithm in a moment, but first an interesting nonconstructive proof.
The Birkhoff-von Neumann theorem states that the convex hull of the permutation matrices is precisely the set of doubly stochastic matrices: matrices with entries in [0,1] with each row and column summing to 1. We will use the equivalent discrete statement that any matrix of nonnegative integers with constant row and column sums can be written as a sum of permutation matrices.6 To prove this by induction (on the constant sum) one need only show that any such matrix is entrywise greater than some permutation matrix. This is
an application of Hall's Marriage theorem, which states that it is possible to arrange suitable marriages between n men and n women as long as any collection of k women can concoct a list of at least k men that someone among them considers an eligible bachelor. Applying this to our nonnegative integer matrix, we can marry a row to a column only if their common entry is nonzero. The constant row and column sums ensure that any k rows have at least k columns they consider eligible.
Now consider the (very large) 0-1 matrix with rows indexed by the (�) hands, columns indexed by the d!l(d - n + 1)! messages, and entries equal to 1 indicating that the cards used in the message all appear in the hand. When we take d to be our upper
6Exercise: Do so for your favorite magic square.
bound of n! + n- 1, this is a square matrix, and has exactly n! 1's in each row and column. We conclude that some subset of these 1's forms a permutation matrix. But this is precisely a strategy for me and my lovely assistant-a bijection between hands and
messages which can be used to represent them. Indeed, by the above paragraph, there is not just one strategy, but at least n!.
Perfection
Technically the above proof is constructive, in that the proof of Hall's Marriage theorem is itself a construction. But with n = 5 the above matrix has 225,150,024 rows and columns, so there is room for improvement. Moreover, we would like a workable strategy, one that we have a chance at performing without consulting a cheat sheet or scribbling on scrap paper. The perfect strategy below I learned from Elwyn Berlekamp, and I've been told that Stein Kulseth and Gadiel Seroussi came up with essentially the same one independently; likely others have done so too. Sadly, I have no information on whether Fitch Cheney thought about this generalization at all.
Suppose for simplicity of exposition that n = 5. Number the cards in the deck 0 through 123. Given a hand of five cards co < c1 < c2 < c3 < c4, my assistant will choose ci to remain hidden, where i = co + c1 + c2 + c3 + c4 mod 5.
To see how this works, suppose the message consists of four cards which sum to s mod 5. Then the hidden card is congruent to -s + i mod 5 if it is ci. This is precisely the same as saying that if we renumber the cards from 0 to 119 by deleting the four cards used in the message, the hidden card's new number is congruent to -s mod 5. Now it is clear that there are exactly 24 possibilities, and the permutation of the four displayed cards communicates a number p from 0 to 23, in "base factorial:" p = d11! + d22! + d33! , where for lex order, di :::; i counts how many cards to the right of the (n- ith) are smaller than it. 7 Decoding the hidden
card is straightforward: take 5p + (-s mod 5) and add 0, 1, 2, 3, or 4 to account for skipping the cards that appear in the message.8
Having performed the 124-card version, I can report that with only a little practice it flows quite nicely. Berlekamp mentions that he has also performed the trick with a deck of only 64 cards, where the audience also flips a coin: after seeing four cards the performer both names the fifth and states whether the coin came up heads or tails. Encoding and decoding work just as before, only now when we delete the four cards used to transmit the message, the deck has 60 cards left, not 120, and the extra bit encodes the flip of the coin. If the 52-card version becomes too well known, I may need to resort to this variant to stay ahead of the crowd.
And finally a combinatorial question to which I have no answer: how many strategies exist? We probably ought to count equivalence classes modulo renumbering the underlying deck of cards. Perhaps we should also ignore composing a strategy with arbitrary permutations of the message-so two strategies are equivalent if, on every hand, they always choose the same card to remain hidden. Calculating the permanent of the aforementioned 225,150,024-row matrix seems like a bad way to begin. Is there a good one?
Acknowledgments
Much credit goes to Art Ber\iamin for popularizing the trick; I thank him, Persi Diaconis, and Bill Cheney for sharing what they knew of its history. In helping track Fitch Cheney from his Ph.D. through his mathematical career, I owe thanks to Marlene Manoff, Nora Murphy, Geogory Colati, Betsy Pittman, and Ethel Bacon, collection managers and archivists at MIT, MIT again, Tufts, Connecticut, and Hartford, respectively. Thanks also to my lovely assistants: Jessica Polito (my wife, who worked out the solution to the original trick with me on a long winter's walk), Ber\iamin Kleber, Tara Holm, Daniel Biss, and Sara Billey.
7Qr, my preference, d, counts how many cards larger than the ith smallest appear to the left of it. Either way, the conversion feels perfectly natural after practicing a few times.
sExercise: Verify that if your lovely assistant shows you the sequence of cards 37, 7, 94, 61 , then the hidden card is the page number in this issue where the first six
colorful algorithms converge:)
VOLUME 24, NUMBER 1, 2002 1 1
Reading Bombelli
FEDERICA LA NAVE AND BARRY MAZUR
r afael Bombelli's L'Algebra, originally written in the middle of the sixteenth cen
tury, is one of the founding texts of the title subject, so if you are an algebraist, it
isn't unnatural to want to read it. We are currently trying to do so.
Now, much of the secondary literature on this treatise concurs with the simple view found in Bourbaki's Elements d'Histoire des Mathematiques:
Bombelli ... takes care to give explicitly the rules for calculation of complex numbers in a manner very close to modem expositions.
This may be true, but is of limited help in understanding the issues that the text is grappling with: if you open Bombelli's treatise you discover nothing resembling complex numbers until page 133,1 at which point certain mathematical objects (that might be regarded by a modern as "complex numbers") burst onto the scene, in full battle array, in the middle of an on-going discussion. Here is how Bombelli introduces these mathematical objects. He writes, ''I have found another sort of cubic radical which behaves in a very different way from the others. "
Ho trovato un'altra sorte di R.c.Iegate molto differenti dall'altre . ...
The cubic radicals that Bombelli is contemplating here are the radicals that occur in the general solution of cubic
polynomial equations in one variable. Bombelli has come to the point in his treatise where he is working with Dal Ferro's formula for the general solution to cubic polynomial equations and considers (to resort to modem language) cubic polynomials with "three real roots ."2 He produces the formula (a sum of cube roots of conjugate quadratic imaginary expressions) which yields ("formally," as we would say) a solution to the cubic polynomial under examination.
Complex numbers, when they occur in Gerolamo Cardana's earlier treatise Ars Magna, occur neatly as quantities like 2 + V-15. But they appear initially in Bombelli's treatise as cubic radicals of the type of quantities discussed by Cardano; a somewhat complicated way for them to arise in a treatise that is thought of as an organized exposition of the formal properties of complex numbers! Why doesn't Bombelli cite Cardano here? Why does he not mention his predecessor's discussion of imaginary numbers? Bombelli is not shy elsewhere of praising the work of Cardano. Why, at this point, does Bombelli rather seem to be announcing a discovery of his own ("I have found ... ")?
Here is a glib suggestion of an answer: Bombelli has no way of knowing, given what is available to him, that his cubic radicals are even of the same species as the complex numbers of Cardano. How, after all, would Bombelli know
10ur page numbers refer to Bortolotti and Forti's 1 966 edition of L'Aigebra. For an account of the history of the publication of this treatise, see below. We have also
listed some of the secondary literature in the bibliography.
2This is what Bombelli's contemporaries called the "irreducible case" (a term still used by Italian mathematicians today).
12 THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
that the cube root of a complex number is again a complex number? Of course one can go in the opposite direction with ease: that is, one can take a complex number z and cube it to get a number y = z3 with known cube root, and one might be lucky in guessing z, given y. Bombelli, for example tells us that the cube root of 2 + 11 v=1 is 2 + v=1 and thereby gets the solution x = 2 + v=1 + 2 - v=1 =
4 to the cubic equation x3 = 15x + 4. But the general problem of extracting cube roots is of a different order. For how would you go about solving the equation
(X + iY)3 = A + iB,
or equivalently, the simultaneous (cubic, of course) equations
without having various eighteenth-century insights at your disposal? There is surely the smell of circularity here, despite the fact that a "modem" can derive some simple pleasure in analyzing the 0-cycle of degree 9 in complex projective 2-space given by the intersection of those two cubics. To Bombelli, his cubic radicals were indeed new kinds of radicals.
Can we be content with this answer? A few paragraphs later Bombelli makes it clear that he
was quite dubious, at first, about the legitimacy of his discovery and only slowly accustomed himself to it; he writes:
[This radical] will seem to most people more sophistic than real. That was the opinion even I held, until I found demonstration [of its existence] . . . 3
What, then, does Bombelli mean by demonstration? What does he mean by existence? As we shall see, Bombelli only ascribes existence, whatever this means, to the yoked sum of two cubic radicals (the radicands being, in effect, conjugate complex numbers). As he puts it,
It has never happened to me to find one of these kinds of cubic root without its conjugate.4
Let us add a further element to this stew of questions: In the "irreducible case," i.e., the case where the cubic polynomial has three real roots, does Bombelli believe that the solution given by his "new kind of cubic radicals" corresponds to any, or all, of the three solutions? (He seems to.) In what sense does Bombelli's general solution lead to a numerical determination of one, or more, of the three roots of the polynomial? If you do not have Abraham de Moivre's insight, or anything equivalent, you may be stymied by the
3Bombelli (1966), p. 1 33.
4Bombelli (1 966), p. 1 34.
problem of "using" the general solution by cubic radicals to help you find, or even approximate, any of the three real numbers that are roots of the cubic polynomial that the "general solution" purports to solve. 5
An evolving theme in Bombelli's thought is the idea of connecting the ancient problem of angle trisection to the problem of fmding roots of cubic polynomials. Of course, the modem viewpoint makes this connection quite clear. Bombelli also develops a method (as he says, "in the plane") for finding a real number solution to a cubic polynomial equation. His method involves making a construction in plane geometry dependent on a parameter (the parameter being the angle that two specific lines in the construction subtend) and then "rotating" one of those lines (this "rotation" effects other changes in his construction) until the lengths of two line segments in the construction are equal; these equal lengths then provide the answer he seeks. Later in this discussion, we refer to this type of construction as a neusis construction. To what extent do these discussions-trisection of angle and neusis construction-play a role in providing a "demonstration" to Bombelli of the existence of his yoked cubic radicals? We discuss this in detail in the latter part of this article.
Tempering any answer that we might offer to any of these questions is the fact that the incubation period for Bombelli's text, and its writing, spanned more than two decades. Bombelli's treatise records the evolution of his thought, and the answers that Bombelli entertains for some of these questions change with time. Reading him may perhaps give us a portrait of an early father of algebra grappling with what it means for a concept to exist. We feel that this portrait deserves to be more fully drawn than has been done.
We are not yet ready to do this, and are only in mid-journey in our reading of Bombelli. Nevertheless we have put this article together in hope that what we have learned so far may be useful to other readers. We wish to thank David Cox for helpful comments and questions regarding earlier drafts.
Bombelli's Writing
Bombelli wrote in Italian (which, according to Dante, is the language of the people). To our knowledge, his is the first long treatise on mathematics written in Italian. He was faced, therefore, with something of a Dante-esque project: to choose words for existing terms (generally from Latin) and to invent Italian words for the various concepts that came along. That his book is in Italian has a mild disadvantage, and a great advantage for a reader. On the one hand many of Bombelli's neologisms never caught on, and
5As de Moivre put it in his article published in 1 738, "There have been several authors, and among them Dr. Wallis, who have thought that those cubic equations,
which are referred to the circle, may be solved by the extraction of the cube root of an imaginary quantity, as of 81 + v' - 2700, without any regard to the table of
sines: but that is a mere fiction; and a begging of the question; for on attempting it, the result always recurs back again to the same equation as that first proposed.
And the thing cannot be done directly, without the help of the table of sines, specially when the roots are irrational; as has been observed by many others." (Abraham
De Moivre, "Of the Reduction of Radicals to more Simple Terms," The Philosophical Transactions of the the Royal Society of London, abridged by C. Hutton, G. Shaw,
and R. Pearson, volume VIII (London: 1 809), 276.)
VOLUME 24, N UMBER 1, 2002 13
they may seem quite strange to a modem. These terms therefore must be carefully deciphered (we give a partial glossary in Appendix B). On the other hand his style is quite personal (putting aside the lengthy computations about cubic irrationalities that are spelled out in prose!). At times the text reads as if it were a private journal. To get a sense of this, see Appendix A for a translation of his introductory remarks. What we know of Bombelli's life comes, it seems, entirely from this treatise. More importantly, as already mentioned, Bombelli's informality allowed him to keep in the text some of his early attitudes, as well as the changes in his outlook over the twenty-year period during which he worked on L 'Algebra.
Bombelli and His Algebra
We do not know precisely where Bombelli was born. In L 'Algebra he calls himself "citizen of Bologna." Bombelli was a member of a noble family from the countryside around Bologna. They came to Bologna at the beginning of the 13th century. At the end of the same century they, being "ghibellini," were forced to leave the city, and only returned in the sixteenth century.
Bombelli was a civil engineer, and in L 'Algebra he mentions his involvement in the project of draining the Chiana swamp in Tuscany. He recounts that during periods of interruption of this project he wrote his book. The treatise L 'Algebra as edited in a complete edition in 1966 consists of two "parts"6 which were, it seems, initially written in 1550.7 After this first manuscript, Bombelli came to know Diophantus's Arithmetic which was in a codex of the Vatican Library. 8 Bombelli then made a general revision of his manuscript and, among other things, included Diophantus's problems in his text. He published none of it until 1572. At that time Bombelli published only the first part. He apologized, saying that he could not publish the other part because it had not yet been "brought to the level of perfection required by mathematics." However, it was surely circulating among scholars, for in Bologna's libraries we still find two copies of the manuscript. The second part of the book was not published and was believed lost until the 1920s when Bortolotti found the complete manuscript (not just the last part, but also the frrst in an unrevised version) in codex B 1560 of the "Biblioteca dell'Archiginnasio di
Bologna. " Here is a run-down of the contents of Bombelli's five
books. As already mentioned, his great innovation was to have "solved" the "irreducible case" of the general cubic polynomial; i.e., the case when the root of Dal Ferro's formula for solving cubic equations involves the square root of a negative number, a thing that at the time was consid-
6Part I consists of three "books"; Part II, of two.
ered a monstrous absurdity (Cardano called the expression containing square roots of negative numbers "sophistic and far from the nature of numbers" and also "wild").
Bombelli gives a definition of variable and notation for exponents. He studies monomials, polynomials, and rules for calculating with them. He treats the equations from the first to the fourth degree, and solves, among other things, all "42" possible cases of quartic equations (improving on the work of Ferraro and Cardano ). Following the practice of the time, he also gives a solid geometric demonstration of the solution of cubic equations in terms of how a cube can be decomposed into two cubes and six parallelepipeds. Moreover, noticing the analogy between this problem and the classic problem of the insertion of two middle proportionals, he also offers his plane geometrical construction of the root of a cubic equation, which we discuss below. This construction is perhaps superfluous for a cubic equation with only one real root, but it is necessary in the irreducible case where the decomposition of the cube is impossible. In doing this Bombelli developed a geometric algebra (he refers to this as algebra linearia, that is to say linear algebra) which has a distinctly cartesian flavor. For at times Bombelli seems to be making the claim that geometry is not necessarily the only way to prove things: rather, certain geometric constructions are grounded in the underlying algebra that represents these constructions. Bombelli addresses the question of the relationship between the problem of the trisection of the angle and that of the solution of the cubic equation in the irreducible case. In his published treatise he expresses his intention to use the solution of the cubic equation in the irreducible case to solve the angle-trisection problem.9 This represents a change of viewpoint from the earlier version of his manuscript, in which Bombelli simply maintained that angle-tri- . section leads to cubic equations that cannot be solved. 10
His treatise contains a collection of problems that include all the problems of the first four books of Diophantus. L 'Algebra remained for more than a century the fundamental text of advanced algebra. It was studied, for example, by Christian Huygens and Gottfried Wilhelm Leibniz.
"Ho trovato un'altra sorte di R.c.legate molto
differenti dall'altre . . . . "
Here is how the text11 continues. (We have shortened it a bit by putting the algebraic formulae in modem notation.)
. . . I have found another kind of cubic root of a polynomial which is very different from the others. This [cubic root] arises in the chapter dealing with the equation of
7Bortolotti reached the conclusion that the manuscript he found in the Library of the Archiginnasio in Bologna (containing the entirety of Bombelli's work, with both
parts, the algebraic and the geometrical, in the first, unrevised version) went back to that date.
81n the introduction of the printed work, Bombelli tells us that he and Pazzi had translated the first five chapters of Diophantus while Pazzi was lector at Rome, i .e. ,
sometime after 1 567.
9Bombelli (1 966), p. 245.
1 0Bombelli (1 966), pp. 639--641 .
' 'Translation of pp 1 33-134 (in the Chapter On the division of a trinomial made by cubic roots of polynomials and number).
14 THE MATHEMATICAL INTELLIGENCER
the kind :il = px + q, when p3/27 > q2/4, as we will show in that chapter. This kind of square root has in its calculation [ algorismo] different operations than the others and has a different name. Since when p3/27 > q2/4, the square root of their difference can be called neither positive nor negative, therefore I will call it "more than minus" when it should be added and "less than minus" when it should be subtracted. This operation is extremely necessary, more than the other cubic roots of polynomials, which come up when we treat the equations of the kind x4 + ax3 + b or x4 + ax + b or x4 + ax3 + ax + b. Because, in solving these equations, the cases in which we obtain this [new] kind of root are many more than the cases in which we obtain the other kind. [This new kind of root] will seem to most people more sophistic than real. This was the opinion I held, too, until I found its geometrical proof (as it will be shown in the proof given in the abovementioned chapter on the plane). I will first treat multiplication, giving the law of plus and minus:12
( + )( +i) = +i (-)( +i) = -i (+)(-i) = -i (-)(-i) = +i ( +i)( +i) = (+i)(-i) = + ( -i)( +i) = + (-i)(- i) = -
Notice that this kind of root of polynomials cannot be obtained if not together with its conjugate. For instance, the conjugate of -\/2 + iv2 will be -\/2 - iv2. It has never happened to me to find one of these kinds of cubic root without its conjugate. It can also happen that the second quantity [inside the cubic root] is a number and not a root (as we will see in solving equations). Yet, [even if the second quantity is a number] , an expression like -\12 + 2i cannot be reduced to only one monomial, despite the fact that both 2 and 2i are numbers.
Commentary
The cube equal to a coefficient times the unknown plus a number refers to the equation which in modern dress is
x3 = px + q.
Here, p is the coefficient and q is the number. Bombelli prefers to think of his equations having only positive num-
121n a more literal translation of Bombelli's words:
Plus times more than minus makes more than minus.
Minus times more than minus makes less than minus.
Plus times less than minus makes less than minus.
Minus times less than minus makes more than minus.
More than minus times more than minus makes minus.
More than minus times less than minus makes plus.
Less than minus times more than minus makes plus.
Less than minus times less than minus makes minus.
bers as coefficients, so will treat separately (in different chapters) equations of the form x3 + px = q, etc. , terms being assembled to the left or right of the equality sign to arrange that p and q are positive. For efficiency, let us cheat, and peek at the modern, but still pre-Galois, treatment of the general cubic equation
x3 = px + q.
If we formally factor the polynomial
x3 - px - q = (x - eJ)(x - 8z)(x - 8s)
as a product of linear factors, we have
el + ez + 8s = 0,
and 11, the discriminant of the polynomial, i.e., the square of
is equal to
which is positive if all three roots eb ez, 8s are real, and is negative if precisely one of them is real. In any event, a "formula" for the real solution(s) to this polynomial is given by jq 1 , � jq 1 , �
x = - + - v -M3 + - - - v -M3 2 6 2 6 '
where if Ll is negative (and we are looking for the unique real solution) the above formula has an unambiguous interpretation as a real number and gives the solution.
If, however, Ll is positive (which is what Bombelli is encountering when he considers the case where the cube of "the third of the coefficient" is greater than the square of 2 3 "half the number," or equivalently, where � - � is nega-
tive and J! - � is imaginary), the above solution, i.e.,
F ' % + j: -;� + o % - j: - ;; involves imaginaries. To the modern eye, this expression is dangerously ambiguous, there being three possible values for each of the cubic radicals in it: to have it "work," of course, you have to coordinate the cube roots involved. That is, to interpret the expression correctly you must "yoke together" the two radicals in the above formula by taking them to be complex conjugates of each other, and then, running through each of the three complex cube roots of q/2 - i v=LV3, you get the three real solutions.
VOLUME 24. NUMBER 1 , 2002 15
p
Figure 1 Geometrical "Demonstration"
Bombelli knows that any cubic polynomial has a root. The (post-cartesian) argument (that a cubic polynomial p( x) takes on positive and negative values, is a continuous function of x, and therefore, as x ranges through all real numbers, it must traverse the value 0 at least once) is not in Bombelli's vocabulary, but as the reader will see, there remains a shade of this argument in Bombelli's geometrical "demonstration." Bombelli convinces himself that cubic polynomials have roots by two distinct methods-the first by consideration of volumes in space, a method that does not work in the irreducible case; and the second by consideration of areas in the plane, a method that does work in the irreducible case. 13
The method by consideration of volumes
Bombelli starts with a cube whose linear dimension let us denote by t. He then decomposes it into a sum of two cubes nesting in opposite comers of the big cube, these being of linear dimensions, say, u and t - u, and three parallelopipeds, following the algebraic formula:
(t - u)3 + 3tu(t - u) = t3 - u3.
Stripping the rest of Bombelli's demonstration of its geometric language, here is how it proceeds. Put p : = 3tu and q : = t3 - u3, and note that the quantity x : = t - u is a solution of the cubic equation
x3 + px = q. Of course, if we had such an equation with given con
stants p, q > 0 which we wished to solve, we would first
q
r
have to arrange to find the t and the u that worked; but ignore this, and let us proceed. Substituting
p u = -3t
in the equation t3 - u3 = q, we get
or
which we think of as a quadratic equation in t3:
and applying the quadratic formula (available, of course, in Bombelli's time) we get
ts = q ± Yq2 + 4ps/27
2 '
i.e., Cardano's formula for the solution x = t - ft of the cubic equations of the form x3 + px = q. All this is performable geometrically to produce the x only if t3 is a real number. That is, this geometric demonstration doesn't work in the irreducible case. 14
1 3For the first method, see Bombelli (1 966), pp. 226-228; for the second method, pp. 228-229.
14This type of "decomposition of the cube" argument had already been used by Cardano in the Ars Magna to explain how, for a particular equation (x6 + 6x = 20),
one can derive his formula; Cardano never considered the irreducible case.
16 THE MATHEMATICAL INTELLIGENCER
The method in the plane
Bombelli's second method resembles some of the neusisconstructions used in questions of angle-trisection in ancient Greek geometry (see below), and indeed does work in the irreducible case. Bombelli promotes this method (invoking the august authority of the ancient authors, who used similar methods) because, he claims, it provides a "geometric demonstration" that his cubic radicals "exist."
By a gnomon let us mean an "L-shaped" figure; i.e., two closed line segments joined at a 90 degree angle at their common point, the vertex. Bombelli uses a construction with two gnomons, one with vertex r and one with vertex unfortunately labeled p in the diagram (taken from his manuscript) shown as Figure 1 .
He will construct such a diagram from the data on his cubic equation x3 = px + q, i.e., from the pair of real numbers p and q; from dimension considerations, we can expect p to appear as an area, and q/p as a linear measurement. Let us calibrate the diagram by putting
lm = unity.
Now by suitably moving the two gnomons, moving the first up and down and pivoting the second about its vertex, Bombelli shows that one can obtain a diagram with
- q la =
p '
and the area of the rectangle abfl equal to p, and moreover, for such a dia�ram, the root x of his equation will appear as the length li.
Neusis-Constructions and the Trisection of Angles
The problem of trisecting a general angle with the aid of no more than an unmarked straightedge and compass, as posed by the ancient Greek mathematicians, is impossible. The fact that (the general solution of) this problem is impossible was established only in 1837 by Pierre Laurant Wantzel, who also made explicit the connection between trisection and solutions of cubic equations. But ancient mathematicians had an assortment of methods of angle-trisection that made use of "equipment" more powerful than mere compass and straightedge. One such method (referred to as neusis: verging, inclination) useful for solving certain problems involves making (as in the gnomon construction of Bombelli's that we have just sketched) a plane geometric construction or, more precisely, a "family of constructions" dependent upon a single parameter of variation. 15 In general, the strategy is to show that by "varying the construction" one can arrange it so that two designated points on a specific line (of the construction) switch their order on the line, under the variation. This then allows one to argue, in the spirit of the modem intermediate-value theorem, that there is a member of the family where the two designated points actually coincide. One then applies the
features of this particular member of the family to the problem one wishes to solve.
In the Book of Lemmas Archimedes (3rd century BCE) trisects a general angle using a neusis construction. (We do not have the original Greek of this work; we have an Arabic translation that does not seem to be completely faithful to the original Archimedean text.) Hippias (end of the 5th century BCE), instead, used a curve that he invented, the so-called Quadratix of Hippias. By means of this curve it is possible to divide a general angle into any number of equal parts. Nicomedes (2nd century BCE) made his conchoid curve by means of a neusis construction and he used the conchoid to solve the problem of trisection. Apollonius (late 3rd to 2nd century BCE) achieved angle-trisection using conics (the two cases we have, transmitted to us by Pappus in his Mathematical Collection, use a hyperbola).
Suggestions
We feel that there are two distinct elements that contribute to Bombelli's "faith" in cubic radicals.
First, Bombelli deals with the "inverse problem," and he does this in two ways: As mentioned, he explicitly tells us, on occasion, what the cube root of a specific number is (the cube root of 2 + 1 1 v=I is 2 + v=l) and thereby explicitly solves an equation (e.g., x = 4 is a solution of x3 = 15x + 4) saying that if one follows his geometrical method for the solution of this problem one obtains that same solution. But he also may simply start with a sum of two yoked cubic radicals,
V a + iVb + V a - iVb,
and discover the cubic equation of which this is a root. 16
Since he has proven by his geometric method that the cubic equation has a real solution (in fact "three" of them), it follows that this sum of two yoked cubic radicals in some sense represents such a solution (and, thus, in some sense, represents a number). But whether it represents one, or all three, of the solutions is not dealt with. It would be difficult, in any case, for us to say what it meant for Bombelli's yoked cubic radicals to represent numbers for him, since they don't lead to the determination or approximation of the number that they represent.
We have put quotation-marks around "three" when we discussed the "three" solutions to the cubic equation in the irreducible case because Bombelli does not consider negative solutions. Nevertheless, by appropriately transforming the equation, Bombelli is able to tum negative solutions of an equation into positive solutions of the transformed equation. See page 230 where Bombelli transforms the equation x3 + 2 = 3x into the equation y3 = 3y + 2, where y = -x, and pp. 230-231 where Bombelli divides x3 - 3x + 2 by x + 2 (y = 2). In his discussion of reducible cases of cubic polynomials, however, Bombelli talked of their (sin-
1 5For neusis see, for instance, Fowler (1 987), 8.2; Heath (1 921) , 235-4 1 , 65-68, 1 89-92, 4 1 2-1 3; Grattan-Guinness (1 997), 85; Bunt, Jones, and Bedient (1 976),
1 03-106; Boyer and Merzbach (1 989), 1 51 and 1 62.
16Cf. Bombelli (1 966), p. 226, the paragraph "Dimostrazione delle R.c. Legate con il +di- e -di- in linea."
VOLUME 24, NUMBER 1. 2002 17
gle, real) root and was surely unaware of the possibility that there might be "complex" interpretations of the relevant "yoked cubic radical" so as to provide the two complex roots of the cubic polynomial.
Second, it seems to us that Bombelli gains confidence in the "existence" of his yoked cubic radicals through his ability to perform algebraic operations with them, and thirdly, by his increased understanding of the relationship between the solution of the general cubic equation and the classical problem of angle-trisection. But it would be good to pin this down more specifically than we have done so far.
REFERENCES
Bombelli, Rafael . L 'A/gebra, prima edizione integra/e. Prefazioni di Et
tore Bortolotti e di Umberto Forti. Milano: Feltrinelli, 1 966.
--- . L 'Algebra, opera di Rafael Bam belli da Bologna. Libri IV e V
comprendenti "La parte geometrica" inedita tratta dal manoscritto B.
1569, [della] Biblioteca deii'Archiginnasio di Bologna. Pubblicata a
cura di Ettore Bortolotti Bologna: Zanichelli, 1 929.
On the mathematical environment at Bombelli 's time in Italy in general
and particularly in Bologna, see:
Bortolotti, Ettore. La storia della matematica nella Universita di Bologna.
Bologna: Zanichelli, 1 94 7 .
Bortolotti, E. "L'Aigebra nella scuola matematica bolognese del sec.
XVI," Periodico di matematica, series IV (5) (1 925).
Cossali , Pietro. Origine, trasporto in ltalia, primi progressi in essa del
l'a!gebra; storia critica di nuove disquisizioni analitiche e metafisiche
arricchita. Parma: Reale Tipografia, 1 797-1 799. 2 vols.
Libri, Guillaume. Histoire des sciences mathematiques en ltalie, depuis
Ia reinaissance des lettres jusqu'a Ia fin du dix-septieme siecle. Vols.
2 and 3. 2nd ed. Halle: Schmidt, 1 865.
For information about Bombelli's life see:
Gillispie, Charles Coulston, editor in chief. Dictionary of Scientific Biog
raphy. New York: Scribners, 1 97Q-1 980. 1 6 vols.
Jayawardene, S. A. "Unpublished Documents Relating to Rafael
Bombelli in the Archives of Bologna," /sis 54 (1 963), 391 -395.
--- . "Documenti inediti degli archivi di Bologna intorno a Raffaele
Born belli e Ia sua famiglia." Atti Accad. Sci. !st. Bologna C!. Sci. Fis.
Rend. 1 0 (2) (1 962/1 963), 235-247.
For the history of algebra during Bornbell i 's age see:
Giusti, E. "Algebra and Geometry in Bombelli and Viete," Boll. Storia
Sci. Mat. 1 2 (2) (1 992), 303-328.
Maracchia, Silvio. Oa Cardano a Galois: momenti di storia dell'algebra.
Milano: Feltrinell i , 1 979.
Reich, K. "Diophant, Cardano, Bombelli, Viete: Ein Vergleich ihrer Auf
gaben," Festschrift fur Kurt Vogel (Munich, 1 968), 1 31 -1 50.
Rivolo, M.T. and Simi , A. "The computation of square and cube roots
in Italy from Fibonacci to Bombell i ," Arch. Hist. Exact Sci. 52 (2)
(1 998), 1 61 -1 93. (Italian)
Sesiano, Jacques. Une introduction a l'histoire de l'algebre. Lausanne:
Presses polytechniques et universitaires romandes, 1 999.
On the relationship between mathematicians and humanists in the re
vival of Greek mathematics:
18 THE MATHEMATICAL INTELLIGENCER
Rose P. L. The Italian Renaissance of Mathematics. Geneva: Librairie
Droz, 1 975.
On the relation between angle trisection and cubic equations in Bombelli
see:
Bortolotti, E. "La trisezione dell'angolo ed il caso irreducible dell'e
quazione cubica neii'Aigebra di Raffaele Bombell i , " Rend. Ace. di
Bologna ( 1 923), 1 25-1 39.
On cubic and quartic equations in Cardano, Bombelli, and the Bologna
school of mathematics see:
Bortolotti, E. " I contributi del Tartaglia, del Cardano, del Ferrari, e della
Scuola Matematica Bolognese alia teoria algebrica delle equazioni
cubiche," Studi e mem. deii'Univ. di Bologna 9 (1 926).
Bortolotti, E. "Sulla scoperta della risoluzione algebrica delle equazioni
del quarto grado," Periodico di Matematica, serie IV (4) (1 926).
Kaucikas, A. P. "Indeterminate equations in R. Bombelli 's Algebra," His
tory and Methodology of the Natural Sciences XX (Moscow, 1 978),
1 38-1 46. (Russian)
Smirnova G. S. "Geometric solution of cubic equations in Raffaele
Bombell i 's 'Algebra, ' " !star. Metoda!. Estestv. Nauk. 36 (1 989),
1 23-129. (Russian)
On Bombelli and imaginary numbers see:
Hofmann, J. E. "R. Bombell i- Erstentdecker des lmaginaren I I , " Praxis
Math. 1 4 ( 10) ( 1 972), 25 1 -254.
--- . "R. Bombell i- Erstentdecker des lmaginaren," Praxis Math.
1 4 (9) (1 972), 225-230.
Wieleitner, H. "Zur Frugeschichte des lmaginaren," Jahresbericht der
Deutschen Mathematiker-Vereinigung 36 (1 927), 74-88.
On Bombelli's L 'Aigebra and its influence on Leibniz see:
Hofmann, J. E. "Bombell i 's Algebra. Eine genialische Einzelleistung und
ihre Einwirkung auf Leibniz," Studia Leibnitiana 4 (3-4) (1 972),
1 96-252.
On the calculation of square roots in Bombelli see:
Maracchia, S. "Estrazione della radice quadrata secondo Bombelli , "
Archimede 2 8 (1 976), 1 80-182.
On Bombelli as engineer see:
Jayawardene, S. A. "Rafael Bombelli, Engineer-Architect: Some Un
published Documents of the Apostolic Camera," Isis 56 ( 1 965),
298-306.
--- . "The influence of practical arithmetics on the Algebra of Rafael
Bombelli, ' ' Isis 64 (224) (1 973), 51 0-523.
Books on Mathematical Problems in the
Ancient World
Ball Rouse W. W. , and H . S. M. Coxeter. Mathematical Recreations
and Essays. New York: Dover, 1 987.
Bold, B. Famous Problems of Geometry and How to Solve Them. New
York: Dover, 1 982.
Boyer, Carl B., and U. C. Merzbach. A History of Mathematics. New
York: John Wiley & Sons, 1 989.
Bunt, Lucas N. H . , P. S. Jones, and J. D. Bedient. The Historical Roots
of Elementary Mathematics . Englewood Cliffs, NJ: Prentice-Hall,
1 976.
Courant, R. , and H. Robbins. What Is Mathematics? An Elementary Ap
proach to Ideas and Methods. New York: Oxford University Press,
1 996.
Dorrie, H. 100 Great Problems of Elementary Mathematics: Their His
tory and Solutions. Trans. David Antin. New York: Dover, 1 965.
Fowler, D. H. The Mathematics of Plato's Academy. A New Recon
struction. Oxford: Clarendon Press, 1 987.
Grattan-Guinness, lvor. The Norton History of the Mathematical Sci
ences. New York: W.W. Norton & Company, 1 997.
Gow, James. A Short History of Greek Mathematics . New York: G.E.
Stechert & Co. , 1 923.
Heath, Thomas. A History of Greek Mathematics. Oxford: Clarendon
Press, 1 921 .
Klein , Jacob. Greek Mathematical Thought and the Origin of Algebra.
Trans. Eva Brann. Cambridge, MA: The M. I.T. Press, 1 968.
Appendix A. Bombelli's Preface
To the reader I know that I would be wasting my time if I tried to use mere finite words to explain the infinite excellence of the mathematical disciplines. To be sure, the excellence of mathematics has been celebrated by many rare minds and honored authors. Nevertheless, despite my shortcomings, I feel obliged to speak of the supremacy, among all the mathematical disciplines, of the subject that is nowadays called algebra by the common people.
All the other mathematical disciplines must use algebra. In fact the arithmetician and the geometer could not solve their problems and establish their demonstrations without algebra; nor could the astronomer measure the heavens, and the degrees, and, together with the cosmographer, find the intersection of circles and straight lines without having been compelled to rely on tables made by others. These tables, having been printed over and over again, and furthermore by people with little knowledge of mathematics, are extremely corrupted. Thus, anyone using them for calculation is certain to make an infinite number of errors.
The musician, without algebra, can have little or no understanding of musical proportion. And what about architecture? Only algebra gives us the way (by means of lines of force) to build fortresses, war machines, and everything that can be measured: solid, and proportions, as occurs when dealing with perspective and other aspects of architecture.
Algebra also allows us to understand the errors that can occur in architecture.
Setting all these (self-evident) things aside, I will say only this: either because of the inherent difficulty of algebra, or because of the confused way that people write about it, the more algebra is perfected the less I see people working on it. I have thought about this situation for a long time and have not been able to figure out why. Many have said that their hesitations with algebra stemmed from the distrust they had of it, not being able to learn it, and from the ignorance that people generally have of algebra and of its
use. But I think rather that these people want only to protect themselves by making such excuses. If they were willing to tell the truth they should rather say that the real cause [of their lack of interest in algebra] is the weakness or roughness of their own minds. In fact, given that all mathematics is concerned with speculation, one who is not speculative works hard, and in vain, to learn mathematics. I do not deny that for students of algebra a cause of enormous suffering and an obstacle to understanding is the confusion created by writers and by the lack of order that there is in this discipline.
Thus, to remove every obstacle to those who are speculative and who are in love with this science, and to take every excuse away from the cowardly and inept, I turned my mind to try to bring perfect order to algebra and to discuss everything about the subject not mentioned by others. Thus, I started to write this work both to allow this science to remain known and to be useful to everyone.
To accomplish this task more easily, I first set about examining what most of the other authors had already written on the subject. My aim was to compensate for what they missed. There are many such authors, the Arab Muhammad ibn Musa being considered the first. Muhammad ibn Musa is the author of a minor work, not of great value. I believe that the name "algebra" came from him. For the friar Luca Pacioli of Bargo del San Sepolcro from the Minorite order, writing about algebra in both Latin and Italian, said that the name "algebra" came from the Arabic, that its translation in our language was "position" and that this science came from the Arabs. This, likewise, had been believed and said by those who wrote after him.
Yet, in these past years, a Greek work on this discipline was found in the library of our Lord in the Vatican. The author of this work is a certain Diophantus Alexandrine, a Greek who lived in the time of Antoninus Pius. Antonio Maria Pazzi, from Reggio, public lector of mathematics in Rome, showed Diophantus's work to me. To enrich the world with such a work, we began to translate it. For we both judged Diophantus to be an author who was extremely intelligent with numbers (he does not deal with irrational numbers, but only in his calculations does one truly see perfect order). We translated five books of the seven that constitute his work We could not finish the books that remained due to commitments we both had. In this work we found that Diophantus often cites Indian authors. Thus, I came to know that this discipline was known to the Indians before the Arabs. A good deal after this, Leonardo Fibonacci wrote about algebra in Latin. After him and up to the above mentioned Luca Pacioli there was no one who said anything of value. The friar Luca Pacioli, although he was a careless writer and therefore made some mistakes, nevertheless was the first to enlighten this science. This is so, despite the fact that there are those who pretend to be originators, and ascribe to themselves all the honor, wickedly accusing the few errors of the friar, and saying nothing about the parts of his work that are good. Coming to our time, both foreigners and Italians wrote about alge-
VOLUME 24, NUMBER 1 , 2002 19
A U T H O R S
FEDERICA LA NAVE
Department of History of Science
Harvard University
Cambridge, MA 02 1 38
USA
e-mail: lanave@fas.harvard.edu
Federica La Nave is a graduate student in history of science.
Her interests include classical philosophy, medieval log ic, and
medieval mus ic . She works on Aristotle, Abelard, Duns Scotus, William of Ockham, and philosophical issues in mathe
matics from the Renaissance to modern times.
bra, as the French Oronce Fine, Enrico Schreiber of Erfurt, and "il Boglione,"17 the German Michele Stifel, and a certain Spaniard18 who wrote a great deal about algebra in his language.
However, truly, there had been no one who penetrated to the secret of the matter as much as Gerolamo Cardano of Pavia did, in his Ars Magna where he spoke at length about this science. Nevertheless, he did not speak clearly. Cardano treated this discipline also in the "cartelli" that he wrote together with Lodovico Ferrari from Bologna against Niccolo Tartaglia from Brescia. In these "cartelli" one sees extremely beautiful and ingenious algebraic problems but very little modesty on the part of Tartaglia. Tartaglia was by his own nature so accustomed to speaking ill that one might think he imagined that by doing so he was honoring himself. Tartaglia offended most of the noble and intelligent thinkers of our time, as he did Cardano and Ferrari, both minds divine rather than human.
Others wrote about algebra and if I wanted to cite them all I would have to work a great deal. However, given that their works have brought little benefit, I will not speak about them. I only say (as I said) that having seen, thus, what of algebra had been treated by the authors already mentioned, I too continued putting together this work for the common benefit. This work is divided in three books.
BARRY MAZUR
Department of Mathematics Harvard University
Cambridge, MA 02138 USA
e-mail : mazur@math.harvard.edu
Banry Mazur is well known to lntelligencer readers for his mathematical contributions, especially to number theory and alge
braic geometry.
The first book includes the practical aspect of Euclid's tenth book, the way of operating with cube roots and square roots; this mode of operating with cube roots is useful when one deals with cubes, that is to say solids. In the second book, I treated all the ways of operating in algebra where there are unknown quantities, giving methods to solve their equations and geometrical proofs. In the third book I posed (as a test for this science) about three hundred problems, so that the scholar of this discipline [algebra] reading them could see how gently one may profit from this science. Accept, thus, oh reader, accept my work with a mind free of every passion, and try to understand it. In this way you will see how it will be of benefit to you. However, I warn you that if you are unfamiliar with the basics of arithmetic, do not engage in the enterprise of learning algebra because you will lose time. Do not condemn me if you fmd in the work some mistakes or corrections that do not come from me but from the printer. In fact, even when all possible care is used, it is still impossible to avoid typographical errors. Equally if you see some impropriety in the framing of my sentences, or a less than lovely style do not consider it [harshly] . . . . My only purpose (as I said earlier) is to teach the theory and practice of the most important part of arithmetic (or algebra), which may God like, it being in his praise and for the benefit of living beings.
1 7Bortolotti, in a footnote on p. 9 of his edition of Bombelli's text, says that "il Boglione" is not identified.
1 8According to Bortolotti, the Spaniard, although not clearly identified, is perhaps the Portuguese Pietro Nunes. See Bombelli (1 966), p. 9.
20 THE MATHEMATICAL INTELLIGENCER
Appendix B. A Glossary of Terms
Agguagliare {equating}: to solve an equation Agguagliatione {the equating}: the solving of an
equation Algorismo {algorithm}: a method for calculating Avenimento {what happens}: the quotient of a division Cavare {to extract}: to subtract Censo : name of ;i2 (used in the manuscript; censo is sub
stituted in the published book by potenza, that is to say, "power")
Creatore {creator}: root Cubato {cubed}: the cube of a number or of x Cuboquadrato {squared cube}: the sixth power Dignita {dignities}: the powers of numbers or of x from
the second power on Esimo {-th}: a word used to express a fraction
For instance 2/4 is 2 esimo di 4 that is "2th of 4", or "two fourths."
Lato {side}: root Nome {name}: monomial Partire {to part}: to divide Partitore {the one who parts}: divisor Positione {position}: equation
Potenza {power}: ;i2 Quadrocubico {square cubic}: sixth power Quadroquadrato {square squared}: fourth power R.c. : "radice cubica," that is to say, cube root R.c.L. or R.c. legata {linked cube root}: cube root of a
polynomial R.q. : square root R.q. legata {linked square root}: square root of a
polynomial R.q.c. or R.c.q. : "radice quadrocubica" and "radice cubo
quadrata," that is to say, sixth root R.R.q. : "radice quadroquadrata," that is to say, fourth root Residua {residue}: a binomial made by the difference of
two monomials. It is thus used for the cof\iugate roots
Ratto {broken}: fraction Salvare {to save}: to put a quantity aside for a moment to
be used later Tanto {an unknown quantity}: x Trasmutatione {transmutation}: linear transformation of
an equation Valuta {value}: the value of x Via {by}: the sign for multiplication
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VOLUME 24, NUMBER 1, 2002 21
11'1Ffii•i§rrF'h£119·1rr1rriil•iht¥J Marjorie Senechal, Editor I
Remembering A. S. Kronrod E. M. Landis and I . M. Yaglom
Translation by Viola Brudno Edited by Walter Gautschi
This column is a forum for discussion
of mathematical communities
throughout the world, and through all
time. Our definition of "mathematical
community" is the broadest. We include
"schools" of mathematics, circles of
correspondence, mathematical societies,
student organizations, and informal
communities of cardinality greater
than one. What we say about the
communities is just as unrestricted.
We welcome contributions from
mathematicians of all kinds and in
all places, and also from scientists,
historians, anthropologists, and others.
Please send all submissions to the
Mathematical Communities Editor,
Marjorie Senechal, Department
of Mathematics, Smith College,
Northampton, MA 01 063 USA
e-mail: senechal@minkowski.smith.edu
Alexander Semenovich Kronrod was born on October 22, 1921 , in
Moscow. Sasha Kronrod discovered mathe
matics when he was a participant in the now legendary study group for schoolchildren that was affiliated with Moscow State University. His teacher, D. 0. Shklyarskii, was a talented young scientist and an outstanding pedagogue. His general method was to encourage students to fmd solutions to difficult problems on their own. In 1938, Kronrod entered the Faculty of Mechanics and Mathematics at Moscow State University, where he immediately became known to the entire faculty, students, and instructors. They were enthralled by his outstanding talent, enormous energy, range of activity, and his sometimes deliberately paradoxical statements-even by his appearance-he was tall and had a beautiful sonorous voice.
While still a freshman, Kronrod did his first independent work. Professor A. 0. Gel'fond, who at that time was Chair of Mathematical Analysis and supervised a student circle, proposed a traditional problem in pre-World War II mathematics (although the problem was not traditional for Alexander Osipovich himself). It was concerned with the description of the possible structure of the set of points of discontinuity of a function that is differentiable at the points of continuity. In 1939, Kronrod's first scientific article, in which this problem was solved, appeared in the journal Izvestiya Akademii Nauk.
The normal course of studies for Kronrod's generation was interrupted by the war. Kronrod petitioned to be sent to the front but was rejected; students at the graduate level were ex-
empt from conscription. In subsequent years, they were sent to military academies. In the early days of the war they were mobilized to build trenches around Moscow. On his return, he renewed his application for enlistment, was accepted, and was sent to the front.
His military career was not easy. During the winter offensive of the Soviet army near Moscow, his bravery resulted not only in his receiving his first military decoration, but also his first severe injury. After he was wounded a second time in 1943, his return to the army became out of the question. He preserved his ability to study mathematics, but not to fight. The last injury made him an invalid; its effects were felt throughout his life.
While still in the hospital, Kronrod returned to a problem proposed to him by M. A. Kreines. The problem was the following: Let the permutation i � ki on the set N = {i } = { 1 , 2, 3, . . . } of natural numbers be such that it changes the sum of some infinite series, L ai * L ak . Does there exist a (conditionally) con�ergent series L bi which the above permutation transforms into a divergent one?
Kronrod greatly extended the scope of the problem. He managed to prove that, with respect to their action on (conditionally convergent) series, permutations fall into several categories. There are permutations mapping some convergent series into divergent ones-Kronrod called these "left." Permutations transforming some divergent series into convergent ones he called "right." Obviously, the inverse of a left permutation will always be a right permutation. The intersection of the sets of right and left permutations form "two-sided" permutations. They can
This article was written shortly after the death of A. S. Kronrod and was intended for publication in the journal
Uspekhi Mathematicheskikh Nauk, but has not been published because of the death of both authors.
W. Gautschi gratefully acknowledges help with the Russian from Alexander Eremenko and Olga Vitek and im·
provements of the English by Gene Golub.
22 THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
A. S. Kronrod.
transform a convergent series into a di
vergent one as well as a divergent se
ries into a convergent one. Permuta
tions which are neither left nor right
Kronrod called "neutral." These per
mutations cannot change the conver
gence of series and, as it turns out, they
cannot change the sum of even one se
ries. The latter follows from the fact
that the set of permutations that can
change the sum of a series (Kronrod
called them "essential") happens to be
a subset of the set of two-sided per
mutations.
The final part of the work contained
a set of effective criteria which permit
deciding to which class a permutation
belongs (left, right, two-sided, neutral,
essential) and an extension of the main
results to series with complex terms.
This extraordinarily fme work, pub
lished in 1945 in the journal Matematicheskii Sbornik, served as his grad
uation thesis. It earned him the prize of
the Moscow Mathematical Society for
young scientists. (We note that, while
it may not have been the first time a
student had been given this award, it
was indeed a rare event. Also, A. S. Kro
nrod was the only person ever to be
awarded this prestigious prize twice.)
In the autumn of 1944, Kronrod re
sumed his 4th-year studies at the Fac
ulty of Mechanics and Mathematics. In
February of the following year, an ex
traordinary event occurred: after a long
absence, the academician N. N. Luzin
returned to lecture at the Faculty. He
announced a course "The theory of
functions of two real variables" and at
the same time started a seminar closely
related to the course.
In those days, Nikolai Nikolaevich
Luzin was perceived by the students as
an almost mythological figure. Most of
the leading scientists of the older and
middle generations were his students.
The famous "Luzitania" (group of
Luzin's pupils) was surrounded by leg
ends. Since he had been absent as a lec
turer during the previous years, a nat
ural gap developed in the sequence of
his students. It appears that A. S. Kro
nrod and G. M. Adel'son-Vel'skii were
his last students. Although Adel'son
Vel'skil may have had other mentors (I.
M. Gel'fand and, in computational
mathematics, the slightly older Kron
rod), for Kronrod, Luzin was the only
mentor. He always was proud of this,
and liked to show a copy of the French
edition of Luzin's famous dissertation
"The Integral and the Trigonometric
Series," which had been presented to
him by the author. In addition, he
fondly remembered Luzin introducing
Kronrod as his student to Jacques
Hadamard.
Luzin's strongest quality had always
been his ability to present pupils with
problems of great general mathemati
cal importance which, when worked
on independently by strong and per
sistent young students, could lead to
the beginning of new directions.
The problem presented to Adel'son
Vel'skll and Kronrod was as follows.
Prove the analyticity of a monogenic
function by methods of the theory of
functions of a real variable without in
voking the Cauchy integral and the the
ory of functions of a complex variable.
Specifically, prove that every function
N(x + iy) = u(x, y) + iv(x, y), where
u(x, y) and v(x, y) satisfy the Cauchy
Riemann conditions, can be developed
into a convergent power series. This
problem was solved by Adel'son-Vel'ski'i
and Kronrod, and even generalized.
They considered arbitrary equations
au = A(x y) .E!:'.. au
= -B(x y) � ax ' ay' ay ' ax
with positive functions A(x, y) and
B(x, y), and established a relationship
between the smoothness of solutions
and the smoothness of the coefficients
A and B. (In the case of the Cauchy
Riemann equations, the coefficients
are identically equal to 1.) The study of
level-curves of functions of two vari
ables, u(x, y) and v(x, y), played an es
sential role in their research, as well as
establishing the maximum principle
for these functions.
This work became the starting point
for studying level-curves of arbitrary
(continuous) functions of two vari
ables; this was done in a subsequent
series of papers by A. S. Kronrod and
G. M. Adel'son-Vel'skii.
VOLUME 24. NUMBER 1 , 2002 23
However, Kronrod did not stop here.
It was not in his character to deal only
with a particular problem; we will speak
below about Kronrod's maximalism (in
life as well as in science). Dealing with
functions of two variables, Kronrod dis
covered that, while the theory of con
tinuous functions of one (real) variable
had achieved some degree of complete
ness at that time, a theory of functions
of two (and more) variables simply did
not exist. Only the most elementary
facts from the theory of functions of one
variable had been extended, and they
did not contain anything "essentially
two-dimensional." If the theory does not
from the one-variable theory to the two
variable theory, features depending on
variation dichotomize, so that for func
tions of two variables it is natural to in
troduce two variations. One of them he
called planar, the other linear. The
boundedness of the planar variation
guarantees the existence almost every
where of an asymptotic total differential.
For a smooth function, this variation
turns out to be equal to the integral of
the absolute value of its gradient, ex
tended over the domain of definition.
The linear variation was basically a
new object. Kronrod introduced the
concept of a monotone function of two
the original function, a metric can be de
fmed and, on the tree, a function. The
linear variation then turns out to be
equal to the usual variation of the func
tion defined on a one-dimensional tree.
The boundedness of both the planar and
linear variation guarantees the exis
tence almost everywhere of the usual
total differential.
Kronrod considered continuous
functions, but the concepts he intro
duced can easily be carried over to the
case of discontinuous functions. He
also outlined a program for investigat
ing functions of many variables, which
was later carried out by his students.
Kronrod introduced the exist, it has to be created. In the
course of the next few years all
of Kronrod's attention was de
voted to exploring this vast prob-
lem area concept of a monotone
function of two variab les . Over four years, Kronrod de
veloped an orderly theory, con
taining properties of functions of two
real variables and their connections
with the concept of variation; it paved
the way for studying functions of many
variables.
At that time, an active group
of students congregated around
Kronrod. (More of Kronrod's
pedagogical activity is dis
cussed below.) Among them
were A. G. Vitushkin, who de
veloped a theory for variations
From the beginning, he avoided us
ing definitions that depend on the choice
of a given orthogonal coordinate system
(e.g., Tonelli variation), and he intro
duced concepts that are invariant with
respect to orthogonal mappings. Varia
tions of functions of two variables are
fundamental concepts for his theory.
Kronrod showed that in the transition
variables, a natural generalization of the
corresponding concept for a function of
a single variable. He proved that the
boundedness of the linear variation per
mits the function to be represented as
a difference of two monotone functions.
For the linear variation itself, he gave a
number of equivalent definitions, one of
which is of particular interest. It turns
out that with a continuous function of
two variables one can associate a one
dimensional tree, the elements of which
are the components of the level-sets of
the function. On them, with the help of
Kronrod's name has become a household word among numelical analysts
because of his work in 1964 on Gaussian quadrature. He had the interest
ing and fruitful idea of extending an n�point Gaussian quadrature 11lle op
timally to a (2n + I)-point nlle by retaining the n Gauss points and adding
n + 1 new points, choosing all 2n + 1 weights in such a way as to achieve
maximum polynomial degree of exactness. This allows a more accurate
approxin1ation to the integral without wasting the n ftmction values already
computed for the Gauss approximation. The new formula, now called the
Gauss-Kronrod formula, is currently used in many software packages as a
practical tool to estimate the enor of the Gaussian quadrature fom1ula
This is particularly tl1le for modem adaptive quadrature routines.
24 THE MATHEMATICAL INTELLIGENCER
Walter Gautschi
Department of Computer Sciences Purdue University
West Lafayette, IN 47907-1 398
USA
e-mail: wxg@cs.purdue.edu
of functions and sets of many variables,
and A. Ya. Dubovitskii, who studied in
detail the set of clitical points for func
tions of many variables and smooth
mappings. In particular, he reproved A.
Sard's theorem, at the time not known
in Moscow, and he also obtained a se
lies of more refmed theorems on the
stl1lcture of the clitical points.
From the modem perspective, half
a century later, it is not A. S. Kronrod's
results themselves that are of the
greatest interest. (They represent an
important but closed phase of devel
opment.) The main value lies in the
apparatus he created for obtaining the
results. For example, Kronrod's one
dimensional tree was used by V. I.
Amol'd to solve Hilbert's 13th problem.
Especially popular nowadays is the
following theorem of Kronrod: Let G c !Rn be a domain andf: G � IR a smooth
function, E1 = {x E Glf(x) = t} the
level-sets of the function !, and ds the
( n - 1 )-dimensional surface element
on E1• Then
meas G = raxf f ( I:!: I ) dt. mmf Et v f
This theorem, for example, lies at the
basis of many modern proofs in the
theory of partial differential equations.
Kronrod's work on the theory of
functions of two variables constituted
the contents of his Masters thesis,
which he defended in 1949 at the
Moscow State University. His official
advisors were M. V. Kel'dysh, A N. Kol
mogorov, and D. E. Men'shov. For this
work he was immediately awarded the
Doctoral degree in physical-mathemat
ical sciences, bypassing the Masters
degree.
The next large problem to attract
Kronrod's attention was the following:
Let S be a given surface with bounded
Lebesgue area, parametrically embed
ded in !R3. Is it true that S has an as
ymptotic tangent plane almost every
where (in the sense of the measure
generated on S by Lebesgue area)? This
converges to the solution of the differ
ential equation, because if the scheme
that was set up is physically correct
and there is no convergence to the so
lution of the differential equation, then
so much the worse for the differential
equation. As a rule, one should not do
a theoretical estimation of the error.
Such an estimation requires the de
scription of a set of functions contain
ing the solution. A priori, this set, as
well as the distribution of solutions in
it, is unknown. Today, all of this seems
trivial, but in those days it sounded
paradoxical. Kronrod devised a series
of algorithms for the fast solution of
bered that at that time (the beginning
of the second half of the forties) there
was still no knowledge in the Soviet
Union of American electronic comput
ers. The project of such a computer
RVM (R for "relay," in contrast to the
E now in use for "electronic")2-was
accepted to go into production.
If this computer had been built
quickly, it would have become the first
digital high-speed computer. Among
other things, with respect to speed of
computation, it would have surpassed
the contemporary American EVMs,
owing to the profound ideas incorpo
rated into its design; in particular, it
remained an unsolved
problem for a long
time; Kronrod found a
positive answer but
did not publish the so
lution. He did so be
cause he had decided
Kronrod and Bessonov conceived
the idea of a u n iversal prog ram
control led dig ital computer.
used the "cascade
method" (a kind of
parallelism, a topical
modern problem) and
the Shannon counter,
which was then
largely unknown in
to break with pure mathematics. That
decision was firm and forever.
To understand what happened, we
must go back a few years. In 1945, dur
ing his fourth-year university studies,
Kronrod started working for the com
puter department of the Kurchatov
Atomic Energy Institute. Initially, the
reason was financial: he was married,
and in 1943 a son was born. In partic
ular, there was a need for accommo
dation. Working for the Institute of
fered a solution. But Kronrod was not
the kind of person who could take his
work lightly. Faced with computa
tional mathematics, he went into it
with great seriousness. He found that
this was an interesting area, quite un
like pure mathematics, in his opinion.
He always stressed that computational
methods must be kept apart from the
orems that are proved about computa
tional mathematics. For example, he
used to say that, when applying finite
difference methods to solve differen
tial equations, the finite-difference
scheme must be set up starting from
the physical problem and not from the
differential equation. And one should
never be interested in whether the so
lution of the finite-difference equations
various problems (e.g., independently
of some other authors, he discovered
the sweep method1).
Thus, Kronrod discovered for him
self a new area of activity. Probably
this was not enough for such a resolute
break with traditional mathematics, in
spite of all the maximalism which, as
has already been said, was one of the
foremost traits in his character.
At that time, besides electric desk
calculators-"mercedes" -tabulators
and sorting machines working with
punched cards were the computational
devices in use. During this period, a
fortunate relationship began to de
velop between Kronrod and Nikolai
Ivanovich Bessonov, a talented relay
engineer. From some tabulators and
supplementary relay machines for mul
tiplying numbers, which he had devel
oped, Bessonov constructed the ma
chine "Combine," on which one could
solve more complex computational
problems. Kronrod and Bessonov at
this point conceived the idea of a uni
versal program-controlled digital com
puter. Apparently, the logical aspect of
the problem was dealt with by Kron
rod, and the design aspect, undoubt
edly, by Bessonov. It must be remem-
the Soviet Union. All of this would have
opened new perspectives and revolu
tionized computational methods.
By the end of the 1940s it was rec
ognized that it was necessary to create,
side by side with the I. V. Kurchatov In
stitute, yet another "atomic" institute,
the guidance of which was entrusted to
A I. Alikhanov. On the recommendation
of I. V. Kurchatov and L. D. Landau,
Alikhanov invited Kronrod to his insti
tute in 1949 and entrusted him with the
direction of the Mathematical Depart
ment, later named the Institute for The
oretical and Experimental Physics
(ITEF). Here, it is appropriate to men
tion yet another aspect of A S. Kron
rod's nature. He was a born organizer.
Being in charge of a department, he was
given the opportunity to organize its
work as efficiently as possible. Compu
tational mathematics, the computer, the
opportunity to organize work in this
area, and the recognition of its useful
ness-all of this took precedence over
his call to pure mathematics; besides, he
was to a large extent a pragmatist.
Upon transferring to ITEF, Kronrod
invited Bessonov to join the staff. The
RVM was being built, but the project
was moving at an agonizingly slow
1The "sweep method" (METOJJ: IIPOfOHKII) is an algorithm for solving linear second-order two-point boundary-value problems or tridiagonal linear systems arising
in the finite-difference solution of them . -W. G.
2The V stands for "vychislitel 'naya" ("computing") and the M for "machine."-W. G.
VOLUME 24, NUMBER 1 . 2002 25
pace. The machine was cheap, and un
fortunately this created an attitude of
low interest toward it. Quite competent
and well-meaning people gave Kronrod
wise advice on how to speed up the
construction. For example, one could
make contacts out of gold, which
would somewhat improve the quality
of the machine, and would make it con
siderably more expensive. This would
radically change the attitude toward
the machine. Kronrod could only laugh
at this kind of advice. His honesty
would never allow him to use such
tricks. By the time the machine was
completed, a project to build the first
electronic computer had already been
started. Thanks to the many rich ideas
incorporated into the design of the
RVM, it would have operated at the
high speed of the EVM, but, of
crease the speed, but in fact brought
down the speed to a very low level. Yet,
the relay machine still remained his fa
vorite accomplishment, bringing tears
when it was dismantled.
During the period 1950-1955, Kron
rod's main activity was finding numer
ical solutions to physical problems. He
collaborated much with physicists, in
particular theoretical physicists,
among whom, with respect to work, he
was closest to I. Ya. Pomeranchuk,
and, on a purely personal level, L. D.
Landau. For his work on problems of
importance to the state he was
awarded the Stalin Prize and an Order
of the Red Banner.
Only in 1955 did a real opportunity
arise for A S. Kronrod to work with an
electronic computer. It was the M-2
mathematics. Then and later, he be
lieved that the theory of functions of a
real variable offers the best method for
encouraging a student's creativity.
Here, in his way of thinking, a minimal
amount of initial knowledge enables
one to derive complex results. Many
mathematicians of the older genera
tion participated in this seminar (E. M.
Landis, A Ya. Dubovitski'i, E. V.
Glivenko, R. A Minlos, F. A Berezin,
A A Milyutin, A G. Vitushkin, R. L. Do
brushin, and N. N. Konstantinov,
among many others).
After the university moved to a new
building, Kronrod quit as the leader of
the seminar. Shortly thereafter, studies
resumed, but were devoted to com
puter principles.
When he started with enthusiasm to
program the M-2 machine, Kro
course, it had no future. On the
other hand, if the computer had
been built more quickly, even
with golden contacts, it would
have repaid the expenses.
An idea is noth ing ; its im
plementation , everyth i ng .
nrod quickly came to the con
clusion that computing is not
the main application of com
puters. The main goal is to
teach the computer to think,
We are talking about this RVM in
such detail in order to underscore one
of A S. Kronrod's leading principles: an
idea is nothing; its implementation,
everything. Even though rich with bril
liant ideas, he did not value them. He
gracefully gave them away left and
right, quite honestly convinced that the
authorship belongs to the one who im
plements them. In this respect, he was
quite the opposite of his teacher, Luzin.
With regard to the RVM, he resolutely
declared Besso nov (definitely a tal
ented person) to be its sole inventor.
Having had a clear and deep insight,
Kronrod quickly realized the advan
tages of electronic computers over re
lay computers. He actively participated
in discussions on building the first
EVM. He was a member of many and
diverse committees planning to build
such a machine at that time. One must
say, though, that, his ideas often being
ahead of their time, he was often left
in the minority in these discussions.
For example, he unsuccessfully in
sisted on hardware support for float
ing-point numbers. However, our first
machines used fixed-point numbers;
operations with floating-point numbers
were implemented by means of soft
ware. This, theoretically, would in-
26 THE MATHEMATICAL INTELLIGENCER
computer constructed by I. S. Bruk, M.
A Kartsev, and N. Ya. Matyukhin in the
laboratory of the Institute of Energy
named after Krzhizhanovski'i and di
rected by I. S. Bruk. This laboratory
later became the Institute for Elec
tronic Control Machines. The mathe
matics/machine interface was devel
oped by A L. Brudno, a great personal
and like-minded friend of Kronrod.
At this point, a new period started
in the life of A S. Kronrod. We will
speak about this later, but to preserve
the chronological order, we will men
tion yet another aspect of his activity.
During the years 1946-1953, he led a
seminar, called the Kronrod circle. At
that time, it was probably not less
known among young mathematicians
than the Luzin seminar. An atmosphere
of enthusiasm always surrounded the
seminars he led. Its participants were
convinced that mathematics was the
most important science and that
A S. Kronrod was one of its prophets.
At the same time, he was not the mas
ter, but simply Sasha, and it so contin
ued to the end of his days. His seminar
studied the theory offunctions of a real
variable, set theory, and set-theoretical
topology. Work continued with the
same fervor, even after he left pure
i.e., what is now called "artificial intel
ligence" and in those days "heuristic
programming."
Kronrod captivated a large group of
mathematicians and physicists (G. M.
Adel'son-Vel'ski'i, A L. Brudno, M. M.
Bongard, E. M. Landis, N. N. Konstan
tinov, and others). Although some of
them had arrived at this kind of prob
lems on their own, they uncondition
ally accepted his leadership. In the
room next to the one housing the M-2
machine, the work of a new Kronrod
seminar started. At the gatherings
there were heated discussions on pat
tern-recognition problems (this work
was led by M. M. Bongard; versions of
his program "Kora" are still function
ing), transportation problems (the
problem was introduced to the semi
nar and actively worked on by
Brudno ), problems of automata theory,
and many other problems.
Kronrod skillfully guided the enthu
siasm of the seminar participants to
ward applications. He proposed to
choose a standard problem, so that an
advance in the solution allowed judg
ment on the level reached by the au
thors in the area of heuristic program
ming. As such a problem, he proposed
an intellectual game. The first problem
chosen and programmed was the card game "crazy eights." This choice (in spite of the smiles it provoked) was not accidental and not meant to be frivolous. It is a complex game with no established theory. Considering the low capabilities of the computer and its limited memory, the game's simple description of positions was very important. The program was written and played. It worked fine as long as there were enough cards remaining and in conditions of "incomplete information." After the game became open and everything was reduced to an enumeration of all possible strategies, the computer's capacity was too limited to handle the extremely large size of the game's tree. (The game was abandoned,
colleagues treated heuristic programming and anything not connected with their immediate needs as mere entertainment.
He organized a chess match between the institute's program and the best (at that time) American program, developed at Stanford University under the guidance of J. McCarthy. Over the telegraph a match of four games was played, ending with a score of 3 to 1 in favor of the institute's program.
However, the Mathematical Department, of course, existed as a service medium for physical problems, and the time has come to say how this work was organized by A S. Kronrod. This may be instructive, for in all scientific
never again to be resumed. It is not clear whether even modem computers have enough capacity for this game.)
In the process of creating the program, general
The programming
m ust be done by the
mathematician .
heuristic programming principles were formulated for the first time. They included a length-independent programmed search (a priori it is not clear whether this is possible or not), algorithms for organizing information, etc. Since the "crazy eights" game clearly did not qualify as a standard text problem because it was a strictly regional (or national) game, Kronrod proposed as a standard another game-chess. Chess is played throughout the world. In the USA, people had already started to create chess-playing programs. Such programs were already developed on special-purpose machines: in the Mathematics Division of the ITEF a first, and then a second M-20 machine was installed. The chess program was written by a group of mathematicians (Adel'sonVel'skii, V. I. Arlazarov, A R. Bitman, and A V. Uskov) which did not include Kronrod himself. Nevertheless, when a difficulty was encountered regarding the development of a general recursive search scheme, he entered the group and invented an improvement which helped to overcome the difficulties. He assumed the role of an organizer. It was necessary, but not easy, to create appropriate working conditions for the chess group at the institute. Most of his
institutes with a need for mathematical service, work is organized differently.
Kronrod believed that a mathematician solving the mathematical aspect of a physical problem should understand this problem, beginning with its formulation, and should understand how the results obtained are going to be used. Moreover, the mathematician must work out the algorithm, usually according to the physical formulation, write the program, and run it. The programming must be done by the mathematician, because only in this way can the optimal variant of the solution be chosen. For this, one needs mathematicians with sufficiently high qualifications, and Kronrod attracted many good graduates from the Faculty of Mechanics and Mathematics to ITEF, also those who specialized in abstract areas. Why precisely people from the Faculty of Mechanics and Mathematics? He liked to quote I. M. Gel'fand: "The objective of the Faculty of Mechanics and Mathematics is to make people capable," meaning that for a mathematician it suffices to formulate the definitions and the rules operating on them.
For a mathematician to be able to program, without expending unneces-
sary efforts, however, one must provide him maximum ease and liberate him from all tasks not requiring his qualifications. The mathematician would use a language that is close to common language, write on a form printed on highquality paper, using a pencil that allowed erasing an unlimited number of times. There was a rich library of standard programs which were easily accessible. A program (or any piece of it) would be sent to the coding center. Coding, checking the code, punching cards, checking the cards-all this did not require the programmer's attention. The next day he would receive two copies of the program without any coding or punching mistakes. The debugging was done in
front of the control panel, and there was no time problem. A programmer was given as much access to the control panel as he needed, and he did not need much. Programs were partitioned into small blocks, each of which could be debugged separately and
usually ran the very first time. A correction could be introduced into a program by pushing a key on the control panel, just as an editor does now. A woman responsible for card-punching worked next to the programmer and could immediately change the respective card. For this, colored cards were used. The next day, a corrected white card took its place in the deck.
Each program was required to undergo a check by hand computation. A general rule, strictly followed, was that a program which worked and produced reasonable answers is not necessarily correct, even if the result is accurate in special cases.
It turned out that the work of the coding and card-punching groups was extremely important in the course of writing a program. These groups consisted of women, since they were believed to be more accurate in this kind of work; on each form for writing a program which was prepared for Kronrod's department, on the bottom was written "program written by (a male name)," "coded by (a female name)," "coding checked by (a female name)," "punched by (a female name)," "punching checked by (a female name)."
How did Kronrod achieve such ac-
VOLUME 24, NUMBER 1 , 2002 27
curate work in all these subdivisions?
First, he selected good female employ
ees; second, he managed to provide
high salaries for them; and finally, he
set the salary in accordance with the
quality of the work done. For error-free
work, he would give a monthly 20%
raise, for two mistakes per month that
were made by a card-punching checker,
this raise was cut in half. For an addi
tional two mistakes per month, there
was no raise at all. (Mistakes on col
ored cards were not counted.) Here,
Kronrod was merciless, but in every
thing not connected with the quality of
work, he was very open and accom
modating. His colleagues liked and re
spected him and took their work to
heart-and there were few mistakes.
Bessonov, retraining himself quickly
in electronics, kept the computers in
exemplary working order. There were
practically no malfunctions. One must
say here that under the guidance of
Kronrod, Bessonov constantly intro
duced improvements to the machines.
In 1963, he completely overhauled the
system of commands, thereby increas
ing the capacity of the machine by a
factor of two.
Kronrod proceeded from the as
sumption that a normal computational
problem must run quickly. There are,
of course, special cases in which
lengthy computations are necessary,
but this is not the rule but a rather rare
exception. The following policy was
adopted: if the debugged program ran
more than 10 minutes, its author was
invited to see the "Senior Council,"
headed by Kronrod. There, the algo
rithms were properly analyzed, and
usually the computing time was short
ened.
All in all, this was similar to a well
organized factory operation. The re
sults were astonishing. On their low
speed machines, the mathematicians
of the ITEF surpassed the West in dif
ficult problems. For example, tracking
observations in scintillating cameras
produced more accurate results in half
the time of a similar program at CERN,
running on a computer 500 times
faster. In a couple of hours during the
night it could compute all that an ac
celerator could do in 24 hours. That is
why there was time to repair and main-
28 THE MATHEMATICAL INTELLIGENCER
tain the machine, which was obligatory
for vacuum tube machines, and also
plenty of time for heuristic and other
problems which we will discuss below.
In the world of Soviet theoretical
physics of that time, a clear tendency
was prevalent: the more talented a the
oretical physicist is, the less computing
is done for him. There was one physi
cist for whom nothing was ever com
puted, namely L. D. Landau. Less gifted
physicists as a rule demanded a lot of
computation, some of them expressing
dismay when asked by mathematicians
about the source of the equations dealt
with, or the utility of the results. We
should say here that Kronrod liked to
quote Hamming: "Before starting a
computation, decide what you will do
with the results." The practice in the de
partment was to check with the math
ematician every physical problem
formulation that demanded a large
amount of computation. Sometimes it
was discovered that a qualitative result
that could be found without computa
tion was sufficient, that the problem
was over- or under-determined, that the
computational errors invalidated the ef
fect of interest, that the problem's for
mulation was not correct, etc. Kronrod
even put a poster on his door: "Not to
be bothered with integral equations of
the first kind!" It did not mean at all that
he thought integral equations of the
first kind could not be solved. For ex
ample, the Mathematics Division of the
ITEF computed shapes of magnetic
poles for several large accelerators.
This leads to a Cauchy problem for the
Laplace equation, which, as is well
known, can be reduced to an integral
equation of the first kind. But that was
a special case-it was really necessary
to do some computing. Incidentally, the
work was done by an excellent mathe
matician, A. M. Il'in.
Returning to A. S. Kronrod, it must
be said that he perfectly understood
that in some cases equations of the first
kind must be solved by virtue of the na
ture of the problem. At the same time
he believed that much more often one
does not need the solution of the first
kind equation itself, but some mean
value. For this mean value, as a rule, a
simple and, importantly, a more cor
rect problem can be formulated.
Two-and-a-half decades have passed.
Generations of electronic computers
have succeeded one another. Their
speed has been increased by many or
ders of magnitude, and their memory
has become practically unlimited.
Along with this, the man/machine in
terface and the type of machine use
have changed. For the most part, the
machines are no longer used for com
puting, but for processing and storing
information. Nevertheless, much of
what was introduced into the practice
by Kronrod is still relevant to this day.
If a mathematician participates (in the
role of computer and programmer) in
solving a natural science problem, he
must begin by understanding the phys
ical, chemical, biological, economical,
etc. formulation of the problem. Col
laborating with the physicist, chemist,
biologist, economist, he must, together
with them (or, if need be, instead of
them) formulate the mathematical
problem, create an algorithm, and
write the program, never ignoring the
fact that whether or not an algorithm
for a serious problem is reasonable can
only be discovered in the process of
writing the program. At the same time,
the mathematician must be provided
with maximum assistance to free him
from tasks that do not require his qual
ifications.
At the end of the fifties, Kronrod be
gan to interest himself in questions of
economics, in particular price forma
tion. He observed that the basic prin
ciples of price formation were wrong.
L. V. Kantorovich came to the same
conclusion, as did other economists. A
USSR Cabinet Ministry commission on
the subject was formed, among which
the mathematicians included Kan
torovich and Kronrod. As a result of
this committee's work, new price for
mation principles were adopted. Their
implementation required computing
the so-called "Leont'ev matrices" of
material expenditure balances across
the country. This colossal computa
tional work was directed by Kronrod
and carried out first on the RVM, and
then on the same two M-20 machines.
Later, the work was further developed
by a pupil of A. S. Kronrod, the now
well-known economist V. D. Belkin.
Another problem which interested
Kronrod in the 1960s was the computerized differential diagnostics for some diseases. In the Cancer Institute named after Gertsen, a laboratory was created, which was headed by P. E. Kunin, a physicist by training and one of Kronrod's students in heuristic programming. The laboratory conducted research, in particular on the differential diagnostics of lung cancer and central pneumonia. (The results were considered crucial for deciding whether surgery was needed.) Kronrod supervised the research. Quite encouraging results were obtained. The sudden death of Kunin cut short this work
During this time, Kronrod organized mathematics courses for high schools and developed teaching methods for them.
After signing a petition in 1968 in support of the prominent dissident and logician Alexander Esenin-Volpin, the son of the famous poet Esenin, Kronrod was summarily fired from his position at ITEF. He later became head of the mathematical laboratory of the Central Scientific Research Institute of Patent Information (CNIIPI). Setting up the mathematical and informational part (and for this, among other things, he needed to create software for Kronrod conditions for the machine "Razdan" located at the CNIIPI and to assemble a cohesive group of mathematicians), Kronrod became interested in matters strictly related to patents and discovered that, here also, radical reforms were needed that would stimulate inventions.
Kronrod proposed a number of measures that would help improve the prevailing situation, and entered the high echelons, where he found understanding. The director of the CNIIPI, who was supportive of Kronrod, departed, and the new director wanted to free himself of such a worrisome colleague. A S. Kronrod left the CNIIPI.
His last employment was at an institute called the Central Geophysical Expedition. Here Kronrod headed a laboratory processing explorationdrilling data. He implemented a series of new computational ideas, but this work, of course, did not match the level of his talent, and so he set new goals for himself.
It must be said that Kronrod's personality attracted many talented people from quite different fields. And while some of them were attracted by his professional competence (e.g., for the prominent oil researcher Lapuk he
had to compute the optimal regime for exploiting oil and gas deposits), communicating with others involved quite different interests. You could meet at his home with the actor Evstigneev, the screenwriter Nusinov, and others. Kronrod could be seen with academician I. G. Petrovski! at the Burdel sculpture exhibition, not discussing mathematical problems, but questions of fine art. Among his friends also were prominent physicians: the surgeon Simonyan, the pediatrician Pobedinskaya, the radiologist-oncologist Marmorshtem, and others.
Having a keen sense of philanthropy, with a strong desire to immediately help people, he was captivated by the professional stories of physicians, sharing their successes and failures. Gradually he understood that saving the terminally ill is the most important thing that can and must be done. At that time, he became acquainted with a Bulgarian doctor, Bogdanov, developer of a medicine called anabol, based on a Bulgarian sour milk extract. This medicine often caused remission in cancer patients. Incidentally, Bogdanov treated i. N. Vekua and S. A Lebedev with anabol.
Kronrod started advertising this medicine. The medicine was not easy to obtain, as it was produced in Bulgaria in limited quantities. Kronrod organized the delivery of this medicine
. for terminally ill patients. But this was not the solution; anabol was rare and expensive. It had to be produced in large quantities and by a simple procedure. Thus, a new medicine appeared, which was sour-clotted milk, based on a Bulgarian milk extract. He gave this medicine the name milil (in honor of Mechnikov Il'ya Il'ich). He developed a simple technology for its production and ways of using the medicine.
Kronrod did not treat patients without a physician. Physicians used milil according to his instructions (there were more and more who came to believe in Kronrod's medicine). The medicine was
used in hopeless cases for patients who were doomed to die. Milil became well known and accepted to some degree: A A Vishnevskll set aside a ward at his institute to treat patients according to the method of A S. Kronrod. Kronrod was promised a laboratory for animal experimentation, but this remained a promise, and he did all the experiments on himself.
No longer a novice in medicine, Kronrod replaced mathematics books with medical books, many of which he obtained from physicians he knew. He already had considerable clinical experience. He kept a large card file on the history of patients' diseases. And he had an important advantage over physicians: he could do a correct statistical inference from the thousands of cards in his file. The well-known therapist I. G. Barenblat (the father of the mechanical engineer G. I. Barenblat) was struck, after a conversation with Kronrod, by his medical erudition. And is it surprising? If a very talented person works hard in a medical field, and if he is helped by good specialists, he is likely to become proficient in it, at least as much as an average, or even a good student in a medical school. But he did not have a medical degree, and milil was not an approved medicine. In the medical field, this could not be tolerated. Recall, for example, the story of artificial pneumothorax. On the other hand, Kronrod did not treat patients without physicians, and was not paid for the treatment. In fact, he spent his fortune on the treatment. (At the end, he was so badly dressed that the laboratory assistants offered him a suit as a birthday gift.) In spite of this, a criminal case was opened against Kronrod and, more seriously, his card files were confiscated. The story had a tragicomic ending. Either the mother or the wife of the prosecutor who brought the case had cancer. And he needed milil. Naturally, the case was dismissed, and the card file was returned. But for A S. Kronrod himself, the story turned into a tragedy. He had a stroke, and he completely lost his speech and his abilities to read and write. Recovery was very slow, but he learned how to speak, read, and write once again. He left his position at the Central Geophysical Ex-
VOLUME 24, NUMBER 1 , 2002 29
pedition. He quit working on mathe
matics. Now he was interested only in
medicine. But at this point he suffered
a second stroke. The situation was pre
carious. The physician believed that a
final stage of agony had started. But
Kronrod was conscious and asked to
be put in a very hot tub and to remain
there for several hours. One of the
prominent neuropathologists re
marked later that this was the only cor
rect solution. This time, he survived.
But he did not survive the third stroke.
He died on October 6, 1986.
Bibliography: Publications of
A.S. Kronrod
1 . A. Kronrod, Sur Ia structure de /'ensemble
des points de discontinuite d'une fonction
derivable en ses points de continuite
(Russian) , Bull. Acad. Sci. URSS, Ser.
Math. [Izvestia Akad. Nauk SSSR] 1 939,
569-578.
2. G.M. Adel'son-Vel'skiy and AS. Kronrod,
On a direct proof of the analyticity of a
monogenic function (Russian), Dokl. Akad.
Nauk SSSR (N.S.) 50 (1 945), 7-9.
3. G.M. Adel 'son-Vel'skiy and AS. Kronrod,
On the level set of continuous functions
possessing partial derivatives, Dokl. Akad.
Nauk SSSR (N.S.) 50 (1 945), 239-241 .
4. G.M. Adel'son-Vel'skiy and A.S. Kronrod,
On the maximum principle for an elliptic
system, Dokl. Akad. Nauk SSSR (N .S.) 50 (1 945), 559-561 .
5 . A. Kronrod, On permutation of terms of nu
merical series (Russian) , Rec. Math. [Mat.
Sbornik] N.S. 18 (60) (1 946), 237-280.
6. AS. Kronrod and E.M. Landis, On level
sets of a function of several variables
(Russian), Dokl. Akad. Nauk SSSR (N.S.)
58 ( 1 947), 1 269-1 272.
7 . AS. Kronrod, On linear and planar varia
tions of functions of several variables
(Russian), Dokl. Akad. Nauk SSSR (N.S.)
66 (1 949), 797-800.
8. A.S. Kronrod, On a line integral (Russian),
Dokl. Akad. Nauk SSSR (N.S.) 66 (1 949),
1 041-1 044.
9. AS. Kronrod, On surfaces of bounded
area (Russian), Uspehi Mat. Nauk (N.S.) 4 (1 949), no. 5 (33) , 1 81 -1 82
1 0. AS. Kronrod, On functions of two vari
ables, Uspehi Matern. Nauk (N.S.) 5 (1 950), no. 1 (35), 24-134.
1 1 . AS. Kronrod, Numerical solution to the
equation of the magnetic field in iron with
30 THE MATHEMATICAL INTELUGENCEA
E. M. LANDIS
E. M. Landis was born in 1 921 in Kharkov
and was raised in Moscow. He was ad
mitted to Mathematics and Mechanics at
Moscow State University in 1 939, but im
mediately had to leave for six years of mil
itary service. Only after the war could he
get back to his studies.
In much of his early research he fol
lowed the interest in real analysis of his
first teacher, A. S. Kronrod. Later, his pri
mary area was partial differential equa
tions, and he had many results and many
students in this area. His achievements in
programming and algorithms were widely
influential as well. He worked at Moscow
State University from 1 953 until his death
in 1 987.
He was a music lover and could often
be found at the Moscow Conservatory.
His paintings appeared in a faculty exhi
bition at the university.
allowance saturation, Soviet Physics Dokl.
5 (1 960), 5 1 3-51 4.
1 2 . AS. Kronrod, Integration with control of
accuracy, Soviet Physics Dokl. 9 (1 964),
1 7-1 9.
1 3 . AS. Kronrod, Nodes and weights of quad
rature formulas. Sixteen-place tables. Au
thorized translation from the Russian, Con
sultants Bureau, New York, 1 965.
1 4. V.D. Belkin , A.S. Kronrod, U.A. Nazarov,
and V.Y. Pan, The rational price calcula
tion based on co.ntemporary economic in
formation, Akad. Nauk SSSR, Ekonomika
i Maternaticeskie Metodi (1 965) 1 , no. 5,
699-7 1 7.
1 5 . V.L. Arlazarov, AS. Kronrod, and V.A. Kron
rod, On a new type of computers. Dokl.
Akad. Nauk SSSR (1 966) 171, no. 2 ,
299-301 .
1 6. AS. Kronrod, V.A. Kronrod, and I .A.
Faradzvev, The choice of the step in the
I. M. YAGLOM
Isaak Moiseevich Yaglom was born in
Ukraine but raised in Moscow. His "can
didate's" and doctoral theses were on
extensions of some very classical geo
metric ideas. Throughout his life he con
tributed to mathematics of this sort and
championed it. Twice subjected to grossly
unfair dismissals from university posts
(1 949 and 1 968), he never lost heart, and
remained a singularly humorous and gen
erous human being. Among his many
books and articles, some of the most ad
mired and widely read are historical es
says and expository texts. He died unex
pectedly in 1 988 of complications following
an uncomplicated operation. Had he
lived, he might feel today that his strug
gle to rehabilitate classical geometry was
emerging victorious.
computation of derivatives (Russian), Dokl.
Akad. Nauk SSSR 194 (1 970), 767-769.
English translation in: Reports of the Acad
emy of Sciences of the USSR 1 94, New
York, 1 970.
1 7 . O.N. Golovin, G.M. Zislin, AS. Kronrod,
E.M. Landis, L.A. Ljusternik, and G .E. Silov,
Aleksandr Grigor'evic Sigalov. Obituary
(Russian), Uspehi Mat. Nauk 25 (1 970), no.
5 (1 55), 227-234.
1 8. AS. Kronrod, The selection of the minimal
confidence region (Russian), Dokl. Akad.
Nauk SSSR 20 (1 972), 1 036.
19 . AS. Kronrod, A nonmajorizable prescrip
tion for the choice of a confidence region
for a given level of reliability (Russian), Dokl.
Akad. Nauk SSSR 208 (1 973), 1 026.
20. AS. Kronrod, A nonmajorizable prescription
for the selection of a confidence region of a
certain form of target function (Russian),
Dokl. Akad. Nauk SSSR 210 (1 973), 1 8-1 9.
MARIA PIRES DE CARVALHO
Chaotic Newton 's Sequences
s a route to ever more exact knowledge, successive approximation has been
a major theme in the development of science. Many algorithms to find ap-
proximations of roots of equations were devised. In all such reasonings we
begin with an idea of where the root lies, albeit less than accurate, and we have
a strategy to improve the estimates. To look up "whale" in a dictionary, the first step is to open the dictionary close to the end, because you have a rough idea where the word is; next, you tum the pages backward or forward till you fmd it, and this is the strategy to improve the first approximation. In the search for zeros of functions, you need to know that a zero exists and how the map behaves in the neighborhood of that zero.
Newton formulated a general and simple method to fmd approximations of zeros of functions. For a real (or complex) function f with a zero at {, and an initial choice x0, Newton suggested the following recurrence formula to obtain better approximations of {:
_ f(xn) Xn+ l - Xn -f'(Xn) ,
which is defined if the derivative off vanishes at no Xn, and which, if convergent, will surely pick up a zero off as its limit. Given x0, the term x1 is obtained by considering the tangent line at (x0, j(x0)) to the graph ofj and intersecting it with the real axis; to get the whole sequence, just iterate this process. Sufficient conditions for the method to work are easy to state, but a major problem arises: the competition among the several zeros of the function. As a consequence, the basin of attraction of each zero (that is, the set of initial conditions x0 such that the corresponding sequence (xn)n E No converges to the specified zero) may have a very complicated boundary, and the dynamics associated to the sequences (Xn)n E No may be highly sensitive to perturbations on initial conditions. These bound-
aries have been a favorite showpiece in popularizing fractals (see for instance [DS]).
But here I will focus on another problem. What happens
if a map f: fR � fR has no real zeros? Newton's sequences (xn)n E No may be defmed, although they will never converge. How do these sequences behave? I will examine here the particular case of the quadratic family x E fR � fc(x) = :i2 + c, where c is a real positive parameter. The natural extension to C of each map of the family has the real line fR X { 0 I as the boundary of the basins of attraction of its two (complex) roots, so its geometry is trivial. However, the sequences (xn)n E No show irregular and unpredictable behavior, which nevertheless has an underlying order that I will describe.
After a clever change of variable, analysis of the sequences (xn)n E No will be straightforward by appealing to some easy techniques and results from dynamical systems and elementary number theory. The main result is that rational initial conditions produce finite or infinite periodic sequences, whereas the irrational ones yield infinite but not periodic sequences. This recalls what happens with decimal or binary expansions (luckily, even the terminology is the same), and the sensitivity with respect to the initial choice x0 is evinced at once. Moreover, the dynamics associated with these sequences is modeled by a left shift on the binary representation of x0 in the new variable.
Let me start by taking a brief tour of discrete dynamical systems. Given a map G : X � X, I may compose G with itself as many times as I please (the n-fold composition of G with itself is denoted by Gn). Therefore for each x in X the sequence (Gn(x))n E No is well defined; it is called the or-
© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 1 , 2002 31
bit of x by G. The set of all orbits is a dynamical system. Dynamical systems form a category in which an isomorphism between two dynamical systems f: X � X and g : Y � Y is given by a homeomorphism h : X � Y such that g o h = h of; such an h is called a conjugacy between! and g. Essentially, the aim of the theory is to know, up to conjugacy, the asymptotic behaviors of each orbit and how they vary with x. The fiXed points are the orbits easier to detect and the ones to look for first; more generally, an orbit is periodic with period p E N if it is a fiXed point of GP; if
aN(b)N-l + · · · + a1b + ao + c (i) + . . . + ck (it + . . . . . .
It will be found useful to discard the integer part and keep information only about the digits ck. Rational numbers have finite or infinite periodic representations in any base, in general not unique; irrationals appear as unique non-periodic infinite representations. To simplify the notation, a periodic sequence of I, say (ai. a2, · · · , ap, a1, a2, · · · ,
nothing is said to the contrary, p is understood to be the smallest period. An orbit is pre-periodic with pre-period n E N0 and period p E N if Gn(x) is a fiXed point for GP.
Rational numbers have ap, · · · · · · ), will be denoted by a1a2 · · · ap, and similarly a pre-periodic binary representation fin ite or infin ite period ic
representations in any base. a n + 2 · · · a n + p a n + l
For maps G defined on subsets of �' the composition of G with itself may be pictured on the graph of G, and this is a good way of guessing how the orbits behave. For instance, consider G : [0, 1] � [0, 1 ] given by G(x) = 1 - x. Then G(x) = x if and only if x = t, for this is the only intersection of the graphs of G and the identity map. If x =I= t, then G2(x) = G(1 - x) = x, so the orbit of x is periodic with period 2: I suggest you check this on the graph of G.
The orbits may present many differences with respect to their topological properties, asymptotic behavior, or cardinality of their range of values. There are dynamical systems that contain essentially all the kinds of orbits that noninjective maps may be expected to have. One such system is based on the space of sequences constructed with the
digits 0 and 1, say I = {0, 1jf'' = { (ab a2, · · · , am · · · ) : aj E {0, 1 } }, with the metric
oo laj - bjl D(z, w) = i� 2j ,
for z = (al, a2, · · · , an, · · · ) and w = (b1, b2, · · · , bn, · · · ). Acting on I, the one-sided full shift map u takes each sequence (a1 , a2, · · · , an, · · · ) to (a2, · · · , an, · · · ). This map is continuous with respect to the above metric; it has periodic points of all periods, because, for each p E N,
uP(a1, a2, · · · , ap, ab a2, · · · , ap, · · · · · · ) = (a1, a2, · · · , ap, a1, a2, · · · , ap, · · · · · · );
and it has dense orbits (e.g., that of the element of I that is obtained by writing down consecutively all possible finite blocks of digits 0 or 1 ordered by their length-see [D] for more details). I will consider each element of I as a binary expansion of a number in [0, 1 ] ; in this process, the finite binary representation (of each dyadic rational) is thought of as having an infinite tail of zeros: thus, 0.01c2) is the element of I given by 01000000 · · · , and is distinct, in I, from 00111111 · · · , although they are expansions of the same number.
In expansion of the real numbers in a given base b, each number is replaced by a sequence aN · · · a0 · c1c2c3 · · · ck • • · · • · (b) with aj, ck in {0, 1, · · · , b - 1} , meaning that the number is given by the sum
32 THE MATHEMATICAL INTELLIGENCER
an+2 . . . an+p . . . . . . will be abbreviated to O.a1a2 · · ·an an+ l an+2 · · · an+p·
When a rational number is written in irreducible form, information on its expansion in a given base can be read from the denominator only. In the case b = 2 it is known that (see [RT]):
(I ) A rational ro E ]0, 1[ has finite binary representation if and only if it is dyadic; that is, it may be written as r0 = k/2n where k, n E N and k is odd.
In this case the (finite) representation of r0 has precisely n digits.
(II ) A rational r0 E ]0, 1 [ has infinite binary representation with a period that starts just after the decimal point if and only if it is an irreducible fraction tJq where q is odd.
Furthermore, the length of the period does not exceed ¢(q), where ¢ is the Euler totient function (for each q E N, ¢(q) is the number of positive integers less than q and co-prime to q); in fact, it divides ¢(denominator) and is independent of the numerator. (For instance, 115 = 0.0011c2) has period 4 = ¢(5) and 1/13 = 0.00 0100111011c2) has period 12 = ¢(13).)
(III ) The denominator is even but not a power of2-that is, r0 = t12nQ, an irreducible fraction where Q is odd and n is a positive integer-if and only if the binary representation is infinite pre-periodic with a pre-period n.
For example, 1/(2 · 5) = 0.0001lc2) has period 4 as 1/5 and pre-period 1.
Cases (II) and (III) merit closer inspection:
(N) If an irreducible fraction of positive integers tJq E ]0, 1 [ has an odd denominator, it may be expressed in the form s/(2P - 1) where s and p are positive integers and are minimal. Once this is achieved, p gives the length of the period of its binary representation.
For example,
1 3 - 1 5 X 63
5 = 24 _ 1
= O.OOllc2); 1:3 = 212 _ 1
= o.0001001110llc2}
(V) If the fraction tlq has an even denominator which is not a power of 2-that is, tlq = tl2nQ with n E N and Q odd-it may be expressed in the form sl2n(2P - 1) where n, s, and p are positive integers, minimal, and p is greater than 1. The integer p is the length of the period of the binary representation of t!q, and n is the pre-period.
For example 1/12 = 1/(22 (22 - 1)) = 0.0001c2)· Let me sketch a proof of these two properties. (V) im
plies (IV) if n is also allowed to be zero; to prove (V), consider the fraction 1/Q and the equations that produce its binary expansion:
1 = Q X 0 + 1
2 X 1 = Q X d1 + r1 2 X r1 = Q X d2 + r2
0 < r1 < Q 0 < r2 < Q
As the remainders r1 are positive integers less than and coprime to Q, they repeat themselves after cjJ(Q) steps, at the most. The first remainder to reappear is precisely 1 because, by (II), the binary representation of 1/Q has a period that starts just after the decimal point. Therefore there exists a positive index p such that rp = 1 , and so the last of the above equations, before they start repeating, is 2 X rp-1 = Q X dp + rp = Q X dp +
1 . Multiply the second equation by 21'-1, the third one by 21'-2 and so on, and add them all to get
21' = Q [21'-1 d1 + 21'-2 d2 + . . . + 2dp- 1 + dp] + 1 .
Therefore
1 [21'- 1 d1 + 21'-2 d2 + . . . + 2dp- 1 + dp]
Q 21' - 1
so
t At - = ---Q 21' - 1 '
At s
A 21' - 1 '
Further, the type of the binary representation of sf(� (2P - 1)) is the same as that of 1/(2n (21' - 1)), and the latter may be obtained from the following calculation:
1
2n (21' - 1)
1 1/21'
2n 1 - 1121'
= ----;;:;: I - = o.o . . . ooo . . . mc2), 1 � ( 1 )j
2 j� 1 21'
where the first block of zeros has size n and the repeating block has p - 1 zeros followed by a single 1. The integer s may change the digits but not the meaning of n and p. Notice that if the denominator is even but not a power of 2, then p must be bigger than or equal to 2. The effect of the power 2n in the denominator is to push the period to the right, creating a pre-period of length n. I suggest you check this on some examples, such as
1
14
1 9 9
2(23 _ 1) = o.oo01c2); 14 =
2(23 _ 1) = 0. 1010c2);
1 1
28 =
22 (23 _
1) = 0.00001 c2)·
Let me summarize for later use:
r0 tE iQ => r0 has a unique representation, infinite, nonperiodic
3k, n E N : r0 = k/2n => has a finite binary representation that terminates at 0 after n
r0 E iQ => digits 3k, n E N 3p E No : r0 = k/(21'(2n - 1)) =>
unique binary representation with pre-period p and period n
It is time to go back to Newton's method and the map f1. If I start with an initial condition x0 E �. then the corresponding Newton's sequence (xn)nENo' if well defined, is real and thus cannot converge: if it did, the recurrence formula Xn+ 1 = (x� - 1)/2xn would imply that the limit L E � verifies the impossible equation 2L2 = L2 - 1. The dynamical system associated with this recurrence formula may be described by the iterates of the map C§ : � - �. C§(t i= 0) = (t2 - 1)/2t, C§(O) = 0. If well defined, the sequence (xn)n E No is the orbit by C§ of xo; however, once an orbit of C§ lands on the fixed point 0, it stops being a Newton's sequence. The map C§ is an odd function, increasing in ] - oo, 0[ and in ]0, + oo[, and is asymptotic to the line y = x/2. It is easy to identify some orbits by observing the graph of C§:
( 1) Consider x0 = 1; then C§(x0) = 0, so C§n(x0) = 0 for n ::::: 1; Xn is not defined for n ::::: 2. I describe this by saying that the orbit of 1 is finite and terminates at 0 after one iterate.
(2) If x0 = 1 + V2, then C§(x0) = 1 and C§2(x0) = 0, so C§n(x0) = 0 for n ::::: 2 although Xn is not defmed for n ::::: 3. This orbit is also finite and terminates at 0 after two iterates.
(3) Take now x0 = 1/\13; then C§(x0) = - l/V3 and C§2(x0) = 1/V3. This is a periodic orbit of period two. The equality C§2(x) = x leads to a polynomial equation of degree 4 with only even exponents; it has no solutions other than llv'3 and - 11V3.
(4) If Xo = V3, then C§(x0) = llv'3 and C§2(C§(xo)) = C§(xo). So x0 is a pre-periodic orbit of period two and pre-period one.
More sophisticated tools are needed to detect other kinds of orbit. The recurrence formula Xn+ 1 = ((xn? - 1)/ (2xn) is similar to the trigonometric formula cotan(28) = (cotan2(0) - 1)/(2 cotan(O)) for 8 E ]0, 1r[ I { 1r/2}. Let x0 =
cotan(11r0) for r0 E ]0, 1 [: this is permissible since cotan: ]0, 1T [ - � is a homeomorphism, and so the topological properties of the orbits of C§ are preserved under this change of variable. Moreover, in this notation, we have C§n (x0) = cotan( 1T2nr0) for each n, provided that 2n11ro is not an integer multiple of 1T. The numbers in ]0, 1 [ that fail
VOLUME 24. NUMBER 1 , 2002 33
to satisfy this requirement for some integer n are just the dyadic rationals; more precisely:
1st Conclusion: r0 = k/2n, with k, n E N and k an odd integer, if and only if the orbit by <§ terminates at 0 after n iterates.
Because k is odd, we have Xn-I = cotan( 'lT2n- I r0) =
cotan( ?Tk/2) = 0 and therefore Xm is not defined for m 2: n; so the orbit of x0 = cotan( ?Tro) by <§ terminates at the fixed point 0 after n iterates. This is the case of ro = 114 =
O.Olc2J, Xo = cotan(?Tro) = cotan(?T/4) = 1 and xi = 0. Conversely, if an orbit of<§ terminates at 0, say <§n(x0) = 0, then cotan( ?T2nr0) = 0 and therefore there exists m E 7L such that 2n?Tr0 = m'lT + 'lT/2. So 2nr0 = m + 112, that is, ro = (2m+ 1)12n+ I.
What real numbers r0 produce periodic or pre-periodic orbits by <§? r0 cannot be dyadic, and there must be N and P such that Cfii+P (x0) = Cf/1 (x0); this implies that r0 =I= k/2n for all integers k and n and cotan( ?T2N+Pr0) = cotan( 'lT2Nr0). Solving this equation, it is found that r0 = ki2N (2P - 1) with k E N, N E N0, P E N and P 2: 2. These are the remaining rationals of ]0, 1 [ (see (IV) and (V) above): they have infinite periodic or pre-periodic binary expansions with period P.
2nd Conclusion: The orbit of x0 by <§ is finite or infinite periodic/pre-periodic if and only if r0 is rational; if such is the case, then the orbit type of x0 is completely determined by the denominator of r0. In particular, if ro is irrational, then Xn is defined for all n E N.
Let me review in this new setting some of the above examples.
(a) ro = 113 = 11(2-2 - 1) = O.Olc2J: then N = 0, P = 2, xo = cotan( ?T/3) = 11v'3, and xi = cotan(27r/3) = - llv'3. The orbit by <§ of Xo is periodic with period P.
(b) r0 = 1/6 = 112(22 - 1) = 0.001c2J: N = 1, P = 2, and x0 = cotan( ?TI6) = V3, xi = cotan(27r/6) = cotan( ?T/3) =
llv'3. The orbit of x0 is pre-periodic with pre-period N = 1 and period P = 2.
(c) ro = 115 = 31(2-4 - 1) = O.OOllc2J: N = 0, P = 4, and xo =
cotan( ?T/5), XI = cotan(27r/5), x2 = cotan( 47r/5), x3 =
cotan(87r/5), x4 = cotan(16?T/5) = x0. The orbit of x0 is periodic with period P = 4.
I suggest you now compare the following diagram with the similar one above.
I r0 $. OJ ==> its orbit by <§ is infinite non-periodic l ::lk, n E N : r0 = k/2n ==> its orbit by <§
ro E Q ==> terminates at 0 after n iterations ::lk, n E N 3p E N0 : r0 = k/2P(2n - 1) ==> its orbit by <§ has pre-period p and period n
Thus the orbit of xo by <§ is completely determined by the binary representation of r0. This also shows that the discrete dynamical system generated by <§ is highly sensitive to initial conditions: the distinction between rational and irrational r0 is enough to produce wide disparities between orbits.
34 THE MATHEMATICAL INTELLIGENCER
Other more particular traits of the orbits for irrational values of r0 can be studied by picking up two clues I left behind:
(1) the function z � cotan( 'lTZ) is periodic of period 1;
(2) iterating x0 by <§ corresponds, in the new variable, to simply doubling the argument of the cotan function.
The first one implies that, when you compute the successive values of cotan( 'lT2nr0), what matters is the fractional part of 2nro (denoted by {2nr0}). If the irrational ro is written in base 2 as ro = 0 . aia2a3 · · · ak · · · (2), this representation is unique, and 2ro = a I . a2a3 · · · ak · · · (2)· Dismissing the integer part, we are left with {2r0} = O.a2a3 · · · ak · · · (2) and, by (2),
( cotan( 'lT2nro))n E No = ( cotan({ 'lT2nro}))n E N0 = ( COtan( 'lT . 0 . an+ I an+2 . . . (2J))n E N0,
which corresponds, up to the action of cotan o ( 'lT X · ), exactly to the iteration an of the shift on the sequence aia2a3 · · · ak · · · . More precisely, the map
]0, 1 [ I {dyadic numbers} � ]0, 1 [ I {dyadic numbers} 0 . aia2 · · · ak · · · (2) � 0 . a2a3 ·. ak · · · · · · C2J
(that is, '!f(t) = 2t if 0 :=::: t < �, '!f(t) = 2t - 1 if� :=::: t < 1) is conjugated by z � co tan( 'lTZ) to the action of <§ on the set of x0 whose orbits by <§ do not terminate at the fixed point 0 after a finite number of iterates; and '!f is the same as the shift map a restricted to the sequences of zeros or ones that are not eventually constant, for the map
h(O . aia2 · · · ak · · · C2J) = aia2a3 · · · ak · · · · · ·
is a conjugacy between the chosen restrictions of '!f and a. Let me illustrate the use of these observations in two
examples:
(i) If r0 = 0.10100100010000 · · · · · · c2J, where each digit 1 is followed by a block of zeros of increasing length, then r0 is irrational and the sequence (xn)nEN = ( COtan( 'lT 2nro))nEN = ( COtan({ 'lT2nro}))n E N iS bounded away from zero, because {2nr0} <0.1010010010 · · · c2J =
-i4 for all n. But, since {2nr0} gets arbitrarily close to 0, this orbit is not bounded from above.
( ii) If r0 is an irrational number whose binary representation is given by a sequence in � with dense a-orbit, then the corresponding sequence (xn)n E No is dense in IR.
If for each dyadic number of ]0, 1 [ I select the binary representation with ending zeros (e.g., writing 112 =
0. 10000 · · · c2J instead of 0.0111 · · · c2J), then the corresponding extension of h is not continuous. However, if I let 'JC(x) = h((li?T)cotan-I x), then the equation a o 'JC(x) = 'JC o <&(x) is still valid for all x =I= 0. This yields the following:
3rd Conclusion: The dynamics of the Newton's sequences (xn)n E N0, for allowed real initial conditions Xo, is determined by the binary representations of the initial conditions in the new variable r0.
I now proceed to check how the parameter c affects the previous calculations. I will show that the dynamics of the corresponding Newton's sequences for parameter c is the same as for c = 1 when c > 0, and changes drastically at c = 0.
Let me rewrite c as ±a2, with a E [0, +oo[. Denote by C£1� the map associated to Newton's method applied to fc, where ± = sign(c): thus C&HO) = 0, C&�(x) = (x2 - a2)f2.x, C&;;(x) = (x2 + a2)!2x. For a fixed sign ± , the family of maps (C&�)a E JO, +oc[ converges pointwise, but not uniformly, to C£10(x) = x/2 as a � 0. The limiting dynamics is uninteresting: for all Xo E IR, the sequence ((C£10)n(xo))nEN has limit 0, the unique fixed-point of C£10.
If a > 0, then for x of=. 0 we have
that is,
This suggests the change of variable
which leads to
and, in general, to
t _ xo o - -, a
t1 =
x1 =
(to)2 - 1
a 2t0
This means that, up to a change of variable, the map C&;i acts as C£1 = C&t, and no further work is needed in this case.
If a > 0 and c = -a2, then fc has two real zeros, a and -a, with basins of attraction given by ]0, +oo[ and ] -oo, 0[, respectively. In fact, the minimum value of C&;;(x) = x2 + a2!2x for x > 0 is a, which is also the unique fixed point of C£1;; in ]0, +oo[; and, since C&;;lla, +ool is a contraction, it follows that, for all initial choices x0 > 0, the sequence (xn)n converges to a. Similar reasoning shows that (xn)n converges to -a for all choices Xo < 0. It is along the imaginary axis that the dynamics of C&c; is chaotic: for, if x0 =
iPo for some Po E IR I {0}, then Newton's recurrence formula Xn+ l = (xn2 + a2)12 Xn becomes
. - ( i Pn)z + a2 - . (Pn)Z - a2 �Pn+ l -
2 . - 1, 2 . �Pn Pn
This means that, in the real variable p, the dynamics is given by Pn+ 1 = C£1� (pn), which has already been analyzed.
It is worth remarking that the conclusions obtained for
A U T H O R
MARIA PIRES DE CARVALHO
Centro de Matematica do Porto
Prac;:a Gomes Teixeira
4099·002 Porto Portugal
e·mail: mpca!Val@fc,up.pt
Maria Carvalho and her twin sister were born in Africa. She
completed her first degree in mathematics at the University of Porto, where she is now an associate professor. Her post
graduate studies were completed at lnstituto de Matematica
Pura e Aplicada, in Rio de Janeiro, where she specialized in
Ergodic Theory and completed her Ph.D. under the guidance
of Ricardo Mane, Maria shares a cat with her husband and is
enthusiastic about l iterature and jazz music,
the quadratic family Cfc)c extend easily to all quadratic polynomials. Given a polynomial p(x) = d2x2 + d1x + d0, with di E IR and d2 =I= 0, the equation p(x) = 0 is equivalent to p(x)ld2 = 0, and so I may assume that d2 = 1. By a simple translation in the variable x, given by x = t + d/2, p becomes
p(t) = t2 + [do - di/4] ,
which belongs to the family Cfc)c. Hence all the previous results hold for this larger family.
Acknowledgments
My thanks to Paulo AraUjo for his help in improving the text.
REFERENCES
[D] Devaney, Robert L, An Introduction to Chaotic Dynamical Systems,
1 989, Addison Wesley,
[DS] Devaney, Robert L,, Keen, Linda (Editors), Chaos and Fractals:
The Mathematics Behind the Computer Graphics, Proceedings of
Symposia in Applied Mathematics, Vol 39 (1 989), American Mathe
matical Society.
[P] P61ya, George. Mathematical Methods in Science, 1 977, The Math
ematical Association of America
[RT] Rademacher, Hans, and Toeplitz, Otto (H. Zuckerman, translator).
The Enjoyment of Math, 1 970, Princeton University Press.
VOLUME 24. NUMBER 1. 2002 35
M a t h e n1 a t i c a l l y B e n t
The proof is i n the pudding.
Opening a copy of The Mathematical
Intelligencer you may ask yourself
uneasily, "What is this anyway-a
mathematical journal, or what?" Or
you may ask, "Where am /?" Or even
"Who am !?" This sense of disorienta
tion is at its most acute when you
open to Colin Adams's column.
Relax. Breathe regularly. It's
mathematical, it's a humor column,
and it may even be harmless.
Column editor's address: Colin Adams,
Department of Mathematics, Bronfman
Science Center, Williams College,
Williamstown, MA 01 267 USA
e-mail: Colin.C.Adams@williams.edu
Colin Ada m s , Editor
F ie lds Medal ist Stripped Colin Adams
March 3, 2005: The International Congress of Mathematics an
nounced today that Wendell Holcomb will be stripped of his Fields Medal after testing positive for intelligence-enhancing drugs. Holcomb has denied the charges. "Just because I never finished high school, and then solved the threedimensional Poincare Conjecture, doesn't mean I took drugs."
When asked how he even knew about the problem, he said, "Nobody told me about it. I just got to thinking. There is a sphere that sits in 3-space, so there must be an analog one dimension up, which I called the 3-sphere. But could a different 3-dimensional space resemble this one in the sense that loops shrink to points, it has no boundary, and it's compact? Or is the 3-sphere the only 3-dimensional object that has those properties? Seemed like a reasonable question at the time."
Unaware that the conjecture was originally made by Henri Poincare 100
years ago, Holcomb quickly proved it was true, scooping generations of mathematicians. He received the Fields Medal in mathematics for his efforts.
Residual amounts of Mentalicid were found in urine samples taken at Princeton University, where Holcomb is now the Andrew Wiles Professor of Mathematics.
"I never gave them urine samples," protested Holcomb.
Sargeant Karen Lagunda of the Princeton Police Department explained.
36 THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
"We have been testing the waste water coming out of the academic buildings for three years now, with the tacit cooperation of the administration. But Holcomb had been hoofing it over to the Seven Eleven and using the facilities there to avoid detection. Ultimately he had one too many slushies and he couldn't wait 'til he got off campus."
"This would explain why he couldn't multiply two fractions on some days, and on others, he would solve conjectures that had been open for fifty years," said the department chair.
The revelations have thrown the mathematical world into chaos. Caffeine has long been used to enhance intellectual alertness. It is acknowledged that without coffee, mathematical productivity would have been half of what it was. But the new class of beta-enhancers that stimulate the transfer of impulses across neurons are in another class altogether.
"These drugs do turn you into a brainiac, no doubt about it," said Carolyn Mischner of the Harvard Medical School, "but they also have a variety of side effects, including seeing double, causing people to drive on the left side of the road, and the eventual degradation of the intellect when the drug is not in use. This causes users to stay on the drug for longer and longer periods. Eventually, the intellect is so diminished that the drug brings one back up to a functional level only, and then not even that."
Holcomb plans to appeal the decision. "This is so unfair. Have you seen my Hula-Hoop? I think my pants are on backward."
The committees for the Nobel prizes in Economics and Medicine have not yet decided whether to strip Holcomb of his prizes in those fields.
JUAN L. VARONA
G raph ic and N u merical Com parison Between Iterat ive Methods
Dedicated to the memory of Jose J. Guadalupe ("Chicho''), my Ph.D. Advisor
let f be a function f : lffi � lffi and ? a root of J, that is, f(?) = 0. It is well known that if we
take x0 close to ?, and under certain conditions that I will not explain here, the Newton
method
f(xn) Xn+l = Xn - ----;-----
( )' n = 0, 1, 2, . . .
f Xn
generates a sequence {xnJ:=o that converges to ?. In fact, Newton's original ideas on the subject, around 1669, were considerably more complicated. A systematic study and a simplified version of the method are due to Raphson in 1690, so this iteration scheme is also known as the Newton-Raphson method. (It has also been described as the tangent method, from its geometric interpretation.)
In 1879, Cayley tried to use the method to find complex roots of complex functions!: C � C. If we take z0 E C and we iterate
fCzn) Zn+l = Zn - ----;-----
( )' n = 0, 1, 2, . . . , ( 1)
f Zn
he looked for conditions under which the sequence {zn}�=o converges to a root. In particular, if we denominate the attraction basin of a root ? as the set of all z0 E C such that the method converges to ?, he was interested in identifying the attraction basin for any root. He solved the problem whenf is a quadratic polynomial. For cubic polynomials, after several years of trying, he finally declined to continue. We now know the fractal nature of the problem
and we can understand that Cayley's failure to make any real progress at that time was inevitable. For instance, for f(z) = z3 - 1 , the Julia set-the set of points where Newton's method fails to converge-has fractional dimension, and it coincides with the frontier of the attraction basins of the three complex roots e2k7Ti13, k = 0, 1, 2. With the aid of computer-generated graphics, we can show the complexity of these intricate regions. In Figure 1 , for example, I show the attraction basins of the three roots (actually, this picture is well known; for instance, it already appears published in [5] and, later, [ 16] and [21]).
There are two motives for studying convergence of iterative methods: (a) to find roots of nonlinear equations, and to know the accuracy and stability of the numerical algorithms; (b) to show the beauty of the graphics that can be generated with the aid of computers. The first point of view is numerical analysis. General books on this subject are [9, 13]; more specialized books on iterative methods are [3, 15, 18]. For the esthetic graphical point of view, see, for instance, [ 16].
Generally, there are three strategies to obtain graphics from Newton's method:
(i) We take a rectangle D c C and we assign a color (or a gray level) to each point z0 E D according to the root at which Newton's method starting from z0 converges;
© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 1, 2002 37
Figure 1 . Newton's method. Figure 2. Newton's method for multiple roots. Figure 3. Convex acceleration of Whittaker's
method.
and we mark the point as black (for instance) if the
method does not converge. In this way, we distinguish
the attraction basins by their colors.
(ii) Instead of assigning the color according to the root
reached by the method, we assign the color according
to the number of iterations required to reach some
root with a fixed precision. Again, black is used if the
method does not converge. This does not single out
the Julia sets, but it does generate nice pictures.
(iii) This is a combination of the two previous strategies.
Here, we assign a color to each attraction basin of a
root. But we make the color lighter or darker accord
ing to the number of iterations needed to reach the
root with the fixed precision required. As before, we
use black if the method does not converge. In my opin
ion, this generates the most beautiful pictures.
All these strategies have been extensively used for poly
nomials, mainly for polynomials of the form zn - 1 whose
roots are well known. Of course, many other families of
functions have been studied. See [4, § 6] for further refer
ences. For instance, a nice picture appears when we apply
the method to the polynomial (z2 - 1)(z2 + 0. 16) (due to
S. Sutherland, see the cover illustration of [ 17]).
2
- 2
- 2 - 1
2
Although Newton's method is the best known, in the lit
erature there are many other iterative methods devoted to
fmding roots of nonlinear equations. Thus, my aim in this
article is to study some of these iterative methods for solv
ing j(z) = 0, where f: IC � IC, and to show the fractal pic
tures that they generate (mainly, in the sense described in
(iii)). Not to neglect numerical analysis, I Will compare the
regions of convergence of the methods and their speeds.
Concepts Related to the Speed of Convergence
Let {znl�=O be a complex sequence. We say that a E [1 , oo) is the order of convergence of the sequence if
. lzn+l - � hm I ria = C, n�co Zn - � (2)
where � is a complex number and C a nonzero constant;
here, if a = 1 , we assume an extra condition lei < 1 . Then,
the convergence of order a implies that the sequence
{zn}�=O converges to � when n � oo. (The definition of the
order of convergence can be extended under some cir
cumstances; but I will not worry about that.) Also, it is said
that the order of convergence is at least a if the constant
C in (2) is allowed to be 0, or, the equivalent, if there ex
ists a constant C and an index n0 such that lzn+ 1 - � ::::::
Figure 4. Double convex acceleration of Figure 5. Halley's method. Figure 6. Chebyshev's method.
Whittaker's method.
38 THE MATHEMATICAL INTELUGENCER
Figure 7. Convex acceleration of Newton's Figure 8. (Shifted) Stirling's method. Figure 9. Steffensen's method.
method (or super-Halley's method).
Clzn - � �a for any n 2: n0. Many times, the "at least" is left tacit. I will do so in this article.
The order of convergence is used to compare the speed of convergence of sequences, understanding the speed as the number of iterations necessary to reach the limit with a required precision. Suppose that we have two sequences {znl�=o and {z;,)�=o converging to the same limit �. and as
where a is the order of convergence. For the methods that I am dealing with here, it is easy to derive both the informational efficiency and the efficiency index from the order. I will do this here for the efficiency index.
The efficiency index is useful because it allows us to avoid artificial accelerations of an iterative method. For in-stance, let us suppose that we have an iterative process
sume that they have, re-spectively, orders of convergence a and a' , where a > a' . Then, it is clear that, asymptotically, the sequence {znl�=o converges to its limit more
The order of convergence is
used to compare the speed of
convergence of sequences.
Zn+ 1 = <fJ(zn) with order of convergence a and we take a new process zo =
zo, z�+1 = <P(<P(z�)). Then it is clear that the new sequence is merely z� = Z2n, but {z�)�=o has order of
quickly (with fewer iterations for the same approximation) than the other sequence.
More refined measures for the speed of convergence are the concepts of informational efficiency and efficiency index (see [ 18, § 1.24 ]). If each iteration requires d new pieces of information (a "piece of information" typically is any evaluation of a function or one of its derivatives), then the informational efficiency is � and the efficiency index is a11d,
convergence a2. However, both sequences {znl�=o and {z;�l�=o have the same efficiency index.
In my opinion, when we have an iterative method Zn+ 1 =
<PCzn), the efficiency index is more suitable than the order of convergence to measure the computer time that a method uses to converge. But, as happens in our case, if <P involves a function f and its derivatives, the efficiency index still has a missing element: it does not take into ac-
2
Figure 10. Midpoint method. Figure 1 1 . Traub-Ostrowski's method and
Jarratt's method.
Figure 12. Inverse-free Jarratt's method.
VOLUME 24, NUMBER 1 . 2002 39
count the computational work involved in computingf, f' , . . . . To avoid this, a new concept of efficiency is given: the computational efficiency (see [ 18, Appendix C]). Suppose that, in a method cfJ related with a function f, the cost of evaluating cfJ is e(f) (for instance, in Newton's method, if the cost of evaluating f and f' are respectively eo and e1, we have e(f) = e0 + e1); then, the computational efficiency of cfJ relative tof is E(cfJ,f) = a11!J(f) where, again, a is the order of convergence. But it is difficult to establish the value of e(f); moreover, it can depend on the computer, so the computational efficiency is not very much used in practice. In the literature, the most used of these measures is the order of convergence; however, this is the one that provides least information about the computer time necessary to fmd the root with a required precision.
Finally, note that, to ensure the convergence of an iterative method Zn+ 1 = cfJ(zn) intended for solving an equation f(z) = 0, it is usually necessary to begin the method from a point z0 close to the solution (. How close depends on cfJ and f Usually the hypotheses of the theorems that guarantee the convergence (I will give references for each method) are hard to check; and, moreover, are too demanding. So, if we want to solvef(z) = 0, it is common to try a method without taking into account any hypothesis. Of course, this does not guarantee convergence, but it is possible that we will find a solution (if there is more than one solution, we also cannot know which solution is going to be found).
Here, I will do some numerical experiments with different functions (simple and hard to evaluate) that allow comparisons of the computational time used. In addition, I will begin the iterations in different regions of the complex plane. This will allow us to measure to some extent how demanding the method is regarding the starting point to fmd a solution. As the fractal that appears becomes more complicated, it seems that the method requires more conditions on the initial point.
The Numerical Methods
In this section, let us consider some iterative methods Zn+ 1 = cfJ(zn) for solving j(z) = 0 for a complex function f : IC --> IC. I only give a brief description and a few references. In all these methods, we take a starting point z0 E IC. • Newton's method: This is the iterative method (1), the
best known and most used, and can be found in any book on numerical analysis. I have already commented on it in the introduction. Its order of convergence is 2.
• Newton's method for multiple roots:
Actually, Newton's method has order 2 when the root off that is found is a simple root. For a multiple root, its order of convergence is 1. This method recovers the order 2 for multiple roots. It can be deduced as follows: iff has a root of multiplicity m 2:: 1 at (, it is easy to check that g(z) =
40 THE MATHEMATICAL INTELLIGENCER
f,�l has a simple root at { Then, we only need to apply the ordinary Newton's method to the equation g(z) = 0.
• Convex acceleration of Whittaker's method [11 ] :
with
j(z)f"(z) L1(z) =
f'(z)2
Whittaker's method (also known as the parallel-chord method, from its geometric interpretation for functions f: !R1 --. IR1, see [ 15, p. 181]) is a simplification of Newton's method in which, to avoid computing the derivative, we make the approximation f'(z) = 1/A with A a constant. We try to choose the parameter A in such a way that F(z) = z - Af(z) is a contractive function, and so will have a fixed point (it is clear that a fixed point for F is a root for f). This is a method of order 1. The convex acceleration is an order 2 method.
• Double convex acceleration of Whittaker's method [ 11 ] :
This is a new convex acceleration for the previous iterative process. It has order 3.
• Halley's method (see [ 18, p. 91] , [3, p. 247], [9, p. 257], [8]):
f(zn) 2 1 Zn+ 1 = Zn - J' (Zn) 2 - Lj(Zn)
= Zn -f'Czn) j"(zn) f(z.) 2j'(zn)
This was presented in about 1694 by Edmund Halley, who is well known for first computing the orbit of the comet that carries his name. It is one of the most frequently rediscovered iterative functions in the literature. From its geometric interpretation for real functions, it is also known as the method of tangent hyperbolas. Alternatively, it can be interpreted as applying Newton's method to the equation g(z) = 0 with g(z) = f(z)lv7'(Z). Its order of convergence is 3.
• Chebyshev's method (see [ 18, p. 76 and p. 81] or [3, p. 246]):
ftzn) ( Lj(Zn) ) Zn+ 1 = Zn -
f'(zn) 1 + --
2- .
This is also known as Euler-Chebyshev's method or, from i� geometric interpretation for real functions, the method of tangent parabolas. It has order 3. (This method and the previous one are probably the best-known order 3 methods for solving nonlinear equations.)
• Convex acceleration of Newton's method, or the superHalley method [7]:
Zn+ 1 = Zn -1 - LJ(Zn)
1 j(Zn) ( 2LJ(Zn) )
= Zn -f' (zn) 1 + 1 - Lj(Zn) .
This is an order 3 method. (Note that, in [3, p. 248], it is called Halley-Werner's method.)
One group of procedures for solving nonlinear equations are the fixed-point methods, methods for solving F(z) = z. The best-known of these methods is the one that iterates Zn+ 1 = F(zn); it is an order 1 method and needs a strong hypothesis on F to converge; that is, it requires F to be a contractive function.
An order 2 method for solving an equation F(z) = z is Stirling's fixed-point method [3, p. 251 and p. 260]. It starts at a suitable point z0 and iterates
Zn - F(zn) Zn+ 1 = Zn -
1 - F'(F(zn)).
If we want to solve an equationfl:z) = 0, we can transform it into a fixed-point equation. To do this, we can take F(z) = z - f(z). It is then clear that F(z) = z �f(z) = 0, so we can try to use a fixed-point method for F. But this is not the only way: for instance, we can take F(z) = z - Af(z) with A =/=- 0 a constant (one example is Whittaker's method, already mentioned), or F(z) = z - 'P(z)f(z) with 'P a nonvanishing function. Also, we can isolate z in the expression f(z) = 0 in different ways (for instance, if we have z3 - z + tan(z) = 0, we can isolate z3 + tan(z) = z or arctan(z -z3) = z). This gives many different fixed-point equations F(z) = z for the same original equationf(z) = 0.
Furthermore, when we try to solvej(z) = 0 by means of an iterative method Zn+ 1 = ¢(zn), like the ones shown above, and {znl�=O converges to �. it is clear that � is a fixed point for ¢ (upon requiring that ¢ be a continuous function and taking limits in Zn+ 1 = ¢(zn)). So, without noticing, we are dealing with fixed-point methods.
But it is interesting to check what happens if we merely use F(z) = z -j(z) without worrying about any hypothesis. In this way, we have
• (Shifted) Stirling's method:
Zn+ 1 = Zn -j'(Zn - j(Zn))
.
Its order of convergence is 2.
In all the methods that we have seen until now, the function f and its derivatives are evaluated, in each step of the method, for a single point. There are other techniques for solving nonlinear equations that require the evaluation off or its derivatives at more than one point in each step. These iterative methods are known as multipoint methods. They are usually employed to increase the order of convergence without computing more derivatives of the function involved. A general study of multi-
point methods can be found in [ 18, Ch. 8 and 9] . Let us look at some of them.
• Steffensen's method (see [15, p. 198] or [ 18, p. 178]):
f(zn) Zn+ 1 = Zn - -( ) g Zn
"th ( ) j(z + j(z)) -f(z) Th" · f th · l l WI g z = f(z) . IS IS one o e simp est mu -
tipoint methods. The iterative function is generated by a derivative estimation: we insert in Newton's method, for small enough h = f(z), the estimate f'(z) = fCz+htf(z) = g(z). This avoids computing the derivative off This is an order 2 method (observe that it preserves the order of convergence of Newton's method).
• Midpoint method (see [ 18, p. 164] or [3, p. 197]):
fl:zn) Zn+ 1 = Zn - '( j(Zn) ) .
f Zn - 2j'(zn)
This is an order 3 method.
• Traub-Ostrowski's method (see [ 18, p. 184] or [3, p. 230]):
f(zn - u(zn)) - fl:zn) Zn+ 1 = Zn - U(Zn)
2j(Zn - U(Zn)) - j(zn)
with u(z) = J,�;l. Its order of convergence is 4 , the highest for the methods that we are studying.
• Jarratt's method [ 12, 2] (for different expressions, see also [3, p. 230 and p. 234]):
1 f(zn) Zn+ 1 = Zn - -;;_u(Zn) + ----------
j'(Zn) - 3j'(Zn - fu(zn))
where, again, u(z) = J,�;l. This is also an order 4 method.
• Inverse-free Jarratt's method (see [6] or [3, p. 234]):
Zn+ 1 = Zn - u(zn) + iu(zn)h(Zn) (
1 - fh(Zn) ),
. j(z) j'(z - �u(z)) -f'(z) With u(z) = f'(z) and h(z) = f'(z) . Also an or-der 4 method.
Fractal Pictures and Comparative Tables
I will now apply the iterative methods that we have seen in the previous section to obtain the complex roots of the functions ( sin(z) )
f(z) = z3 - 1 and.f'(z) = exp 100 (z3 - 1).
It is clear that the roots off* are the same as the roots of J, that is, 1 , e271i13 and e471i13. But the function f* takes much more computer time to evaluate. Moreover, the successive derivatives off are easier and easier, contrary to the general case. This does not happen with f*. So, f* can be a better test of the speed of these numerical methods in gen-
VOLUME 24, NUMBER 1, 2002 41
Table 1 . Function f and rectangle Rb
Nw
NwM
CaWh
DcaWh
Ha
Ch
CaN/sH
Stir
Steff
Mid
Tr-Os
Ja
lfJa
Ord
2
2
2
3
3
3
3
2
2
3
4
4
4
Eff
1 .41
NC 1/P
0.00267 7.52
T P/S 1/S
1 .26 0.00381 7.93 1 . 1 7 0.857 0.904
1 .41 24.5 1 8.9 3.23 0.309 0.778
1 .44 0. 1 25 6.5 1 .41 0. 71 1 0.615
1 .44 0 4.38 0.901 1 . 1 1 0.646
1 .44 0.0492 6.27 1 . 1 1 0.902 0. 752
1 .44 0 3.82 0.81 5 1 .23 0.623
1 .41 86.6 36.4 4.71 0.212 1 .03
1 .41 85 35.7 5.79 0.1 73 0.820
1 .44 4 .62 6.32 1 . 1 0.91 1 0.766
1 .59 0 3.69 0.696 1 .44 0. 705
1 .59 0 3.69 0.699 1 .43 0. 702
1 .59 1 .62 7.45 1 .41 0.71 1 0.705
eral. (Note that many of these iterative methods are also adapted to solve systems of equations or equations in Banach spaces. Here, to evaluate Frechet derivatives is, usually, very difficult.)
I take a rectangle D c IC and I apply the iterative methods starting in "every" z0 E D. In practice, I will take a grid of 1024 X 1024 points in D as z0. Also, I will use two different regions: the rectangle Rb = [ -2.5, 2.5] X [ -2.5, 2.5] and a small rectangle near the root e27Ti13 ( = -0.5 + 0.866025i), the rectangle R8 = [ -0.6, -0.4] X [0. 75, 0.95]. The first rectangle contains the three roots; the numerical methods starting from a point in Rb can converge to some of the roots, or perhaps diverge. However, R8 is near a root, so it is expected that any numerical method starting there will always converge to the root.
In all these cases, I use a tolerance E = 10-8 and a maximum of 40 iterations. The three roots are denoted by �k = e2k7T'i13, k = 0, 1, 2, and ¢ is the iterative method to be used. Then, I take z0 in the corresponding rectangle and iterate Zn+l = c/J(Zn) up to lzn - �kl < E for k = 0, 1 or 2. If we have not obtained the desired tolerance with 40 iterations, I do not continue, but declare that the iterative method starting at z0 has failed to converge to any root.
Table 2. Function f and rectangle Rs
Nw
NwM
CaWh
DcaWh
Ha
Ch
CaN/sH
Stir
Steff
Mid
Tr-Os
Ja
lfJa
Ord
2
2
2
3
3
3
3
2
2
3
4
4
4
Eff
1 .41
1 .26
1 .41
1 .44
1 .44
1 .44
1 .44
1 .41
1 .41
1 .44
1 .59
1 .59
1 .59
NC
0
0
0
0
0
0
0
0
0
0
0
0
0
42 THE MATHEMATICAL INTELLIGENCER
1/P
2.97
2.97
3.23
2
2
2
2
4 . 1 5
3.44
2
1 .96
1 .96
1 .99
T
1 . 1
1 .39
1 . 1
1 .03
0.914
1 .06
1 .36
1 .42
0.898
0.925
0.928
0.969
P/S
0.91 0
0.71 9
0.91 1
0.974
1 .09
0.946
0.733
0.706
1 . 1 1
1 .08
1 .08
1 .03
1/S
0.910
0.781
0.613
0.656
0.737
0.636
1 .02
0.82
0.749
0.714
0.712
0.690
Table 3. Function f* and rectangle Rb
Nw
NwM
CaWh
DcaWh
Ha
Ch
CaN/sH
Stir
Steff
Mid
Tr-Os
Ja
lfJa
Ord
2
2
2
3
3
3
3
2
2
3
4
4
4
Eff NC 1/P T P/S 1/S
1 .41 3.06 8 . 17
1 .26 2.86 8.2 1 .4 7 0.681 0.683
1 .41 33.2 1 9.9 3.58 0.279 0.679
1 .44 1 8. 1 1 1 1 .88 0.532 0. 71 4
1 .44 0.321 4.48 0.91 8 1 .09 0.597
1 .44 1 1 .5 9 . 1 1 1 .56 0.641 0.7 1 4
1 .44 1 .92 4.59 0.907 1 . 1 0 0.61 9
1 .41 87.7 36.5 4.04 0.248 1 . 1 0
1 .41 84.5 35.6 3.39 0.295 1 .28
1 .44 5.61 6.57 1 .21 0.824 0.662
1 .59 1 . 1 0 4.03 0.677 1 .48 0. 729
1 .59 0.965 3.99 0. 777 1 .29 0.628
1 .59 19 1 1 .2 1 .71 0.584 0.797
With these results, combining f and f* with Rb and R8, I compiled four tables. In them, the methods are identified as follows: Nw (Newton), NwM (Newton for multiple roots), CaWh (convex acceleration of Whittaker), DcaWh (double convex acceleration of Whittaker), Ha (Halley), Ch (Chebyshev), CaN/sH (convex acceleration of Newton or super-Halley), Stir (Stirling), Steff (Steffensen), Mid (midpoint), Tr-Os (Traub-Ostrowski), Ja (Jarratt), IfJa (inversefree Jarratt).
For each of them, I show the following information:
• Ord: Order of convergence. • Eff: Efficiency index. • NC: Nonconvergent points, as a percentage of the total
number of starting points evaluated (which is 10242 for every method).
• VP: Mean of iterations, measured in iterations/point. • T: Used time in seconds relative to Newton's method
(Newton = 1). • PIS: Speed in points/second relative to Newton's method
(Newton = 1). • 1/S: Speed in iterations/second relative to Newton's
method (Newton = 1).
Table 4. Function f* and rectangle R5
Nw
NwM
CaWh
DcaWh
Ha
Ch
CaN/sH
Stir
Steff
Mid
Tr-Os
Ja
lfJa
Ord
2
2
2
3
3
3
3
2
2
3
4
4
4
Eff
1 .41
1 .26
1 .41
1 .44
1 .44
1 .44
1 .44
1 .41
1 .41
1 .44
1 .59
1 .59
1 .59
NC
0
0
0
0
0
0
0
0
0
0
0
0
0
1/P
2.97
2.97
3.22
2
2
2
2
4 . 13
3.43
2
1 .96
1 .96
1 .99
T
1 .50
1 .67
1 . 1 3
1 . 1 0
1 .06
1 . 1 2
1 .38
1 .06
1 .02
0.909
1 .04
1 .05
P/S 1/S
0.666 0.666
0.599 0.649
0.883 0.594
0.906 0.61
0.944 0.636
0.895 0.602
0.724 1 .01
0.945 1 .09
0.979 0.659
1 . 1 0.727
0.959 0.634
0.955 0.639
To construct the tables, I used a C + + program in a Power Macintosh 82001120 computer. In the tables, I show the time and speed relative to Newton's method, so that this will be approximately the same in any other computer. In our computer, the absolute values for Newton's method are the following:
• For Table 1, 137.467 sec, 7627.86 pt/sec and 57336.9 it/sec. • For Table 2, 59.1667 sec, 17722.4 pt/sec and 52610.2 it/sec. • For Table 3, 410.683 sec, 2553.25 pt/sec and 20870.6 it/sec. • For Table 4, 150.083 sec, 6986.63 pt/sec and 20737 it/sec.
In any case, a computer programming language that permits dealing with operations with complex numbers in the same way as for real numbers (such as C + + or Fortran) is highly recommended.
With respect to the time measurements, it is important to note that, for each iterative method Zn+ 1 = c:f>(zn), I have written general procedures applicable to generic f and its derivatives. That means, for instance, that when I usef*, I d t · lify f t
· J*CzJ Al ·f b ·
o no s1mp any ac or m Cf*J'CzJ . so, 1 a su expresswn of (f*) ' has already been computed in f* (say, sin(z)) in the generic procedure to evaluate J, its value is not used, but computed again, in the procedure that calculates generic j1 • If we were interested only in a particular function! (or if we wanted a figure in the fastest way), it would be possible to modify the procedure that iterates Zn+ 1 = c:f>(zn) for J, adapting and simplifying its expression.
Now, let us go back to the other target of this paper: to compare the fractal pictures that appear when we apply different iterative methods for solving the same equation
f(z) = 0, where f is a complex function. Figures 1 to 12 show the pictures that appear when we
apply the iterative methods to fmd the roots of the functionj{z) = z3 - 1 in the rectangle Rb· I have used strategy (iii) described in the introduction. Respectively, I assign cyan, magenta, and yellow for the attraction basins of the three roots 1, e27Ti13, and e47T'i13, lighter or darker according to the number of iterations needed to reach the root with the fixed precision required. I mark with black the points z0 E Rb for which the corresponding iterative method starting in z0 does not reach any root with tolerance w-3 in a maximum of 25 iterations.
In the final section of this article, I show the programs that I have used and similar ones that allow us to generate both gray-scaled and color figures. Of course, it is also possible to use the function f* or the small rectangle Rs (or any other function or rectangle); this will only require small modifications to the programs.
Although an ordinary programming language is typically hundreds of times faster, to generate the pictures it is easier if we employ a computer package with graphics facilities, such us Mathematica, Maple, or Matlab. The graphics that I show here were generated with Mathematica 3.0 (see [20]); in the next section, I show the programs used to obtain the figures.
Note that both Traub-Ostrowski's method and Jarratt's method for j{z) = z3 - 1 lead to the iterative function
1 + 12z3+54z6+ 14z9 c:f>(z) = 6zz+42zs+sszs • Hence the fractal figure for both
of them is the same (Figure 1 1), and the same happens for the data of Tables 1 and 2.
The tables and the figures provide empirical data. From them, and the indications given here, we can guess the behavior and suitability of any method depending on the circumstances. This is good entertainment.
Stirling's and Steffensen's methods are a case apart. First, they are the most demanding with respect to the initial point (in the tables, see the percentage of nonconvergent points; in the figures, see the black areas). And, second, in their graphics, the symmetry of angle 2 7TI3 that we observe in the other methods does not appear (with respect to symmetry of fractals, see [ 1]).
Mathematica Programs to Get the Graphics
In this section, I explain how the figures in this article were generated. To do this, I show the Mathematica [20] programs used.
First, we need to define function f and its derivatives. This can be done by using f [ z_ ] : = z " 3 - 1 1
df [ z_] : = 3 * z " 2 and d2 f [ z_ ] : = 6 * z, but it is faster if we use the compiled versions
f = Comp i l e [ { { z I _Complex} } I z " 3 - 1 ] ;
df = Comp i l e [ { { z i _Complex} } I 3 * z " 2 ] ;
d2 f = Comp i l e [ { { z I _Complex } } I 6 * z ] ;
Of course, any other function, such as f*(z) = exp (sin�l ) (z3 - 1), can be used.
The three complex roots off are
Do [ root [ k ] = N [ Exp [ 2 * ( k- 1 ) * Pi * I / 3 l l �
{ k l 1 1 3 } ]
I use the following procedure which identifies which root has been approximated with a tolerance of w-3, if any.
rootPo s i t i on = Comp i l e [ { { z I _Complex } } I
Which [ Abs [ z - root [ 1 ] ] < 1 0 . 0 " ( - 3 ) 1 3 ,
Abs ( z - ro o t f [ 2 ] ] < 1 0 . 0 " ( - 3 ) , 2 ,
Abs [ z - ro o t f [ 3 ] ] < 1 0 . 0 " ( - 3 ) , 1 , True , 0 ] ,
{ { roo t f [ _ ] , _ Complex } }
l We must define the iterative methods, that is, the dif
ferent Zn+ 1 = c:f>(zn)· For Newton's method, this would be
i terNewton = Comp i l e [ { { z , _Complex} } ,
z - f [ z J I df [ z J J and, for Halley's method,
i terHal l ey = Comp i l e [ { { z , _Complex} } ,
Block [ { v = df [ z ] } , z - 1 . 0 I ( v/ f [ z ] - ( d2 f [ z ] ) I ( 2 . 0 * v ) ) ]
(observe that an extra variable v is used so as to evaluate d f [ z J once only). The procedure is similar for all the other methods in this paper.
VOLUME 24, NUMBER 1 , 2002 43
The algorithm that iterates the function i terMethod to
see if a root is reached in a maximum of l im iterations is
the following:
i terAlgori thm [ i t erMethod_ , x_ , y_ , l im_ ] .
Block [ { z , c t , r } , z = x + y I ; c t = O ;
r = roo tPosi t i on [ z J ;
Whi le [ ( r = = 0 ) && ( c t < l im) ,
++ct ; z = i terMethod [ z ) ;
r = rootPo s i t i on [ z )
l ; I f [ Head [ r ) = = Whi c h , r = O ) ;
( * " Whi ch" uneva luated * )
Return [ r )
Here, I have taken into account that sometimes Mathe
matica is not able to do a numerical evaluation of z. Then
it cannot assign a value for r in rootPo s i t i on. Instead,
it returns an unevaluated Whi ch. Of course, this corre
sponds to nonconvergent points.
We are going to use a limit of 25 iterations and the com
plex rectangle [ -2.5, 2.5] X [ -2.5, 2.5]. To do this, I define
the following variables:
l imi terat i ons = 2 5 ;
xxMin = - 2 . 5 ; xxMax = 2 . 5 ;
yyMin = - 2 . 5 ; yyMax = 2 . 5 ;
Finally, I defme the procedure to paint the figures ac
cording to strategy (i) described in the introduction. White,
33% gray and 66% gray are used to identify the attraction
basins of the three roots 1, e271i13 and e471i13• The points for
which the iterative method does not reach any root (with
the desired tolerance in the maximum of iterations) are pic
tured as black The variable points means that, to gener
ate the picture, a points X points grid must be used.
p l o t Frac tal [ i terMe thod_ , points_] : =
Den s i ty P l o t [ i t e rA l g o r i thm [ i t e rMetho d ,
x , y , l imitera t i ons ) ,
{ x , xxMin , xxMax } , { y , yyMin , yyMax } ,
PlotRange � { 0 , 3 } , P l o tPoints � point s ,
Mesh � False
I I Timing
Note that I I Timing at the end allows us to observe the
time that Mathematica employs when pl otFrac t a l is
used.
Then a graphic is obtained in this way (the example is
a black-and-white version of Figure 1 ):
p l o tFractal [ i terNewton , 2 5 6 )
When we use the functions that have been defined, over
flow and underflow errors can happen (for instance, in
Newton's method, j'(z) can be null and then we are divid
ing by zero, although that is not the only problem). Math
ematica informs us of such circumstances; to avoid it, use
the following before calling p l o tFrac tal:
O f f [ General : : ovf l ) ; O f f [ General : : un f l ) ;
O f f [ Infinity : : inde t )
44 THE MATHEMATICAL INTELLIGENCER
Also, the previous problems, and some others, sometimes
force Mathematica to use a noncompiled version of the
functions. Again, Mathematica informs us of that circum
stance; to avoid it, use
O f f [ Comp i l edFunc t i on : : c c c x ) ;
O f f [ Comp i l edFunc t i on : : c fn ) ;
O f f [ Comp i l edFunc t i on : : c f c x J ;
O f f [ Comp i l edFunc tion : : c f ex ) ;
O f f [ Comp i l edFunc t i on : : crcx) ;
O f f [ Comp i l edFunc t i on : : i l sm )
Perhaps some other O f f are useful depending on the func
tion! and the complex rectangle used.
To obtain color graphics, I use a slightly different pro
cedure to identify which root has been approximated; this
is done because we also want to know how many iterations
are necessary to reach the root. I use the following trick:
in the output, the integer part corresponds to the root and
the fractional part is related to the number of iterations.
i t erCol orAlgori thm [ i terMethod_ ,
x_ , y_ , l im_ ] . -
Block [ { z , c t , r } , z = x + y I ; ct = O ;
r = roo t P o s i t i on [ z ) ;
Whi l e [ ( r == 0 ) && ( c t < l im ) ,
+ +c t ; z = i terMethod [ z ) ; r = rootPo s i t i on [ z J
l ; I f [ Head [ r ) == Whi ch , r = O ) ;
( * "Which" unevaluated * )
Return [ N [ r+c t l ( l im+ O O . O O l ) ] J
To assign the intensity of the color of a point, I take into
account the number of iterations used to reach the root
when the iterative method starts at that point. I use cyan,
magenta, and yellow for the points that reach, respectively,
the roots 1 , e271i13 and e471i13; and black for nonconvergent
points. To do this, I use
and
col orLevel = Comp i l e [ { { p , _Rea l } } ,
0 . 4 * Frac t i onalPart [ 4 *p ] J
fractalC o l o r [ p_ ] . -
Bl ock [ { pp = colorLevel [ p ) } ,
Swi tch [ IntegerPar t [ 4 *p ) ,
3 , CMYKColor [ 0 . 6 +pp , 0 . , 0 . , 2 * pp ) ,
2 , CMYKColor [ 0 . , 0 . 6 +pp , 0 . , 2 * pp ] ,
1 , CMYKCo l or [ 0 . , 0 . , 0 . 6 +pp , 2 * pp ) ,
0 , CMYKColor [ 0 . , 0 . , 0 . , 1 . ]
(In the internal behavior of Mathematica, when a function
is going to be pictured with Dens i ty P l o t , it is scaled to
[0, 1 ] . However, i terCol orAlgori thm has a range of
[0, 4]; this is the reason for using 4 * p in some places in
col orLevel and frac talCol or. Also, note that c o l
orLevel can be changed to modify the intensity of the colors; for other graphics, it is a good idea to experiment by changing the parameters to get nice pictures.)
Finally, a color fractal will be pictured by calling the procedure
plotCol orFrac tal [ i terMethod_ , points_]
Dens i tyPl ot [
i terC o l orAlgori thm [ i terMethod , x , y ,
l imitera t i ons ] ,
{ x , xxMin , xxMax } , { y , yyMin , yyMax } ,
P l o tRange � { 0 , 4 } ,
P l o t Points � po int s , Mesh � Fal s e ,
Col orFunction � frac ta l C o l or
I I T iming
For instance,
p l o tC o l o rFrac ta l [ i terNewt on , 2 5 6 ]
is just Figure 1 .
Families of Iterative Methods
There are many iterative methods for solving nonlinear equations in which a parameter appears; one speaks of families of iterative methods.
One of the best-known is the Chebyshev-Halley family
f(zn) ( 1 LJ(Zn) ) Zn+ l = Zn - f'(zn) 1 + 2 1 - f3L./._zn) '
with {3 a real parameter. These are order 3 methods for solving the equation f(z) = 0. Particular cases are {3 = 0 (Chebyshev's method), {3 = 112 (Halley's method), and {3 = 1 (super-Halley's method). When {3 � -oo, we get Newton's method. This family was studied by W. Werner in 1980 (see [ 19]), and can also be found in [3, p. 219] and [10]. It is interesting to note that any iterative process given by the expression
f(zn) Zn+ l = Zn -
f'(zn) H(L./._zn)),
where function H satisfies H(O) = 0, H' (0) = 1/2 and IH"(O)i < oo, generates an order 3 iterative method (see [8]). The Chebyshev-Halley family appears by takingH(x) = 1 + 1 :r 2 1 - (3:1:•
A multipoint family (see [ 18, p. 178]) is
f(zn) Zn+l = zn - -( ) g Zn
f(z + [3f(z)) - f(z) • with g(z) = f3f(z) and {3 an arb1trary constant
({3 = 1 is Steffensen's method). Its order of convergence is 2.
An order 4 multipoint family was studied by King [14] (see also [3, p. 230]):
Zn+ 1 = Zn - U(Zn) f(Zn - u(zn)) fCzn) + f3f(zn - u(zn))
f'(zn) f(zn) + ({3 - 2)j(Zn - u(zn)) '
where {3 is an arbitrary real number and u(z) = f,��)· TraubOstrowski's method is the particular case {3 = 0.
Finally, here is another order 4 multipoint family:
where {3 is a parameter and u, h denote u(z) = ft{;; and
h(z) = rl.z - f,ucz)j -f'Cz) . Here for {3 = 0 we getJarratt's method f'(z) ' ' (actually, in [12] a different family appears; the method that I am calling Jarratt's method is a particular case of both families). For {3 = -3/2, we get the so-called inverse-free Jarratt's method.
Uniparametric iterative methods offer an interesting graphic possibility: to show pictures in movement. We take a fixed function and a fixed rectangle, and we represent the fractal pictures for many values of the parameter. This then generates a nice moving image that shows the evolution of the fractal images when the parameter varies. Unfortunately, it is not possible to show moving images on paper. To generate them in a computer, one can use small modifications of the Mathematica programs from the previous section, using also the Mathematica commands An
imate or ShowAnima t i on. Later, it is possible to export these images in Quick-Time format (so that Mathematica will not be necessary for seeing them). Of course, this requires a large quantity of computer time, but as computers become faster and faster this is less of a problem.
A U T H O R
JUAN L. VARONA
Departamento de Matematicas y Computaci6n
Universidad de La Rioja
26004 Logroiio
Spain
e-mail: jvarona@dmc.unirioja.es
Juan L. Varona is a native of La Rioja, a region of Spain known
hitherto mostly for its wines. He studied mathematics at
Zaragoza, and went on for his Ph.D. at Cantabria, also in
Spain . His research is mainly in Fourier analysis, but also in
computational number theory. One of his more "serious" hob
bies is developing tools for writing Spanish in TeX/LaTeX.
VOLUME 24, NUMBER 1 , 2002 45
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Equations (Proc. , Bremen, 1 980), E. L. Allgower, K. Glashoff and H .
0. Peitgen, eds. , Lecture Notes in Math. 878 (1 981 ) , 427-440.
20. S. Wolfram, The Mathematica Book, 3rd ed. , Wolfram Media/Cam
bridge University Press, 1 996.
21 . J. W. Neuberger, The Mathematical lntelligencer, 21 (1 999), no. 3,
1 8-23.
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 California, Los Angeles and
Wilfried Schmid, Harvard University, Cambridge, MA (eds.)
Mathematics Unl imited 200 1 and Beyond
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as we enrer a new millennium. True ro irs tide, the book extends beyond the spectrum of mathematics ro include contributions from other related sciences. You will enjoy reading the many stimulating conrribution and gain insights into the astounding progress of mathematics and the perspectives for its future. One of the ediror , Bjorn Engquist, is a world-renowned researcher in computational science and engineering. The second editor, Wilfried Schmid, is a disringuished mathematician at Harvard University. Likewise, the authors are all foremost mathematicians and scientists, and their biographies and phorograph appear at the end of the book. Unique in both form and conrenr, this i a «must-read" for every mathematician and scientist and, in particular, for graduates still choosing their specialty.
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46 THE MATHEMATICAL INTELLIGENCER
Contents: Amman, S. : Nonlinear Continuum Physics. • Babuska, I./Tinsley Oden, J. : Computational Mechanics: Where is it Going? • Bailey, D.H./Borwein J.M.: Experimental Mathematics: Recent Developments and Future Outlook. • Darmon, H.: p-adic L-functions. • Fairings, G.: Diophantine Equations. • Farin, G.: SHAP • Jorgensen, J ./Lang, S.: The Hear Kernel All Over rhe Place. • Kllippelberg, C.: Developments in Insurance Mathematics. • Koblitz, N.: Cryprography. • Marsden, j./Ccndra, H./Ratiu, T.: Geometric Mechanics, Lagrangian Reduction and onholonomic Systems. • Roy, M.-F. : Four Problems in Real Algebraic Geometry. • Serre, D.: Systems of Conservation Laws: A Challenge for the XX!st Century. • Spencer, J.: Discrete Probability. • van der Geer, G.: Error Correcting Codes and Curves Over Finite Fields. • von Storch,
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li.l$?ffli•i§rr6hl£11i.Jih?"Ji D irk H uylebrouck, Editor I
Homage to Emmy Noether Istvan Hargittai and
Magdolna Hargittai
Does your hometown have any
mathematical tourist attractions such
as statues, plaques, graves, the cafe
where the famous conjecture was made,
the desk where the famous initials
are scratched, birthplaces, houses, or
memorials? Have you encountered
a mathematical sight on your travels?
lf so, we invite you to submit to this
column a picture, a description of its
mathematical significance, and either
a map or directions so that others
may follow in your tracks.
Please send all submissions to
Mathematical Tourist Editor,
Dirk Huylebrouck, Aartshertogstraat 42,
8400 Oostende, Belgium
e-mail: dirk.huylebrouck@ping.be
Let us add a few words to Alice Sil
verberg's informative article about
the birthplace of Ernmy N oether [ 1 ] .
We have long admired Emmy N oe
ther's contributions to the general con
cept of symmetry [2, pp. 200-201 ] . Her
man Weyl said in his memorial address
at Emmy Noether's funeral [3], "She
was a great mathematician, the great
est, I firmly believe, that her sex has
ever produced, and a great woman." Al
bert Einstein expressed a similar opin
ion in a letter to The New York Times
upon her death on May 4, 1935 [ 4, p.
75] , "In the judgment of the most com
petent and living mathematicians,
Fraulein Noether was the most signifi
cant creative mathematical genius thus
far produced since the higher educa
tion of women began."
She did seminal work in the field of
the theory of invariants, in spite of all
the difficulties she had to face. First,
she had difficulties in getting into the
university to study. Later she had to
work free, and for a long time she
could not get her habilitation (a higher
doctorate needed for an independent
university teaching position) as it was
"declared impossible because of legal
requirements. " According to regula
tions in effect in Germany in the
1910s, habilitation could only be
granted to male candidates. David
Hilbert and Felix Klein tried to help,
but without success. According to
Weyl, the non-mathematician mem
bers of the Philosophical Faculty, to
which the mathematicians belonged,
argued that the soldiers coming back
from the war should not find them
selves "being lectured at the feet of a
woman." This is when Hilbert made
his famous statement [5, p. 14] , "I do
not see that the sex of the candidate
is an argument against her admission
as Privatdozent. After all, we are a
university, not a bathing establish
ment."
48 THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
Eventually she was allowed to lec
ture under Hilbert's name in Gottingen.
Finally, in 1919, she gave her habilita
tion lecture with the title "Invariante
Variationsprobleme." In this lecture,
she summarized her work concerning
the connection of symmetry and the
conservation laws of physics. She
proved that all conservation laws are
connected with a certain type of sym
metry (invariance ), and she stated that
"the converses of these theorems are
also given"; that is, for every symmetry
there is a conservation law. Following
her habilitation, she eventually re
ceived teaching rights in Gottingen, al
though she never became full profes
sor.
We knew that Emmy Noether was
buried at Bryn Mawr College in Penn
sylvania, and when, on March 7, 1999,
we were in the neighborhood, we vis
ited Bryn Mawr College as a tribute to
her. It was an unusually cold day and
we did not encounter anybody on the
grounds, but we found the beautiful
courtyard of the Cloisters of the M.
Carey Thomas Library, and we also
found the stone in its pavement with
the inscription E N 1882-1935. Noe
ther's ashes are under this stone.
But why was she at Bryn Mawr?
Noether was Jewish, and in 1933 anti
Semitic policies went into effect in Ger
many. She was stripped of her univer
sity position in Gottingen in April 1933
at the same time as Richard Courant
and Max Born. When it became known
that Noether had lost her position,
Bryn Mawr College expressed an in
terest in having her, at least on a tem
porary basis. The Rockefeller Founda
tion provided Bryn Mawr with financial
assistance. Oxford University was also
interested in getting Emmy N oether,
but in October 1933 Noether accepted
Bryn Mawr's offer. "Were it not for her
race, her sex, and her liberal political
opinions (they are mild), she would
Courtyard of the M. Carey Thomas Library, Bryn Mawr College (photograph by I. and M. Hargittai).
have held a first rate professorship in Germany . . . " wrote one of her sup
porters [6] . Clark Kimberling describes
Noether's productive last years at Bryn
Mawr [5, 31-46]. Noether died follow
ing a tumor operation. According to
Kimberling, a few weeks before her
death Emmy Noether remarked to a
colleague "that the last year and a half
had been the very happiest in her
whole life. For she was appreciated in
Bryn Mawr and Princeton as she had
never been appreciated in her own
country" [5, p. 39].
Acknowledgment: We thank Profes
sor Victor Donnay of the Department
of Mathematics, Bryn Mawr College for
excellent directions.
The stone in the pavement under which Emmy Noether's ashes rest (photograph by I. and M .
Hargittai).
REFERENCES
[1 ] Alice Silverberg, "Emmy Noether in Erlan
gen." The Mathematical lntelligencer (2001 )
Vol. 23, No. 3 , 44-49.
[2] I. Hargittai, M. Hargittai, In Our Own Image:
Personal Symmetry in Discovery. Kluwer/
Plenum, New York, 2000.
[3] H. Weyl, "Emmy Noether," Memorial Ad
dress, reprinted in A. Dick, Emmy Noether:
1882-1935. Birkhauser, Boston, 1 981 , pp.
1 1 2-1 52.
[4] A. Einstein, The Quotable Einstein, col
lected and edited by A. Calaprice, Prince
ton University Press, Princeton, New Jer
sey, 1 996, p. 75.
[5] C. Kimberling, "Emmy Noether and Her In
fluence," in Emmy Noether: A Tribute to Her
Life and Work. J.W. Brewer, M.K. Smith,
eds., Marcel Dekker, New York, 1 981 , p. 14 .
[6] From Solomon Lefschetz's letter of Decem
ber 31 , 1 934, as quoted in [5] pp. 34-5.
Istvan Hargittai
Budapest University of Technology and
Economics
e-mail: Hargittai@tki .aak.bme.hu
Magdolna Hargittai
Ebtvbs University and Hungarian Academy
of Sciences
e-mail: Hargittaim@ludens.elte.hu
H-1 521 Budapest, Hungary
VOLUME 24, NUMBER 1 , 2002 49
STEPHEN BERMAN AND KAREN HUNGER PARSHALL
Victor Kac and Robert Moody : The i r Paths to Kac- Moody Lie Algebras
uilding on the late-nineteenth-century researches of Sophus Lie and Wilhelm Killing,
Elie Cartan completed the classification of the finite-dimensional simple Lie alge
bras over the complex numbers C in his 1894 thesis [ 8]. 1 Surprisingly, these fall into
jive classes: the four ''great classes" consisting of the classical simple Lie algebras,
and the class of the five "exceptional algebras. " Relative to the great classes, for l 2:: 1, the (l + 1) X (l + 1) matrices of trace zero give a model for the simple Lie algebra of type A1, while the orthogonal or symplectic Lie algebras similarly supply a model for the others, namely, the algebras of type B1 (l 2:: 2), C1 (l 2:: 3), and D1 (l 2:: 4). Of the exceptional algebras, the simplest, G2, can be realized as the Lie algebra of derivations of the octonions.2 For the others, E6 , E7,
E8, and F4, however, questions of existence and of finding models are highly non-trivial and deeply influenced the development of Lie theory.
Killing and Cartan approached their analysis of the finite-dimensional simple Lie algebras over 1[: by considering each Lie algebra as a decomposable entity. They aimed to
effect a decomposition that not only revealed the internal structure of the Lie algebra but also efficiently synthesized the information so obtained.3 Their attack was, in broad terms, linearly algebraic; they used what would now be termed "generalized eigenspace decompositions"-root space decompositions in their setting-relative to the socalled Cartan subalgebra, a maximal abelian diagonalizable subalgebra of the Lie algebra. They then distilled, from the root system derived from this decomposition, the fundamental system of simple roots associated with the Lie algebra. They used the latter to define a "finite Cartan matrix," namely, an integral matrix satisfying the properties (a), (f3), and ( y) (see the next section below). This realized their dual goals: they had uncovered the internal structure
Stephen Berman gratefully acknowledges support from the National Sciences and Engineering Research Council of Canada as well as the hospitality of the Mathe
matics Department of the University of Virginia. Both authors thank Victor Kac, Robert Moody, and George Seligman for their cooperation during the preparation of this
paper.
1 For an historical treatment of these developments, see [21 ]. For standard modern mathematical references on Lie groups and Lie algebras, see [22] and [24], re
spectively.
2The octonions were discovered by John Graves late in 1 843; he wrote of his finding to the discoverer of the quaternions, Sir William Rowan Hamilton, entrusted their
publication to him, and unfortunately did not see his work in print. Early in 1 845, Arthur Cayley discovered the octonions independently and published his result im
mediately [9]. To use terminology that would only develop in the early twentieth century, Graves and Cayley had hit upon the first known noncommutative, nonasso
ciative algebra.
3The process sketched here is fundamental. For the precise definitions and for further details, see [24, pp. 1 -72].
50 THE MATHEMATICAL INTELUGENCER © 2002 SPRINGER-VERLAG NEW YORK
Victor Kac and Robert Moody.
of the Lie algebras, and they had succeeded in efficiently
and completely encoding in the finite Cartan matrix the per-
tinent structural information about the Lie algebra. The
classification then proceeded by enumerating these matri-
ces. This was subsequently schematized further in terms of
the Dynkin diagrams associated with each of the matrices;
the precise composition of the finite Cartan matrix is, fol-
lowing a number of conventions, recoverable from its as-
sociated Dynkin diagram.4 Thus, the nine types of simple
algebras correspond to the nine types of finite Cartan rna-
trices, which, in tum, correspond to the nine types of fmite
Dynkin diagrams given in Figure 1 .
Cartan quite naturally followed this early work with a
classification of the [mite-dimensional irreducible represen-
tations associated with the [mite-dimensional simple Lie al-
gebras over C [6]. Once again, his classification involved a
fundamentally linearly algebraic decomposition. In this case,
however, the decomposition was into weight spaces, thereby
generalizing the root space decomposition in the Lie algebra
setting. He showed that the representations were in one-to-
one correspondence with the so-called dominant highest
weights. Moreover, just as the root spaces were the funda-
mental building blocks of the fmite-dimensional simple Lie
algebras, the weight spaces played that key role in the as-
sociated theory of fmite-dimensional irreducible represen-
tations. To "know" the representations (in Cartan's theory)
was thus to "know" the so-called dominant integral highest
weight. This, however, did not readily yield knowledge of
the dimensions of all of the weight spaces. Hermann Weyl's
4See [24, pp. 56-63] for the conventions and the exact associations.
stunning discovery in 1926 effectively provided this addi
tional level of familiarity [51] . Informally, and in more mod
em terms, Weyl gave a polynomial expression in several vari
ables, the coefficients of which gave the dimensions of the
weight spaces involved in the decomposition. In light of
Weyl's result, then, to "know" the weight spaces was to
"know" his so-called character formula. 5
Here, we sketch the lines of research that led from these
problems of proving existence and of fmding realizations
of simple Lie algebras-first over the complex numbers but
later over other fields-to the recognition in the 1960s and
the development in the 1970s of a new kind of algebra, the
Kac-Moody Lie algebra.
The Work of Claude Chevalley and Harish-Chandra
The line of research from Lie through Killing and Cartan to
• • • • Ae
• • •===7• Be
• • ------�· Ce
• • ! • De
• • ! • • E6
• • ! • • • E1
• • • • I • • Es
• •===7• • F4
·� G2
Figure 1 . Dynkin diagrams, the finite case.
5See [24, pp. 1 38-1 40] for a modern statement and proof of Weyl's character formula.
VOLUME 24, NUMBER 1 , 2002 51
Weyl on the simple finite-dimensional Lie algebras over C and their irreducible representations had further natural
extensions in light of concurrent mathematical develop
ments. In particular, as field theory developed following
Ernst Steinitz's groundbreaking paper of 1910 [48] , mathe
maticians began to study Lie-theoretic objects over other
fields, especially over the real field IR.6 Questions of exis
tence and of finding realizations became even more diffi
cult and detailed in this broader field-theoretic context. For
example, satisfying knowledge about the situation over
number fields was only obtained in the last half of the twen
tieth century. Researchers like A. Adrian Albert, Hans
Freudenthal, Nathan Jacobson, George Seligman, and
Jacques Tits made fundamental contributions to this the
ory and influenced those who sought to give various mod
els for these finite-dimensional simple Lie algebras over
fields of characteristic 0. Their approach to providing mod
els often hinged on showing that the algebras are isomor
phic to certain Lie algebras of matrices with coordinates
Chevalley's note was read by Elie Cartan at the meeting
of the Paris Academy of Sciences on 29 November 1948. In
it, Chevalley identified two "holes" in Lie theory by pre
senting them in the context of the then-recent history of
the subject. Chevalley remarked that, in his thesis, Elie Car
tan had established, using a case-by-case analysis, that
there was one and only one simple Lie algebra corre
sponding to each of the nine types of fmite Cartan matri
ces. Van der Waerden pursued this line of research. Using
results of Weyl, he proved a priori that there can exist no
more than one type of algebra for a given simple system
(hence finite Cartan matrix) [49]. Chevalley also singled out
the "elegant construction" [ 10, p. 1 136 (our translation)]
Ernst Witt had given in 1941, showing the existence of the
five exceptional types [52] . This historical sketch pointed
Chevalley to the first hole that needed filling, namely, an a priori proof of the existence of all of the finite-dimensional
simple Lie algebras over C [ 10, p. 1 136] . He also noted that
the analogous question could be posed for the irreducible
coming from various types of
non-associative algebras. A dif
ferent and technically daunting
tack, however, establishes the
existence of the finite-dimen
sional simple Lie algebras
without presenting a particular
realization. Various existence
schemes have indeed been put
The n ine types of s im ple
algebras correspond to
the n ine types of fin ite
Cartan matrices .
representations of these alge
bras, even though this had been
settled earlier by Cartan and
Weyl either using a case-by-case
argument or by means that were
not entirely algebraic. Thus, the
second hole to be filled involved
giving an a priori-that is,
purely algebraic-proof of their
forth, but the most successful and penetrating one issued
from work of Claude Chevalley [ 10] and Harish-Chandra
[20] in the late 1940s and early 1950s. 7 In 1948, Chevalley published a very short yet highly
suggestive note "Sur la classification des algebres de Lie
simples et de leurs representations" that indicated a way
to construct simultaneously the finite-dimensional sim
ple Lie algebras and all of their finite-dimensional irre
ducible representations [ 10 ] . Whereas Killing and Cartan
had developed a process that went from the finite-di
mensional simple Lie algebra to the finite Cartan matrix,
Chevalley and Harish-Chandra reversed the process.
Theirs was a constructive scheme that began with the fi
nite Cartan matrix and produced the finite-dimensional
simple Lie algebra. Moreover, whereas Weyl's results had
hinged on what Chevalley termed "the transcendent the
ory of compact groups" [ 10, p. 1 137 (our translation)] ,
the reverse process of Chevalley and Harish-Chandra
"made algebraic" the results of Lie theory, avoided the
tedious case-by-case analyses, and penetrated even more.
deeply than their predecessors the Lie algebra structure.
In the bargain, the question of showing existence and of
giving models also played out for irreducible represen
tations.8
6Again, Carlan was one of the pioneers. See [7].
existence [10, p. 1 137]. Chevalley proceeded to outline, but
only in very broad terms, a method for dealing with these
lacunae. He did not provide proofs.
New developments followed almost immediately. When
the Academy met nine days later on 8 December, Cartan
presented the following addendum from Chevalley: "In a
recent note, I outlined an a priori algebraic proof of the
existence of the irreducible representations of a given sim
ple Lie algebra, given a dominant highest weight. I have
learned that another proof of the same theorem has been
obtained simultaneously and independently by Barish
Chandra working at the Institute for Advanced Study in
Princeton. Harish-Chandra's proof furnishes, at the same
time, an upper bound on the degree of the irreducible rep
resentations in question" ( 1 1 ] (our translation).
Chevalley had come to know Harish-Chandra at Prince
ton University during the 1947-1948 academic year when
he found the young Indian physicist in his course on Lie
groups and Lie algebras [5, p. 9] . Harish-Chandra had
earned his Ph.D. in physics under Paul Dirac at Cambridge
University in 1947 and had accompanied his adviser to the
Institute for Advanced Study later that year [50]. While in
Princeton, Harish-Chandra came to realize that his talents
lay more in mathematics than in physics. "I once com-
7 Andre Weil is reported to have said that "he knew only two mathematicians for whom technical difficulties simply did not exist, namely Chevalley and Harish·Chan
dra" (4, p. 920].
8See the recent works by Knapp (35] and by Goodman and Wallach (18] for excellent modern-day accounts of rnuch of Lie theory. The latter work, especially, juxta
poses the algebraic, analytic, and topological approaches to the theory.
52 THE MATHEMATICAL INTELLIGENCER
plained to Dirac about the fact that my proofs were not rigorous," Harish-Chandra is reported to have said. When Dirac replied, "I am not interested in proofs but only in what nature does," Harish-Chandra realized that he "did not have the mysterious sixth sense which one needs in order to succeed in physics and so [he] soon decided to move over to mathematics" [23, pp. 7-8] (see also (36]). So whereas Dirac had been Harish-Chandra's mentor in physics, Chevalley quickly became his mentor and early guide in Lie theory. As Harish-Chandra's Lie-theoretic result of 1948 attests, he was a quick study.
Harish-Chandra did not publish this work until 1951, and then it was in the context of a long and very wide-ranging paper [20] (see below). Relative to the theory of Lie algebras, though, he prominently acknowledged Chevalley and his work after giving his own sketch of the recent history of the area. Not surprisingly, Harish-Chandra, like Chevalley, found the origins of the ideas in the work of Cartan and W eyl. "The representation theory of semisimple Lie algebras over the field of complex numbers," according to Harish-Chandra, "has been developed by Cartan and Weyl. However some of Cartan's proofs . . . make explicit use of the classification of semisimple Lie algebras and in fact require a verification of the asserted statement in each case separately. Weyl . . . has given alternative proofs of these results by making use of general arguments depending on the theory of representations of compact groups . . . . His proofs therefore are necessarily of a nonalgebraic nature" [20, p. 28]. In his paper, Harish-Chandra thus "propose[d] to give 'general' algebraic proofs of some of these results " and he noted that his "work overlaps considerably wi;h some recent results of Chevalley [C]. In particular the formulation of Theorem 1 and some of the ideas in the proof are due to him" [20, p. 28] .
The paper [C] was Chevalley's "Sur la classification des algebres de Lie simples et de leurs representations" [ 10]. Its Theorem 1 asserted the existence of both the finite-dimensional simple Lie algebras and their finite-dimensional irreducible representations. Harish-Chandra later informed his readers that in his original attack on this problem he had only been interested in the representations; Chevalley's work, however, had significantly influenced his own. As he put it, "in my original proof I had considered the second question alone. The idea of dealing with both questions simultaneously is due to Chevalley [C] who obtained independently a proof of the theorem . . . . I present here a modified version of my original proof so as to be able to consider the two questions together. But in this modification I have adopted several of Chevalley's ideas" (20, p. 30].
In his paper, Harish-Chandra worked over an algebraically closed field of characteristic zero. He began with an integral l X l matrix A = (AiJ) having the following three properties:
(a) Aii = 2; AiJ :=::: 0, i =F j; AiJ = 0 <=> AJi = 0; (/3) det A =F 0; and
( Y) the Weyl group associated with A ( defmed immediately below) is a finite group.
He then considered an [-dimensional vector space with basis a1, . . . , at and defined l linear transformations rr, . . . , r1 by
ri(aj) = aJ - AJiai, 1 :=::: i, j :=::: l.
The Weyl group of A is then the group generated by r1, . . . , rt. Using generators ei, fi, hi, 1 :=::: i :=::: l, he gave a construction that explicitly showed the existence of the simple Lie algebras as well as of their irreducible representations.9 As Harish-Chandra noted, however, "(t]he proof is rather long but otherwise not very complicated. It depends on the consideration of the representations of a certain infmite dimensional associative algebra A. We shall have to prove a series of lemmas about left ideals in this algebra, some of which are very simple but are nevertheless essential" [20, p. 31 ] .
More specifically, the construction involved taking a free associative algebra on the set {eiJi, hi II :=::: i :=::: l} and a natural representation for it acting on another free associative algebra. This permitted the factorization of both of these objects using either certain definite relations or (in some instances) more abstractly given objects. One of the types of relations that played a particularly important role was of the form
(1) (ad ei)-Aji+l e1 = 0 = (adfirAji+ 1 jj, 1 :=::: i, j, :=:::z, i * j,
where (ad x)(y) = [x y] for x, y in the Lie algebra with product [ · · ] . Here, the Cartan matrix associated with the Lie algebra in question is A = (AiJ) for 1 :=::: i, j :=::: l, where l is the rank of the Lie algebra. Harish-Chandra credited Chevalley for a key lemma concerning these elements [20, p. 36] .
Harish-Chandra's paper [20] contained much more than this construction, however. It presented his now-famous work relating characters of the irreducible representations to the universal enveloping algebra and, in particular, his construction and analysis of the properties of what is now called the "Harish-Chandra homomorphism." It also contained results about representations of both the groups and algebras acting on Hilbert spaces. Of broad scope, this paper became one of the foundational pillars of the theory of harmonic analysis on semisimple Lie groups ( cf. (23]). Its breadth and import were almost immediately recognized; Harish-Chandra won the Cole Prize of the American Mathematical Society for it in 1954.
The method developed independently by Chevalley and Harish-Chandra was ultimately presented, with simplifications and modifications, by Nathan Jacobson in Chapter 7 of his influential 1962 text, Lie Algebras [25]. As Jacobson explained in opening that chapter, "Harish-Chandra's proof of these results is quite complicated. The version which we shall give is a relatively simple one which is based on an explicit definition of a certain infmite dimensional Lie al-
9Here and throughout, we have adopted the now-standard notation and terminology of [24] or [45] rather than that used by Harish-Chandra.
VOLUME 24, NUMBER 1 , 2002 53
gebra [" [25, p. 207]. Like Barish-Chandra, Jacobson began with an integral l X l matrix A = (AiJ) satisfying properties (a), (/3), and (y) above, but, in his exposition, these three criteria appear almost as an axiom scheme. Moreover, he replaced Barish-Chandra's associative algebra A with a free Lie algebra on 3l free generators
(2)
and worked over a general field of characteristic zero. The construction followed much more easily in this set-up. Jacobson factored the free Lie algebra by the relations
(hi, hJ) = 0, [k;, ej) = AjieJ, lhi, .fjl = -AJJJ, and [ei. fil = oiJhi
to obtain a Lie algebra L After studying representations for [, he proceeded to factor [ by the intersection of the kernels of all of its finite-dimensional irreducible representations. This resulted in the desired finite-dimensional simple Lie algebra with Cartan matrix A. Interestingly, Jacobson explicitly noted the single use of axiom (y) in the construction [25, p. 220). Coming as it did at the very end of his construction, this remark almost challenged the reader to study algebras that come from matrices satisfying only axioms (a) and (/3).
The line of research and exposition stemming from the work of Chevalley and Barish-Chandra came to a natural conclusion in 1966 when Jean-Pierre Serre gave a presentation in [ 46] for all of the finite-dimensional simple Lie algebras over IC, a result now known as Serre's theorem. Specifically, he showed that if the Lie algebra [ above is factored by the ideal generated by the elements in (1), then the resulting Lie algebra is none other than the finitedimensional simple Lie algebra with Cartan matrix A, the same matrix with which the construction began. Given the earlier developments, the proof was not too complicated; the extra ingredient depended on a clever argument involving the roots, and Serre clearly credited the work of Chevalley, Barish-Chandra, and Jacobson. 10 If Serre's work represented a natural conclusion to a line of mathematical results extending back to the nineteenth century, however, it also marked a natural beginning for what would become a very prominent theme in both the mathematics and physics of the latter part of the twentieth century, namely, the theory of Kac-Moody Lie algebras.
New Algebras Emerge
In the fall of 1962, Robert Moody entered the graduate program in mathematics at the University of Toronto. There, he came under the influence both of the geometer, H. S. M. Coxeter, and of the algebraist and student of Nathan Jacobson, Maria Wonenburger. In Coxeter's lectures on regular polytopes, Moody encountered reflection groups; in Wonenburger's course during the 1964-1965 academic year on Lie algebras from Jacobson's book [25], the very same
1 °For an exposition of this work, see [24) and [46).
Robert Moody and his mentor, Maria Wonenburger.
groups arose. As Moody has put it, "by good fortune then I was presented with the same groups, but in very different contexts, and I asked what was probably a very naive question: if there were Lie algebras for finite Coxeter groups (at least the crystallographic ones), why not also for the Euclidean ones?" [44]. 1 1 When he mentioned this question to his adviser, Wonenburger, she directed him to Chapter 7 of Jacobson's book [25] , what he called "a wonderful piece of intuition on her part" [44].
By 1966, Moody had answered his question in light of this intuition in his doctoral dissertation. He announced his thesis results in a 1967 article in the Bulletin of the American Mathematical Society communicated on 3 October 1966 [40]. There, he sketched the classification of, as well as the structure theory for, what are now termed "affme Kac-Moody Lie algebras." Following the path that Jacobson had laid out in Chapter 7 of [25], Moody presented his algebras in terms of generators and relations that corresponded to what he called the "generalized Cartan matrices" of type 2; those of type 1 were simply the usual finite Cartan matrices. Using the notational scheme Coxeter had developed for the non-affine context [ 12, p. 142) in the setting of his generalized Cartan matrices of affine type (see note 13 below), Moody not only effected a classification that drew crucially from both Coxeter's Regular Polytopes [ 13) and his Generators and Relations for Discrete Groups [ 12), but also defined the Weyl groups associated with the new algebras. (He would change his notation in later papers.)
Moody followed his announcement with substantial treatments in 1968 of "A New Class of Lie Algebras" [42] and then again in 1969 of "Euclidean Lie Algebras" [39] that provided complete and detailed proofs of his new results. As he explained, these "two papers [are] devoted to the study of certain types of Lie algebras (generally infinite-dimensional) which are constructed from matrices (called
1 1The finite Coxeter groups are a slightly broader class of groups than the finite Weyl groups defined in the preceding section. Finite Weyl groups are thus finite Cox
eter groups, specifically, the so-called crystallographic finite Coxeter groups Moody refers to here.
54 THE MATHEMATICAL INTELLIGENCER
generalized Cartan matrices) closely resembling
Cartan matrices" [42, p. 2 1 1] . In [42], he "con
struct[ed] the Lie algebras, derive[d] their basic
properties, and construct[ed] a symmetric in
variant form on those Lie algebras derived from
the so-called symmetrizable generalized Cartan
matrices" [42, p. 21 1] . After showing that these
algebras are almost always simple, he turned to
the subclass of what he called "Euclidean Lie al
gebras"12 and classified those as well [42, pp.
226-229]. Today, the algebras in the broader
class are known as "Kac-Moody Lie algebras,"
while those in the subclass are termed "affine
Kac-Moody Lie algebras."13 Moody focused in
on the latter more tightly in [39], completing the
classification proof begun in [42] and providing
realizations for the newly named Euclidean Lie
algebras.
Also in the mid-1960s but a half a world away,
another student, Victor Kac, was working at
Moscow State University under the direction of
E. B. Vinberg. Kac had gone to Moscow State as Victor Kac and his advisor, E. B. Vinberg.
an undergraduate in 1960 and had begun attend-
ing the Lie groups seminar that Vinberg ran jointly with A.
L. Onishchik as early as his second year. Vinberg was an ac
tive and a talented mentor, guiding Kac even as an under
graduate to the question of generalizing compact Lie groups
in the same way that Coxeter groups generalize finite W eyl
groups. By 1965, Kac had earned his bachelor's and master's
degrees and had begun working in earnest toward his doc
torate. In the meantime, Jacobson's Lie Algebras [25] had
come out in Russian translation in 1964; Vinberg had pointed
out the construction in Chapter 7; and Kac and Vinberg had
recognized the implications of Jacobson's construction on
the generalization Kac had worked on for his undergraduate
diploma. Kac and Vinberg began working on this new class
of algebras-what would come to be called Kac-Moody Lie
algebras-beginning in the fall of 1965. By 1966, Vinberg had
proposed that Kac work on the following thesis problem: to
"find a classification of simple infinite-dimensional Lie alge
bras that would include the algebras from Jacobson's book
and the Cartan (type] Lie algebras" [27].
By 1967, Kac had proven his main results and, like Moody,
had given a preliminary announcement of them in print.
Kac's note, "Simple Graduated Lie Algebras of Finite
Growth," was originally communicated in Russian on 7 July
1967 to the journal Functional Analysis and Its Applications [31] . Not surprisingly given its title, this paper did not
emphasize what would later be called the affine Kac-Moody
Lie algebras, rather it stressed their role in his classification
of simple graded Lie algebras of finite growth14 at the same
time that it presented and named the affme diagrams. 15 With
this announcement out of the way, however, "it took the
whole of 1967 to write down the detailed account" [27]. As Kac described it, "[e]very week I came to Vinberg's home to
show him the progress in writing and at least half of it would
be demolished by him each time" [27]. By 1968, however, the
dissertation was complete; Kac had earned his Ph.D.; and he
had published, again in Russian, both an announcement of
another main result [28]-how to use the results of the first
note [31] to classify symmetric spaces-and a fuller account
of his thesis research as a whole [32].
The latter paper, "Simple Irreducible Graded Lie Al
gebras of Finite Growth, appeared in Izvestiya. A work
of enormous scope and breadth, it treated the so-called
"algebras of Cartan type" which had originally been stud-
1 2The term, in fact, was well chosen since the algebras have a natural finite root system (possibly non-reduced) attached to the root system of the algebra. Moody jus
tified his choice of terminology explicitly in [39, p. 1 433]: "The use of the adjective ' Euclidean' in the present context comes from the fact that the Weyl group of a Eu
clidean Lie algebra is isomorphic to the Coxeter group with corresponding diagram . . . which in turn is the group generated by the reflections in the sides of a Eu
clidean simplex."
13Just as the structural information about the finite-dimensional simple Lie algebras is totally contained in their associated finite Cartan matrix, so the generalized Car
tan matrices encode the structural information about the Kac-Moody Lie algebras. The generalized Cartan matrix for a Kac-Moody Lie algebra satisfies (a) above; the
generalized Cartan matrix for an affine Kac-Moody Lie algebra satisfies (a) together with one additional condition. namely. there exist 0 :s d1, d2, . . . , d1 E 7l. (not all
zero) such that (Aij) tirnes the I x I column vector of the d/s yields the zero vector. Matrices of the latter type are called "generalized Cartan matrices of affine type" or
simply "affine Cartan matrices. "
,.,Finite growth" is a key technical condition that bounds the size of the root spaces of the algebra.
15As would be expected, Kac's terminology here differed from Moody's. Kac's notation was close to that standard today (see, for example, [30]) and conveyed the al
gebra that one must start with in order to construct a realization of the algebra in question. Moody's notation, on the other hand, emphasized the algebra's related fi
nite root system. For a comparison of the two different notations employed by Kac and Moody, see [1 , p. 678].
VOLUME 24. NUMBER 1. 2002 55
AP)
A�1l (£ > 2)
B?) (£ > 3)
cPl (£ > 2)
D�l) (£ > 4)
G�l)
pjll
E�l)
e{==9e 1 1 1 �� 1 1 • ! 1
• 1 2 2
e-=7e--1 2
• ! 1 • 1 2 2
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• • 1 2
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p • 3 2
1 1 --e-=7• 2 2
e( • 2 1 L 2 1
• 2
• 1 • • • ! 2
• 1 2 3 4 3
• • • • • 1 2 3 4 5 Figure 2. Dynkin diagrams, the infinite-dimensional setting.
ied by Elie Cartan in relation to pseudogroups. At the
time Vinberg set Kac to work on his thesis problem, Vic
tor Guillemin and Shlomo Sternberg [ 19], as well as
Isadore Singer and Sternberg [47], were doing pertinent,
related research, and Vinberg recommended that Kac
read their papers. The only problem was that Kac did not
really read English at that point, so the going was tough,
and Kac was making little progress [27). A chance meet
ing with I. M. Gelfand in the spring of 1966 turned things
around, however. Gelfand gave Kac numerous reprints
and told him to study them carefully. One of them, "Sur
les corps lies aux algebres enveloppantes des algebres de
56 THE MATHEMATICAL INTELLIGENCER
• 2
! 3 6
A�2)
A(2) (£ > 2) 2£ -
A���1 (£ > 3)
D(2) (£ > 2) £+1 -
E�2)
Di3l
• 1
• • 4 2
�· 2 1 �----2 2
• ! 1 • 1 2 2
�--1 1 • • -�· 1 2 3
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2
ec • 2 1
---�· 2 1 • • 1 1
• 1
Lie" by Gelfand and A. A. Kirillov [ 17], presented the no
tion of growth of an algebra in a Lie-theoretic setting.
This struck a loud chord. As Kac put it, "(i]n a split sec
ond it had become clear what I should be doing: I should
classify simple Lie algebras of finite Gelfand-Kirillov di
mension!" [27]. This notion of finite Gelfand-Kirillov di
mension (or finite growth), like the affine Kac-Moody Lie
algebras, proved central to the classification he ulti
mately gave in [32] .
Thus, beginning in the mid-1960s, Moody in Canada and
Kac in Russia worked simultaneously and independently to
extend the construction Jacobson had presented in Chap-
ter 7 of his book [25] to the infinite-dimensional setting.
Both dropped axiom ( y) and recognized that axiom (/3) was
expendable as well, and, quite remarkably, both were led
to study the particular subclass of algebras associated with
the affine diagrams in Figure 2. Still, while this construc
tion represented by no means the main thrust of their early
work, 16 both singled out the particular subclass now
termed "the affine Kac-Moody Lie algebras," giving realiza
tions and obtaining deep structural information about
them. The import of this aspect of their work was not im
mediately recognized, however.
An Area Is Born: Kac-Moody Lie Algebras
Moody and Kac both followed their initial series of papers
with some additional research on their new class of algebras,
although neither worked solely on such questions. In 1969,
Kac published a paper in Russian on "Some Properties of
to a certain product over the positive roots, Macdonald's
affme analogue involved what he termed "an extra factor"
[37, p. 92] on the product side of the equation. Specializing
his formula to specific affine root systems unexpectedly
yielded classical number-theoretic identities such as Jacobi's
triple product identity from the theory of theta functions and
Ramanujan's T-function (see [37, pp. 91-95] for an overview
of the results). Thus, Macdonald had found a natural con
text within the theory of affine W eyl groups and their root
systems for a number of previously isolated number-theo
retic results. As he also noted, the classification of affine root
systems that he presented in the fifth section of his paper
was identical to that given by Moody in [42] and [39] in the
context of Euclidean Lie algebras [37, p. 94] (Fig. 2).
It did not take Kac and Moody long to pick up on Mac
donald's results and to recognize their implications for the
new class of algebras they had discovered in their thesis
Contragredient Lie Alge
bras" [33] (his name at the
time for the more general
class of the new algebras), as
well as a short note in Russ
ian and in English transla
tion on "Automo:rphisms of
Finite Order of Semisimple
Lie Algebras" [26]. The latter
Beginning i n the m id- 1 960s,
Moody in Canada and Kac
in Russia worked s imulta
neously and independently .
research. Again, they made
their discoveries indepen
dently. Kac submitted his
original note on "Infinite
Dimensional Lie Algebras
and Dedekind's 17-Function"
in Russian on 14 February
1973; it appeared both in
Russian and in English
provided an application of affine Kac-Moody Lie algebras to
the theory of finite-dimensional simple Lie algebras. The
years from 1969 to 1971 found him principally embroiled in
the theory of finite-dimensional Lie algebras of characteris
tic p, however. Moody, on the other hand, analyzed the "Sim
ple Quotients of Euclidean Lie Algebras" in a paper in 1970
[43] but also worked on other Lie-theoretic topics.
Beyond Kac and Moody, the algebras had generated a
bit of interest almost exclusively within a small circle of re
searchers centered on Moody's adviser, Maria Wonen
burger. She as well as her two students, Stephen Berman
and Richard Marcuson, produced a number of papers in the
early 1970s in which they developed the theory further, al
though this hardly constituted a groundswell of activity. 17
The new class of algebras was interesting enough, but at
this point it had no natural context.
That changed after 1972 and the publication of Ian Mac
donald's surprising paper on "Affine Root Systems and
Dedekind's 17-Function" [37]. In that work, Macdonald em
ployed the algebraic and combinatorial tools afforded by the
affme Weyl group and its corresponding root system to prove
an analogue in this affme setting of the so-called W eyl de
nominator formula [37, p. 1 16]. 18 Whereas Weyl's formula
equated a certain sum over the elements in the Weyl group
translation the following year [29]. In just over two short
pages, Kac not only gave a natural explanation of Mac
donald's "extra factor" in terms of the imaginary roots of what was not yet called the affine Kac-Moody Lie algebra,
but he also sketched the proof of his character formula for
the more general class of Lie algebras defined by sym
metrizable generalized Cartan matrices. A result truly re
markable for its generality, the now so-called Weyl-Kac
character formula would soon deeply influence develop
ments in the area. The immediate result of Kac's note, how
ever, was to place Macdonald's number-theoretic results
squarely and naturally in the context of the theory of the
new algebras he and Moody had discovered and developed.
Almost simultaneously, Moody also recognized how to
interpret Macdonald's "extra factor," and submitted a pa
per on "Macdonald Identities and Euclidean Lie Algebras"
to the Proceedings of the American Mathematical Society on 13 November 1973 [41] .
As Moody explained,
A feature of the Macdonald identities . . . is the appear
ance of a factor . . . whose description is quite awkward
and whose meaning is very obscure. Our intention here
is to show that it is possible to place the identities in the
16 1n 1 968, 1. L. Kantor presented a construction of infinite-dimensional simple graded Lie algebras that is similar in spirit to that of Kac and Moody [34]. Unlike Kac and
Moody, however, Kantor did not focus in on the all-important affine Kac-Moody Lie algebras.
1 7See [2] for the references.
1 8The affine Weyl group is an immediate generalization of the finite Weyl group (as defined in the second section above); it is defined in terms of the affine Carlan ma
trices as opposed to the finite Carlan matrices. The affine Carlan matrices are recoverable from the sixteen types of Dynkin diagrams in Figure 2, just as the finite Car
tan matrices are recoverable from the Dynkin diagrams in Figure 1 . It was precisely this correspondence that prompted Moody's initial choice of terminology for the
affine Kac-Moody l..ie algebras. Compare note 1 2 above.
VOLUME 24, NUMBER 1 , 2002 57
context of Euclidean Lie algebras, whereupon the meaning of [the factor] becomes obvious and the identities take on a simpler and even more beautiful appearance.
In their new form, the identities give a marvellous relationship between the Weyl group, the root system, and the dimensions of the root spaces. It is not unreasonable to expect that similar identities may hold for all the Lie algebras determined by arbitrary Cartan matrices [41, p. 43] .
B y the time Moody received the proof sheets of his paper, he had seen the Russian version of Kac's note [29] and had recognized that Kac had, in fact, proven the latter result when the Cartan matrices were symmetrizable. Moody acknowledged that Kac's "work establishes the Macdonald identities . . . by techniques which are intrinsically related to the corresponding Lie algebras" [41, p. 51] . Thus, Moody, like Kac, realized that here was the context-and a fascinating one at that-that these Lie algebras had lacked. Now all they needed was a name, but Moody seemed to sense that as well.
Although he used his former nomenclature "Euclidean Lie algebras" in the title of [41 ] , Moody coined a fanciful new term for the wider class of all Kac-Moody Lie algebras in the same paper, "heffalump Lie algebras" [41 , p. 44). Moody had been poring through the densely packed pages of Freudenthal and de Vries's book, Linear Lie Groups, at the same time that he had been reading A. A. Milne's classic, The World of Pooh, to his children. In Milne's story, he read of the mysterious heffalump, an elusive elephant-like creature that Pooh and Piglet try unsuccessfully to catch. In Freudenthal and de Vries, he encountered the so-called "hef-triples," derived from the fact that the usual basis for the smallest finite-dimensional simple Lie algebra \3{2 (C) is denoted h, e, j [ l5, p. 497]. For fixed i, the ei, ji, hi in (2) above are hef-triples in the sense of Freudenthal and de Vries. Thus, the hef-triples also arise in the construction that mimics Jacobson's but that drops the finiteness condition ( y); that is, they also arise in the context of the new class of algebras discovered by Moody and Kac. Since these new algebras are usually infmite, they are elephantine; since they were little understood at the time, they seemed elusive. They had the same characteristics as that heffalump that had evaded Pooh and Piglet [38).
Although Moody's terminology did not catch on, 19 the algebras that Kac and Moody had discovered attracted increasing attention following their linkage to Macdonald's results. In particular, Howard Garland at Yale and James Lepowsky then at the Institute for Advanced Study recast Kac's proof of the character formula [29] in a homological setting in their 1976 paper on "Lie Algebra Homology and the Macdonald-Kac Formulas" [ 16). As they put it, "[t]he main purpose of the present paper is to generalize B. Kostant's fundamental result . . . on the homology (or cohomology) of nilradicals of parabolic subalgebras in certain modules, from
19See [3], however, for at least one paper that adopted it.
58 THE MATHEMATICAL INTELLIGENCER
(finite-dimensional) complex semisimple Lie algebras to the Kac-Moody Lie algebras defined by symmetrizable Cartan matrices" [ 16, p. 37) . This was the first use of the term "KacMoody Lie algebras" in the literature. Within a decade, it would not only be universally adopted, it would come to define a vibrant and burgeoning subfield of mathematics with deep and surptising physical applications [ 14).
Kac and Moody independently discovered their new class of algebras during the course of their doctoral research. Both drew on groundbreaking work of Chevalley and Harish-Chandra as filtered through Jacobson's influential textbook. Both recognized the special nature of the affine Kac-Moody Lie algebras and gave the standard realizations of them. Both appreciated the implications that their algebras held for Macdonald's results. Neither, however, would likely have predicted in the mid-1970s that the algebras they had isolated would so quickly define an area of such spectacular growth and influence in both physics and mathematics. Still less would they have suspected the kudos the field would elicit. Their algebraic work led to deep results in physics and won for them the prestigious
Wigner Medal in 1994; it also paved the way for the definition and development of vertex operator algebras that
won the highly prized Fields Medal for Richard Borcherds in 1998. Kac and Moody sensed the importance of affine Kac-Moody Lie algebras early on. The centrality of these algebras in both mathematics and physics attests to the power of that intuition.
REFERENCES
[1] Bruce N. Allison, Stephen Berman, Yun Gao, and Arturo Pianzola,
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A U T H O R
STEPHEN BEAMAN
Department of Mathematics and Statistics
University of Saskatchewan
Saskatoon, SK S7N 5E6
Canada
e-mail: berman@snoopy.usask.ca
Stephen Berman got his undergraduate degree from Worcester
Polytechnic Institute, and his Ph.D. in 1 97 1 from Indiana Univer
sity under the supervision of Maria J. Wonenburger. Since that time
he has been on the faculty of the University of Saskatchewan. His
mathematical interests are in infinite-dimensional Ue theory, rep
resentation theory, and vertex operator algebras. His hobbies in
clude playing the guitar and T'ai Chi Ch'uan.
KAREN HUNGER PARSHALL
Departments of History and Mathematics
University of Virginia
Charlottesville, VA 22904-4 1 37
USA
e-mai l : khp3k@virgin ia.edu
Karen Parshall, alter receiving a B.A. (1 977) in French and mathe
matics and an M.S. (1 978) in mathemat ics, both at the University
of Virgin ia, has worked i n history of mathematics. Her doctoral su
pervisors at the University of Chicago were Yitz Herstein in math
ematics and Allen G. Debus in history of science. She has been
on the Virginia faculty since 1 988. She is currently working on a bi
ography of James Joseph Sylvester, the subject of one of her sev
eral lntelligencer articles (see vol. 20, no . 3, pp. 35-39).
Berman and Parshall both also enjoy canoeing. In fact, it was on a canoe trip in northern Saskatchewan with their partners, while enjoy
ing the northern lights, that they got the idea of writing the present paper.
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VOLUME 24. NUMBER 1. 2002 59
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60 THE MATHEMATICAL INTELLIGENCER
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[52] Ernst Witt, Spiegelungsgruppen und Aufzahlung halbeinfacher Liescher Ringe, Hamburger Abhandlungen 14 (1 941 ) , 289-322.
l'iilfW·\·1·1 David E. Rowe , Editor I
On the Myriad Mathematica l Trad it ions of Ancient Greece David E. Rowe
Send submissions to David E. Rowe,
Fachbereich 1 7 - Mathematik,
Johannes Gutenberg University,
055099 Mainz, Germany.
To exert one's historical imagina
tion is to plunge into delicate de
liberations that involve personal judg
ments and tastes. Historians can and
do argue like lawyers, but their argu
ments are often made on behalf of an
image of the past, and these historical
images obviously change over time.
Why should the history of mathemat
ics be any different?
When we imagine the world of an
cient Greek mathematics, the works of
Euclid, Archimedes, and Apollonius
easily spring to mind. Our dominant
image of Greek mathematical tradi
tions stresses the rigor and creative
achievement that are found in texts by
these three famous authors. Thanks to
the efforts of Thomas Little Heath, the
English-speaking world has long en
joyed easy access to this trio's major
works and much else besides. Yet our
conventional picture of Greek mathe
matics has drawn on little of this plen
tiful source material. Our image of
Greek geometry, as conveyed in math
ematical texts and most books on the
history of mathematics, has tended
to stress the formal structure and
methodological sophistication found in
a handful of canonical works-or,
more accurately, in selected portions
of them. Even the first two books of
Euclid's Elements, which concern the
congruence properties of rectilinear
figures and culminate in theorem II 14
showing how to square such a figure,
have often been trivialized. Many writ
ers have distilled their content down to
a few definitions, postulates, and ele
mentary propositions, intended merely
to illustrate the axiomatic-deductive
method in classical geometry.
Talk of the origins of Greek mathe
matics shows similar selectivity. The
discovery of incommensurables, though
shrouded in mystery, presumably took
place around the time of Plato's birth.
Two younger contemporaries, Theaete
tus and Eudoxus, both of whom had
ties with the Academy, are credited
with having developed theories that
bear on this problem. These were the
basis for the mature theories found in
Euclid's Elements: Theaetetus's classi
fication scheme for ratios of lines ap
pears in Book X, the longest and most
technically demanding of the thirteen
books, whereas Book V presents Eu
doxus's general theory of proportions,
which elegantly skirts the problem of
representing ratios of incommensu
rable magnitudes by providing a gen
eral criterion for determining when
two ratios are equal (Definition V.5). A
standard picture of the activity that led
to this work has a group of mathemati
cians huddled over a diagram at Plato's
Academy during the early fourth cen
tury. Some of these geometers have fa
miliar names, and a few even appear in
Plato's Dialogues, which contain sev
eral vivid scenes and vital clues for his
torians of mathematics. A few of its pas
sages have provided some of the most
tantalizing tidbits of information that
have come down to us.
Particularly famous is the pas
sage in Plato's Theaetetus where the
young mathematician recounts how
his teacher, Theodorus, had managed to
prove the irrationality of the sides of
squares with integral non-square areas,
but only up to the square of area 17.
Given that Theaetetus is credited with
having solved this problem on his way
to developing the massive theory of ir
rational lines that received its final form
in Book X of Euclid's Elements, the sig
nificance of the historical events Plato
alludes to in this passage has long been
clear. Little wonder that experts like the
late Wilbur Knorr were tempted to tease
out of it as much as they could, begin
ning with the obvious question: why did
Theodorus stop with the square of area
17? Knorr and numerous others have of
fered ingenious speculations about what
went wrong with Theodorus's proof.
Needless to say, such efforts to recon
struct Theodorus's argument on the ba
sis of the meager remarks contained in
the Platonic passage are driven by math
ematical, not historical imagination. A
© 2002 SPRINGER· VERLAG NEW YORK. VOLUME 24. NUMBER 1. 2002 61
mundane historical interrogation of the
famous passage leads to quite a differ
ent thought. What if Theodorus simply
gave up after finding separate proofs for
the earlier cases? Maybe the number 17
had no special significance at all!
For David Fowler, these and other
sources raised, but did not answer, a
related historical question: how did the
geometers of Plato's time (427?-347?)
represent ratios of incommensurable
magnitudes? Fowler was by no means
the first to ask this question, but what
interests us here is the way he went
about answering it. He naturally reex
amined the sources on the relevant pre
history. But inquisitive minds have a
way of turning over new stones before
all the old ones can be found, and so
Fowler's inquiry became broader.
What were the central problems that
preoccupied the mathematicians in
Plato's Academy? This world is lost,
but it has left quite a few tempting
mathematical clues, and Fowler makes
the most of them in an imaginative at
tempt to restore the historical setting.
In The Mathematics of Plato 's Academy, he offers an unabashed recon
struction of mathematical life in an
cient Athens, replete with fictional
dialogues. Accepting the limitations
imposed by the scanty sources, he
gives both his historical and mathe
matical imagination free reign, and pro
duces a new picture of mathematical
life in ancient Athens.
Ironically, we seem to know more
about the activities of the mathemati
cians affiliated with Plato's Academy
than we do about those of any other time
or place in the Greek world, even the
museum and library of Alexandria,
where many of the mathematical texts
that have survived the rise and fall of civ
ilizations and empires were first written.
The Alexandrian mathematicians dedi
cated themselves to assimilating and
systematizing the work of their intellec
tual ancestors. But we know next to
nothing about their lives and how they
went about their work Even the famous
author of the thirteen books known to
day as Euclid's Elements remains a
shadowy figure. Was he a gifted creative
mathematician or a mere codifier of the
works of his predecessors? Is it even
plausible that a single human being
62 THE MATHEMATICAL INTELLIGENCER
could have written all the numerous
works that Pappus of Alexandria later
attributed to Euclid? On the basis of in
ternal evidence alone, it seems unlikely
that the Data and the Elements were
written by the same person. But what
about all the other mostly nameless
scholars who surely must have mingled
with Euclid in Alexandria shortly after
Alexander's death? Perhaps our Euclid
was actually a gifted administrator who
worked at the library and headed a re
search group to produce standard texts
of ancient mathematical works. Is it too
farfetched to imagine Euclid as the an
cient Greek counterpart to the twentieth
century's Bourbaki?
But leaving these biographical spec
ulations aside, we can easily agree that
the Elements established a paradigm for
classical Greek geometry, or what came
to be known as ruler-and-compass
geometry. Indeed, synthetic geometry in
the style of Euclid's Elements continued
to serve as the centerpiece of the Eng
lish mathematical curriculum until well
into the nineteenth century. For Anglo
American gentlemen steeped in the
classics, no formal education was com
plete without a sprinkling of Euclidean
geometry. This mainly meant mimicking
an old-fashioned style of deductive rea
soning that many believed disciplined
the mind and prepared the soul to un
derstand and appreciate Reason and
Truth. With David Hilbert's Grundlagen der Geometrie, published in 1899, the
Euclidean style may be said to have
made its peace with mathematical
modernity. Hilbert upgraded its struc
ture and redesigned its packaging, but
most of all he gave it a new modernized
system of axioms. Within this universe
of "pure thought," Greek mathematics
could still retain its honored place. En
shrined in the language of modem
axiomatics, it took on new form in
countless English-language texts that
presented Greek geometry as a wa
tered-down version of Heath's Euclid.
The history of mathematics abounds
with examples of this kind: a good the
orem, so the adage goes, is always
worth proving twice (or thrice), just as
a good theory is one worthy of being
renovated. In the case of an old
warhorse like Euclidean geometry, we
take this for granted. But if mathemati-
cians will never tire of modernizing
older theories, we might still do well to
ask what consequences this activity has
for historical understanding. The re
flection is required most urgently for
Euclid's Elements, a work that has gone
through more shifts of meaning and
context than any other. Reading Euclid
(carefully) had profound consequences
for Isaac Newton, who soon thereafter
immersed himself in the lesser-known
works of ancient Greek geometers. He
emerged a different mathematician, set
on defending the Ancients against Mod
erns like Rene Descartes, who claimed
to have found a methodology superior
to Greek analysis. We need not puzzle
over why Newton wrote his Principia in the language of geometry, once we
understand his strong identification
with what he understood by the prob
lem-solving tradition of the ancient
Greeks. Nothing rankled him more than
Cartesian boasting about how this tra
dition had been supplanted by modem
analysis.
For ourselves, looking from a post
Hilbertian perspective, the question can
be posed like this: If we continue to view
Greek mathematics through the prism of
Euclid's Elements, and to view the Elements mainly as a model of axiomatic
rigor, what effect will this have on our
conception of the more remote past in
which Greek mathematics grew? One of
the more obvious consequences has
been the glorification of the ancient
Greeks at the expense of other ancient
cultures. This theme has been the sub
ject of much bickering ever since the
publication of Martin Bernal's Black Athena. I will not enter this fracas here;
it does suggest, however, that our pic
tures of ancient mathematics are in the
process of change, and this applies to the
indigenous traditions of Greece as well
as to interaction with other cultures.
By accenting the plural in traditions,
I mean to emphasize that there were
several different currents of Greek
mathematical thought. They continued
to flourish in the Hellenistic world and
beyond: we should not imagine Greek
mathematics monolithically, as if a sin
gle mathematical style dominated all
others.
Nor should we overestimate the
unity of Greek mathematics even
within the highbrow tradition of Eu
clid, Archimedes, and Apollonius. In
his Conica and the other minor works,
Apollonius systematically exploits an
impressive repertoire of geometrical
operations and techniques in order to
derive a series of complex metrical the
orems whose significance is often ob
scure. In this respect, his style con
trasts sharply with Euclid's Elements. When we compare the works of
Apollonius and Euclid with those of
Archimedes, whose inventiveness is
far more striking than any single styl
istic element, the contrasts only widen.
Unlike Apollonius, Archimedes appar
ently had little interest in showcasing
all possible variant results merely to
demonstrate his arsenal of techniques.
He was first and foremost a problem
solver, not a systematizer, and many of
the problems he tackled were inspired
by ancient mechanics. Ivo Schneider
has suggested that Archimedes's early
career in Syracuse was probably closer
to what we would today call "mechan
ical engineering" than to mathematics.
Not that this was unusual; practical
and applied mathematics flourished in
ancient Greece, and again in early
modern Europe when Galileo taught
these subjects as professor of mathe
matics at the University of Padua,
which belonged to the Venetian Re
public. Like Venice, Syracuse had an
impressive navy, and we can be fairly
sure that Archimedes spent a consid
erable amount of time around ships
and the machines used to build them.
From these, he must have learned the
principles behind the various mechan
ical devices that Heron and Pappus of
Alexandria would later describe and
classify under the five classical types
of machines for generating power.
Archimedes was neither an atomist
nor a follower of Democritus. Never
theless, the parallels between these two
bold thinkers are both striking and sug
gestive. In one of his flights of fancy,
Archimedes devised a number system
capable of expressing the "atoms" in the
universe. For this purpose he took a
sand grain as the prototype for these
tiny, indivisible corpuscles. Archimedes
must have seen Democritus's atomic
theory as at least a powerful heuristic
device in mathematics. Democritus had
introduced infmitesimals in geometry,
and by so doing had found the volume
of a cone, presumably arguing along
lines similar to the ideas that led
Bonaventura Cavalieri to his general
principle for finding the volumes of
solids of known cross-sectional area.
As is well known, Eudoxus is cred
ited with having introduced the "method
of exhaustion" in order to demonstrate
theorems involving areas and volumes
of curvilinear figures, including the re
sults obtained earlier by Democritus.
Archimedes used the Eudoxian method
with impressive virtuosity, but because
this technique could only be applied after one knew the correct result, he had
to rely first on ingenuity to obtain pro
visional results. His inspiration came
from mechanics. By performing sophis
ticated thought experiments with a fic
titious balance, Archimedes could
"weigh" various kinds of geometrical ob
jects as if they were composed of "geo
metrical atoms" -indivisible slivers of
lower dimension. As he clearly realized,
this mechanical method was a definite
no-no for a Eudoxian geometer, but he
also knew that there was "method" to
this madness, since it enabled him to
"guess" the areas and volumes of curvi
linear figures such as the segment of a
parabola, cylinders, and spheres. As Heath once put it, here we gain a glimpse
of Archimedes in his workshop, forging
the tools he would need before he could
proceed to formal demonstration.
Going one step further, he carried
out thought experiments inspired by a
problem of major importance to the
economic and political welfare of Syra
cuse: the stability of ships. Archimedes's
idealized vessels had hulls whose cross
sections were parabolic in shape, en
abling him to determine the location of
their centers of gravity precisely. Had
he performed a similar service in
seventeenth-century Sweden for King
Gustav Adolfus, the latter might have
been spared from witnessing one of the
great blunders in maritime history: the
disaster that befell his warship, the
Vassa, which flippe<;l over and sank in
the harbor on her maiden voyage. (If you've ever visited the Vassa Museum
in Stockholm, you'll realize that it
wouldn't have taken an Archimedes to
guess that this magnificent vessel was
likely to keel over as soon as it caught
its first strong gust of wind.)
Archimedes's work presumably was
related to his other duties as an advisor
to the Syracusan court, which later
called upon him when the city was be
sieged by the Roman armies of Marcel
lus. Plutarch immortalized the story of
how Archimedes single-handedly held
back the Roman legions with all manner
of strange, terrifying war machines.
These legendary exploits inspired Italian
Renaissance writers to elaborate on
Archimedes's feats of prowess as a military engineer. No longer content with
mechanical contraptions, the new-age
Archimedes devises a system of mirrors
that could focus the sun's rays on the
sails of Roman ships, setting them all
ablaze. These mythic elements reflect
the imaginative reception of Archimedes
during the Renaissance as a symbol of
the power of human genius, a central
motif in Italian humanism. Within the
narrower confines of scientific thought,
the reception of Archimedes's works un
derwent a long, convoluted journey dur
ing the Middle Ages, so that by Galileo's
time they had begun to exert a deep
influence on a new style of mathemat
ics. By the seventeenth century, the
Archimedean tradition had become
strongly interwoven with the Euclidean
tradition, but these two currents were by
no means identical from their inception.
Another major significant tradition
within Greek mathematics can be
traced back to Pythagorean idealism,
which continued to live on side-by-side
with the rationalism represented by
Euclid's Elements. If the Pythagorean
dogma that "all is number" could no
longer hold sway after the discovery
of incommensurable magnitudes, this
does not mean that all traces of
Pythagorean mathematics vanished.
Far from it: we have every reason to
believe that the Pythagorean and Eu
clidean traditions interpenetrated one
another, influencing both over a long pe
riod of time. Euclid's approach to number
theory in Books VII-IX differs markedly
from that found in the Arithmetica of
Nicomachus of Gerasa, who continued
to give expression to the Pythagorean
tradition during the first century A.D.
Still, the distinctive Pythagorean doc
trine of number types (even and odd,
VOLUME 24, NUMBER 1, 2002 63
perfect, etc.) can be found in both Eu
clid and Nicomachus, albeit in very different guises. Thabit ibn Qurra knew both works and assimilated these arith
metical traditions into Islamic mathe
matics. Finding Nicomachus's treatment of amicable numbers inadequate
(Euclid ignores it completely), Thabit
developed this topic further. Al-Kindi
later translated the Arithmetica into
Arabic and applied it to medicine.
These two writers thus helped perpet
uate and transform the Pythagorean
mathematical tradition within the
world of Islamic learning.
Taking Pythagorean cosmological
thought into account, we seen an even
deeper interpenetration of mythic elements into the Euclidean tradition. For
Plutarch, a writer whose imagination
often outran his critical judgment, Eu
clid's Elements was itself imbued with
Pythagorean lore. He linked Euclid's beautiful Proposition VI 25 with the
creation myth in Plato's Timaeus, a
work rife with Pythagorean symbolism. Plato's Demiurge, the Craftsman of
the universe, fashions his cosmos out of chaos following a metaphysical princi
ple, one that Plutarch identified with the
orem VI 25: given two rectilinear figures,
to construct a third equal in area to the
first figure and similar to the second. In other words, Euclid's geometrical craftsman must transform a given quantity of
matter into a desired form. But we need no Plutarchian wings
of imagination to see that Euclid's Elements contain numerous and striking
allusions to Pythagorean/Platonic cosmological thought, as noted by Proclus
and other commentators. The theories
of constructible regular polygons and polyhedra appear in Books N and XIII,
respectively, thereby culminating the
first and last major structural divisions
in the Elements (Books I-N on the
congruence properties of plane figures; Books XI-XIII on solid geometry). In
both cases, the figures are constructed
as inscribed figures in circles or spheres, the perfect celestial objects
that pervade all of Greek astronomy
and cosmology. Perhaps most striking of all, in Book XIII, which ends by prov-
64 THE MATHEMATICAL INTELLIGENCER
ing that the five Platonic solids are the
only regular polyhedra, Euclid deter
mines the ratio of the side length to the radius of the circumscribed sphere ac
cording to the classification scheme pre
sented in Book X for incommensurable
lines. This body of mathematical knowl
edge shows its connection with the doc
trine of celestial harmonies, an idea
whose origins are obscure, but which un
doubtedly stems from Pythagoreanism.
The doctrine that the heavens pro
duce a sublime astronomical music through the movements of invisible
spheres that carry the stars and planets continued to ring forth in the works of
Plato and Cicero. Johannes Kepler went
further, proclaiming in Harmonice Mundi (1619) the underlying musical,
astrological, and cosmological signifi
cance of Euclid's Elements. For him, Book N, on the theory of constructible
polygons, contained the keys to the plan
etary aspects, the cornerstone of his "scientific" astrology. Historians of sci
ence have long overlooked the inspira
tion behind Kepler's self-acknowledged magnum opus from 1619, preferring in
stead to emphasize his "positive contri
butions" to the history of astronomy, namely Kepler's three laws. Few seem to have been puzzled about the connec
tion between these laws and Kepler's
cosmological views as first set forth in
Mysterium Cosmographicum (1596),
where he tries to account for the dis
tances between the planets by a famous
system of nested Platonic solids. Kepler
published his first two laws (that the
planets move around the sun in elliptical orbits, and that from the sun's posi
tion they sweep out equal areas in equal times) thirteen years later in Astronomia Nova (1609), which presents the astronomical results of his long struggle to
grasp the motion of Mars. The third law
(that for all planets the ratio of the
square of their mean distance to the sun to the cube of their period is the same
constant) only appeared another ten
years later in Harmon ice Mundi. Unlike the first two astronomical laws, the third had a deeper cosmic significance for Ke
pler, who never abandoned the cosmological views he advanced in 1596. In-
deed, for him the third law vindicated
his cosmology of nested Platonic solids by revealing the divine cosmic har
monies that God conceived for this system as elaborated by Kepler in Book V
of Harmonice Mundi. Kepler knew Euclid's Elements per
haps better than any of his contemporaries, and his imagination ran wild with
it in Harmonice Mundi. Like so many
early moderns, he saw his work as the
continuation of a quest first undertaken
by the ancient Greeks. Kepler believed
that the Ancients had already discovered
deep and immortal truths, none more
important than those found in the thir
teen books of the Elements. And since
truth, for Kepler, meant Divine Truth, he
saw his quest as inextricably interwoven
with theirs. His historical sensibilities
were shaped by a profound religious
faith that led him to identify his Christ
ian God with the Deity that pagan
Greeks described in the mythic language
of Pythagorean symbols. We gasp at the gulf that separates our post-historicist
world from Kepler's naive belief in a
transcendent realm of bare truth. We
can only marvel in the realization that it
was Kepler's sense of a shared past that
enabled him to compose his Harmonice Mundi while contemplating the truths
he thought he saw in the works of an
cient Greek writers.
These brief reflections suggest some
broader conclusions for the history of
mathematics: that mathematical knowl
edge, as a general rule, is related to var
ious other types of knowledge, that its
sources are varied, and that the form and content of its results are affected by the
cultures within which it is produced.
Those who have produced mathematics have done so in quite different societies,
within which these producers have had
quite varied functions. Western mathematics owes much of course to ancient
Greek mathematicians, but even within
the scope of the Greeks' traditions we
encounter considerable variance in the styles and even the content and pur
poses of their mathematics. For this reason, we should avoid the temptation to
reduce Greek mathematics to one dominant paradigm or style.
ATHANASE PAPADOPOULOS
Mathematics and Mus ic Theory: From Pythagoras to Rameau
usic theory is a wide and beautiful subject, and some basic math-
ematical ideas are inherent in it. Some of these ideas were intra-
duced in music theory by mathematicians, and others by musi-
cians with no special mathematical skill. This paper describes some
of the connections between music theory and mathemat
ics. The examples are chosen mainly from the works of
Pythagoras and of J. Ph. Rameau, who both were impor
tant music theorists, although the former is usually known
as a mathematician, and the latter as a composer.
Before going into the works of Pythagoras and Rameau,
I present, in the next section, a summary history of the re
lation between music and mathematics.
A Few Historical Markers
I start with Greek antiquity.
It is well known that the schools of Pythagoras, Plato, and
Aristotle considered music as part of mathematics, and a
Greek mathematical treatise from the beginning of our era
would usually contain four sections: Number Theory, Geom
etry, Music, and Astronomy. This division of mathematics,
which has been called the quadrivium1 (the "four ways"),
lasted in European culture until the end of the middle ages
(ca. 1500). One can see bas-reliefs and paintings represent
ing the four branches of the quadrivium on the walls or pil
lars of cathedrals in several places in Europe (see for instance
the pictures in [1 ]). The situation changed with the Renais
sance, when theoretical music became an independent field,
but strong links with mathematics were maintained. 2 Several important mathematicians of the seventeenth
and eighteenth centuries were also music theorists. For in-
1This terminology is due to Boethius (ca. 480--524 AD), who worked on the translation and the diffusion of Greek science and philosophy in the Latin world. He is re
sponsible in particular for a Latin translation and a commentary of the mathematical treatise of Nichomachus. Boethius considered the study of the quadrivium to be
a prerequisite for philosophy, and this idea was at the basis of Western European curricula for almost ten centuries.
"The AMS subject classification i1as a section called Astronomy, but none called Music.
© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 1, 2002 65
stance, the first book that Rene Descartes wrote is on mu
sic (Compendium Musicae, 1618). Marin Mersenne wrote
several treatises on music, among them the Harmonicarum Libri (1635) and the Tmite de l 'harmonie universelle (1636), and he had an important correspondence on that
subject with Descartes, Isaac Beekman, Constantijn Huy
gens, and others. John Wallis published critical editions of
the Harmonics of Ptolemy (2d c. AD), of Porhyrius (3d c.
AD), and of Bryennius (a Byzantine musicologist of the four
teenth century). Leonhard Euler published in 173 1 his Tentamen novae theoriae musicae ex certissimis harmoniae principiis dilucide expositae. Jean d'Alembert wrote in
1 752 his Elements de musique theorique et pr-atique suivant les principes de M. Rameau and in 1754 his Rejlexions sur la musique; and there are many other examples.
Music theory as well as musical composition requires a
certain abstract way of thinking and contemplation which
are very close to mathematical pure thought. Music makes
use of a symbolic language, together with a rich system of
notation, including diagran1s which, starting from the
eleventh century (in the case of Western European music),
are similar to mathematical graphs of discrete functions in
two-dimensional cartesian coordinates (the x-coordinate
representing time and the y-coordinate representing pitch).
Music theorists used these "cartesian" diagrams long be
fore they were introduced in geometry. Musical scores from
the twentieth century have a variety of forms which are
close to all sorts of diagrams used in mathematics. Besides
abstract language and notation, mathematical notions like
symmetry, periodicity, proportion, discreteness, and conti
nuity, among others, are omnipresent in music. Lengths of Until well into the Renaissance, the term "musician" re
ferred to music theorists rather than to mu-
sical performers. Research and teaching in
music theory were much more prestigious
occupations than musical composition or
performance. Some famous mathemati
cians were also composers or performers,
but this is another subject. 3 J. Ph. Rameau, who is certainly the great
est French musicologist of the eighteenth
century, wrote in his Traite de l'harmonie reduite a ses principes naturels (1722):
La musique est une science qui doit avoir des regles certaines; ces regles doivent etre tirees d'un principe evident, et ce principe ne peut guere nous etre connu sans le secours des mathematiques. Aussi dois-je avouer que, nonobstant toute l'experience que je pouvais m'etre acquise dans la musique pour l'avoir pratiquee pendant une assez longue suite de temps, ce n'est cependant que par le secours des mathematiques que mes idees se sont debrouillees, et que la lumiere y a succede a une certaine obscurite dont je ne m 'apercevais pas auparavant.
Music is a science which must have deter
mined rules. These rules must be drawn from
a principle which should be evident, and this principle cannot be known without the help
of mathematics. I must confess that in spite
of all the experience which I have acqllired in
music by practising it for a fairly long period,
it is nevertheless only with the help of math
ematics that my ideas became disentangled
and that light has succeeded to a certain dark
ness of which I was not aware before.
3For instance. Pythagoras. according to his biographers, be·
sides being a geometer, a number-theorist, and a musicol-
p R E1 F A C E. lui efl natNrcUe , •fin qru i'effirit en confoi�c lu proprim", aujfi focilcment que /'oreillc les fent.
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ogist , was a composer. and he also played several instru- Figure 1 . Glowing words from Rameau's Traite de /'harmonia reduite a ses principes
ments; see e.g., [7]. Chapter XV, p. 32. naturels.
66 THE MATHEMATICAL INTELLIGENCER
musical intervals, rhythm, duration, tempi, and several
other musical notions are naturally expressed by num
bers. The mathematical use of the word "harmonic"
(for instance in "harmonic series" or "harmonic analy
sis") has its origin in music theory. The composer Mil
ton Babbitt, who taught mathematics and music the
ory at Princeton University, writes in [2] that a musical
theory should be "statable as a connected set of axioms, definitions and theorems, the proofs of which
are derived by means of an appropriate logic."
It is important to realize that there are contribu
tions in both directions. On the one hand, mathe
matical language and mathematical ideas have shaped the language and the concepts of music the
ory. This is illustrated in the work of Rameau dis
cussed below, but there are several other instances.
For example, Milton Babbitt uses group theory and
set theory in his theoretical musical teaching and in
his compositions. Olivier Messiaen speaks of "sym
metric permutations." Some pieces of Iannis Xenakis
are based on game theory, others on probability the
ory; and so on. 4 On the other hand, questions and
problems arising in music theory have constituted, at
several points in history, strong motivation for in
vestigations in mathematics (and of course in
physics). For example, phenomena like the produc
tion of beats or the production of the harmonic fre
quencies were noticed and discussed by music theo
rists several decades before they were explained by
mathematical and physical theories. Some of the the
ories developed in the seventeenth century by Wal
lis, J. Sauveur, and others were essentially motivated
by these phenomena. I shall discuss the question of
the harmonic frequencies in the last part of this arti-
I Z. �e cS" I t
� : s . 9
<SO !) ToNvs . 8 o �· DJA:trs:rs
cle. It is also fair to acknowledge that there are in- Figure 2. The hammers of Pythagoras, according to Gafurius (1492).
stances where music theorists have used mathemat-
ical notions in an intuitive manner, before these notions
had been shaped and refined by mathematicians. One such
example is the use of logarithms, also discussed below.
Now let us start from the beginning, that is, with
Pythagoras.
Pythagoras and the Theory of Musical Intervals
Historians of science usually agree that Pythagoras (sixth
c. BC) is at the origin of mathematics as a purely theoreti
cal science. 5 At the same time, Pythagoras is regarded as
the first music theorist (from the point of view of European
music). The major musical discovery of Pythagoras is the
relation of musical intervals with ratios of integers. This is
described by Jamblichus ([7], Chap. XXVI, p. 62) in these
terms: Pythagoras was "reasoning with himself, whether it
would be possible to devise instrumental assistance to the
hearing, which could be firm and unerring, such as the sight
obtains through the compass and the rule." Walking
through a brazier's shop, Pythagoras heard the different
sounds produced by hammers beating an anvil. He realized
that the pitch, that is, the musical note, that was produced
by a particular hammer, depended only on the weight of
the hammer and not on the particular place where the ham
mer hit the anvil, or on the magnitude of the stroke.
Pythagoras realized also that the compass of a musical in
terval between two notes produced by two different ham
mers depended only on the relative weights of the ham
mers, and in particular that the consonant musical
intervals, which in classical Greek music were the intervals
of octave, of fifth and of fourth, correspond, in terms of
4The idea of using mathematical theories in musical composition is not new. Athanasius Kircher, a seventeenth-century mathematician at the Court of Vienna, wrote a
treatise on musicology, Misurgia Universafis (1 622) in which he described a machine, Area Musicarithma, which produces musical compositions based on mathemat
ical structures.
5The theories and results which Pythagoras and his school developed were not intended for practical use or for applications, and it was even forbidden for the mem
bers of the Pythagorean school to earn money by teaching mathematics, and the exceptions confirm the rule: Jamblichus (see [7], Chap. XXIV, p. 48) relates that "the
Pythagoreans say that geometry was divulgated from the following circumstances: A certain Pythagorean happened to lose the wealth that he possessed; and in con
sequence of this misfortune, he was permitted to enrich himself from geometry."
VOLUME 24, NUMBER 1. 2002 67
I I I I ·�fourth� I I I E-<-- fifth I ' <:: octave
Figure 3. The classically "consonant" intervals.
weights, to the numerical fraction 2/1, 3/2, and 4/3, respec
tively. Thus, Pythagoras thought that the relative weights
of two hammers producing an octave is 2/1, and so on. As soon as this idea occurred to him, Pythagoras went home
and performed several experiments using different kinds
of instruments, which confirmed the relationship between
musical intervals and numerical fractions. Some of these
experiments consisted of listening to the pitch produced
by the vibrations of strings that have the same length; he
had suspended the strings from one end and attached dif
ferent weights to the other end. Other experiments involved
strings of different lengths, which he had stretched end-to
end, as in musical instruments. He also did experiments on
pipes and other wind instruments, and all these experi
ments confirmed him in his idea that musical intervals cor
respond in an immutable way to definite ratios of integers,
whether these are ratios of lengths of pipes, lengths of
strings, weights, etc. 6
Theon of Smyrna, in Part 2, Chapter XIII of his mathe
matics treatise [ 12], describes other experiments which il
lustrate this relation between musical intervals and quo
tients of integers. He relates, for instance, that the
Pythagoreans considered a collection of vases, filled par
tially with different quantities of the same liquid, and ob
served on them the "rapidity and the slowness of the move
ments of air vibrations." By hitting these vases in pairs and
listening to the harmonies produced, they were able to as
sociate numbers to consonances. The result is again that
the octaves, fifths, and fourths correspond respectively to
the fractions 2/1, 3/2 and 4/3, in terms of the quotients of
levels of the liquid.
These experiments were repeated and reinterpreted by
the acousticians of the seventeenth century. The ideas and
observations of Pythagoras and his school established the
relation between musical intervals and ratios of integers.
Logarithms
The arithmetic of musical intervals involves in a very nat
ural way the theory of logarithms. For an example, we re
turn for a moment to Jamblichus, who relates in Section
XV of [7] that Pythagoras defmed the tone as the difference between the intervals of fifth and of fourth. (The defmition
may seem circuitous, but it becomes natural if we recall
that the defmitions of musical intervals had to be based on
those of consonant intervals, which are naturally recog
nisable by the ear.) The point now is that the fraction as
sociated to the tone interval is not the difference 3/2 - 413,
but the quotient (3/2)/( 4/3) = 9/8.
It is natural to define the compass of a musical interval
as the number (or the fractions of) octaves it contains.
Thus, when we say that two notes are n octaves apart, the
fraction associated to the interval that they define is 2n. The
definition of the compass can be made in terms of fre
quency, and in fact one usually defines the pitch as the log
arithm in base 2 of the frequency. (Of course, the notion of
frequency did not exist as such in antiquity, but it is clear
that the ancient Greek musicologists were aware that the
lowness or the highness of pitch depends on the slowness
or rapidity of the air vibration that produces it, as explained
in Theon's treatise [12], Chapter XIII.) The relation of mu
sical intervals with logarithms can also be seen by consid
ering the lengths of strings (which in fact are inversely pro
portional to the frequency). For instance, if a violinist (or
a lyre player in antiquity) wants to produce a note which
is an octave higher than the note produced by a certain
string, he must divide the length of the string by two.
Thus, music theorists dealt intuitively with logarithms
long before these were defmed as an abstract mathemati
cal notion. (It was only in the seventeenth century that log
arithms were formally introduced in music theory, by Isaac
Newton, and then by Leonhard Euler and Jacques Lam
bert.) The theory of musical intervals is a natural example
of the practical use of logarithms, an example easily ex
plained to children, provided they have some acquaintance
with musical intervals.
6We must note that the experiment with the hanging weights is considered to be a mistake of Pythagoras, or an extrapolation due to Pythagoras's disciples, or a mis
interpretation of what Pythagoras really said. This mistake was noticed by Vincenzo Galilei (the father of Galileo Galilei). Vincenzo was a most cultivated person, in par
ticular a music theorist and a music composer. He did the experiment with the hanging weights and realized that to produce the intervals of octave, fifth, and fourth,
the ratios of the pairs of weights should be respectively 4/1 , 9/4, and 1 6/9, which are the squares of the numbers which occur in the experiments involving the lengths
of strings. Galilei was proud of that discovery (and of the discovery of a mistake in the theory of Pythagoras), and he published it in his famous musical treatise, the
Oiscorso intorno aile opere de Gioseffo Zarlino. The physical reason behind this fact is that the frequency of a vibrating string, while it is proportional to the length of
the string, is proportional to the square root of the tension. Nonetheless, the relation between musical intervals and ratios of integers is still there, even though it is not
so direct in all cases. We note too that the same experience with the hanging weights is described by Vincenzo's son, Galileo (see [5], p. 98 to 1 1 0).
68 THE MATHEMATICAL INTELLIGENCER
Music in the Mathematical Treatise
of Theon of Smyrna
It is interesting to go through the music theory part of a
mathematics treatise of the classical Greek era. I consider
here the section on Music (Part 2) of Theon's treatise [12]. This section deals with the definition and the combinations
of musical intervals, with proportions, musical units, and
so on. It involves non-trivial arithmetic, and Theon, in this
section, often refers to the discoveries made by Pythago
ras and the Pythagoreans.
The title of Part 2 of Theon's mathematical treatise is "A
book containing the numeric laws of music." In the intro
duction, he says, "Harmony is spread in the world, and of
fers itself to those who seek it only if it is revealed by num
bers." The first part of this sentence, that "Harmony is
spread in the world," has been repeated throughout the
erect them unnatural and a threat to their philosophical sys
tem, based on positive integers. The adjective "irrational"
which they introduced clearly indicates this. It is also well
known that the Pythagoreans wanted to keep the existence
of irrational numbers (the discovery of which is attributed
to Pythagoras himself) a secret. Jamblichus relates in [7] Chapter XXIX (p. 126) that "he who first divulgated the the
ory of commensurable and incommensurable quantities, to
those who were unworthy to receive it, was so hated by
the Pythagoreans that they not only expelled him from their
common association, and from living with them, but also
constructed a tomb for him." The reasons why ancient Greek music used semitones
of 16/15 or 25/24 are certainly related to the fact that these
intervals are acceptable by the ear. But it is also a fact
that the ancient Greek musicologists liked to deal with suages, and it was at the ba
sis of a strong feeling of
cosmic structure and or
der. There are important
philosophical and esoteric
traditions behind this idea,
which led eventually to
explanations of physical
phenomena, like the mo
For the Pythagoreans , deal
ing with i rrational numbers
wou ld have been incom pat i
ble with their ph i losophy.
perparticular ratios derivedfrom 2, 3, and 5, that
is, fractions of the form
(n + 1)/n with numerator
and denominator having
only 2, 3, and 5 as prime
factors. Pythagorean num
ber symbolism is involved
here, but that subject is
tion of planets. Famous adepts and advocates of such tra
ditions include, after Pythagoras himself, Plato, Boethius,
Copernicus, and Kepler (see for instance [8], Book V, where
Kepler gives a relation between the eccentricities of the or
bits of the planets and musical intervals). The second part
of Theon's sentence, that "harmony is revealed by num
bers," has also been repeated throughout the ages, for in
stance in the citation of Rameau mentioned earlier and in
the following citation of Gottfried Wilhelm Leibniz, from
his Principles of nature and of grace (1712): "Musica est exercitium arithmeticae occultum . . . " (Music is a secret
exercise in arithmetic).
Let us look at the treatment of semitones in Theon's
treatise. There are several kinds of semitones used in an
cient Greek music, two of which are the "diatonic semi
tone" and the "chromatic semitone," the values of which
are, respectively, 16/15 and 25/24. One could expect that
there is a semi tone whose value is equal to half of the value
of a tone, in the sense that if we concatenate two such semi
tones, we obtain a tone. This is not the case for any of the
semitones used by the Pythagoreans, however. Indeed, by
the discussion on logarithms above, we know that if the
semitone were half of the tone, then its numerical value
should have been V9!8, which is an irrational number. For
the Pythagoreans, dealing with irrational numbers would
have been incompatible with their philosophy. Theon
writes in §VIII of Part 2 that "one can prove that" the tone,
the value of which is 9/8, cannot be divided into two equal
parts, "because 9 is not divisible by 2." Of course, this is
nonsense: the point is not to divide 9 by 2, but to take the
square root of9/8. Although Pythagoras and his school were
aware of the existence of irrational numbers, they consid-
beyond the scope of this paper. The following is a list of
"useful" musical intervals, which was known to Gioseffo
Zarlino and Descartes:
2/1 octave
3/2 fifth 413 fourth
5/4 major third
6/5 minor third
9/8 major tone
10/9 minor tone
16/15 diatonic semitone
25/24 chromatic semitone
81180 comma of Didymus.
There is a discussion of this list in both [6] and [9]. Many
years after this list was known to music theorists, C. St0rmer
proved that this is a complete list of the superparticular ra
tios derived from the prime numbers 2, 3, and 5 [11 ] .
Scales
Scales are building blocks for musical compositions. (This
is true at least in tonal music, that is, in almost all pre-twen
tieth-century European music.) I shall talk in this section
about the arithmetic of scales, and I remark by the way that
in addition to this arithmetic, there is a more abstract re
lation between scales and mathematics, namely in the con
text of formal languages. Classical musical compositions
are based on scales, fragments of which appear within a
piece in various forms, constituting a family of privileged
sequences of musical motives. This fact has been exploited
VOLUME 24, NUMBER 1 , 2002 69
and systematically generalized in certain twentieth-century compositional techniques (for instance, serial music), which are related to mathematics, but which are beyond the subject matter of this paper.
The major part of post-Renaissance Western European classical music uses a very limited number of scales; in fact, since the general acceptance of the tempered scale in the eighteenth century, there are basically two scales, the major and the minor scale. The tempered scale (the one we play on a piano keyboard), is based on the division of the octave into 12 equal intervals, the unit being the tempered semitone, the value of which is equal therefore to 12\12. Any two major (respectively, minor) tempered scales are translations of each other on the set of pitches. (In musical terms, these translations are called transpositions.) This was not the case in pre-Renaissance music.
In contrast, the theory of harmony in classical Greece included a complicated and very subtle system of scales. Greek mathematical treatises usually contain a description of scales in terms of fractions, with a discussion of the logic behind the definitions. For instance, the scale which is known today as the "scale of Pythagoras" is defined by the following sequence of numbers:
1, 9/8, 81/64, 413, 3/2, 27/16, 243/128, 2.
These numbers can be regarded as representing ratios of lengths of strings, the nth number being the ratio of a pair of strings having the same section and stretched at the same tension, producing the interval between the first and the nth note. Thus, for instance, the interval between the first and the last note in the list is an octave, the interval between the first and the fourth note is a fourth and the interval between the first and the fifth note is a fifth, as expected, since the Pythagorean scale needs to contain these three consonant intervals. The intervals between consecutive notes, except those between the third and the fourth and the seventh and the eighth, have the value 9/8. The intervals which we have excluded have the common value 256/243, which corresponds to another semitone. The scale of Pythagoras sounds approximately, but not exactly, like our tempered major scale. The semitone which is used in our tempered scale, 12V2, is closer to the diatonic semitone, 16/15, than to the other two which we encountered.
tion of the scale of Pythagoras. One starts by assigning the values 2, 3/2, and 4/3, respectively, to the eighth, fifth, and fourth notes in the list. The rest of the values are obtained by an iterative process involving fifths whose values are 3/2
(such fifths are called pur·e fifths). Thus, for example, if we start from the first note (with value 1) and concatenate two pure fifths, we obtain an interval of ninth, with value 3/2 X 3/2 = 9/4, which is greater than 2 (as expected, since this interval is larger than an octave). To come back inside our octave, we divide by two, obtaining the value 9/8. In the same way, the value 27/16 is found as (3/2)3 divided by 2,
and so on. Unfortunately the process gives an infinite number of notes, but it is reasonable to stop after the octave has been divided into these seven intervals.
The scale of Pythagoras has beautiful properties. One is that all fifths and all fourths are pure, their common values being 3/2 and 4/3. For instance, the value of the interval between the second and the fifth note is (3/2)/(9/8) =
4/3. This is a remarkable property which does not follow obviously from the construction.
There is a logic behind the defini- Figure 4. Jean-Philippe Rameau. Portrait by Jean Bernard Restout, titled "The inspired poet."
70 THE MATHEMATICAL INTELUGENCER
Providing a scale with the maximum number of pure in
tervals was a domain of research of early music theory. In sixteenth-century Western European music, the intervals
of minor and major third began to be considered as con
sonant, and the scale of Pythagoras was less suitable for
new harmonies that involved many of the new intervals.
(The value of a pure major third interval is 5/4, whereas in
the scale of Pythagoras the value of the interval between
the first and the third notes is 81/64, which is a little bit
greater than 5/4). A scale which was useful in that respect
is the one named after Gioseffo Zarlino, a famous sixteenth
century Venetian musicologist. Zarlino's scale makes a
compromise between pure thirds, pure fourths, and pure
fifths. The sequence of numbers is
1, 9/8, 5/4, 413, 3/2, 5/3, 15/8, 2.
Some of the fifths in this scale are pure, but not all of them.
For instance, the value of the interval between the second
and the sixth note is 40/27, which is strictly less than 3/2. The value of the difference is (3/2)/( 40/27) = 81/80, the
Didymus comma, which is an audible interval.
It is impossible to have only pure intervals in a scale,
unless the scale is short. Aristoxenus (fourth c. BC) made
a systematic theory of scales based on "tetrachords," scales
consisting of four notes corresponding to different divi
sions of the fourth by tones and semitones. A long scale
would be obtained by concatenating tetrachords.
Let us return for a moment to the scale of Pythagoras.
Problems are encountered as soon as one needs to con
catenate several such scales, for instance in order to play
musical instruments whose ranges cover several octaves.
For example, one would expect that the concatenation of
12 fifths gives 7 octaves (as is the case for instruments like
the guitar or the harpsichord). This cannot be the case if
one uses the scale of Pythagoras, since (3/2)12 is not equal
to (2/1)1. The interval with value (3/2)12 is larger than the
one with value (2/1F The difference is a small (but never
theless audible) interval, (3/2)12/27. This small interval is
called a "Pythagorean comma."
Similar problems occur in all the other scales based on
pure intervals. For instance, we would expect that the con
catenation of 4 fifths gives an interval of 2 octaves and one
major third. If we do the computation in Zarlino's scale, we
find that this is not the case, and the difference is the Didy
mus comma (81180). It is worthwhile to mention here that music theorists in
ancient China encountered similar arithmetical problems
in their theory of scales.
It should be clear now that the definition of a scale in
volves some arbitrariness and depends strongly on which
intervals we insist be pure. One solution to the problem
was, instead of making a restricted choice, to keep differ
ent possibilities. This is one of the reasons why there are
so many scales in antique Greek music. In this music, dif
ferent scales were adapted to different melodies and dif
ferent types of instruments. The choice of scale for a mu
sical piece determined much of the character of the piece
and of its psychological effects on the listener. (This is also
related in Jamblichus [7].) This subtle dependence of the
piece upon the scale lasted in European music until the
adoption of the tempered scale. For instance, Rameau gives
a list of characteristics of different tonalities in his Traite de l'harmonie reduite a ses principes naturels, Book II,
Chapter 24 (Vol. 1 of [10]).
Rameau and the Harmonic Sequence
Like Pythagoras 2000 years before him, the composer and
theoretician Jean-Philippe Rameau made a real synthesis be
tween music as an art whose aim is to express and to cre
ate emotions, and music as a mathematical science with a
deductive approach and rigorous rules. Pythagoras estab
lished the important relation between musical intervals and
pairs of integers, Rameau went a step further and gave a mu
sical content to the whole sequence of positive integers.
One of the main ideas for Rameau is that the infinite se
quence of integers is contained, in a beautiful way, in na
ture, as a sequence of frequencies.
When a sonorous body (Rameau's terminology: "corps sonore") vibrates, it creates a local periodic variation of the
pressure of air. This vibration propagates as an acoustic
wave. It hits our ear drums, and we hear a musical note.
The musical note produced by a vibrating string (bowed or
plucked), consists usually in a superposition of a funda
mental tone and overtones. The frequencies of the over
tones, which are called the harmonic frequencies, are in
tegral multiples of the frequency of the fundamental tone.
The sequence of harmonic frequencies is naturally param
etrized by the positive integers. For instance, the frequency
of the note C1 (which corresponds to the lowest C key on
a piano keyboard) is (approximately)! = 33 Hz (cycles per
second). The frequencies of the corresponding overtones
are therefore
j, 2j, 3j, 4j, 5j, 6j, . . .
whose values in Hz are
33, 66, 99, 132, 165, 198, . . .
The corresponding sequence of notes is
In principle, one can hear the first four or five overtones on
an instrument like an organ. (Mersenne, in his Harmonie Universelle, says that he can hear the first nine overtones.)
Rameau's theoretical work is based on scientific dis
coveries in acoustics which were made in the seventeenth
century, in particular by the mathematician Joseph
Sauveur. The phenomenon of "harmonics" in music had
been noticed long before Rameau, but Rameau was the one
who used it as the basis of a coherent theoretical teaching
of music, in particular in his Traite de l'harmonie reduite a ses principes naturels.
Rameau's textbooks on music theory (about 2000 pages) include the basics of figured bass, accompani
ment, chords, modulation, and composition techniques.
All the theories he developed are based on simple rules
VOLUME 24, NUMBER 1 , 2002 71
T R A I T E' DE L,H A R. M 0 N I E, Rameau liked to consider the har
monic sequence of frequencies emitted
by a sonorous body as a proof that the
principles of music theory are contained
in nature. Later on (starting from the
year 1750), and especially in his Nouvelles rejlexions sur le principe sonore, Rameau argued that since the funda
mental objects of mathematics are de
rived from the sequence of positive in
tegers, and since this sequence is
contained in music, then mathematics it
self is part of music. These reflections
provoked a dispute between Rameau
and eighteenth-century French mathe
maticians, like L . B. Castel and J.
d'Alembert, and with the ency
clopaedists, like Denis Diderot, Jean
Jacques Rousseau, and Friedrich von
Grimm. The details of the controversy
are worth studying, but they cannot be
included in this short report. A very
strong hostility followed several years
of friendship and mutual praise between
Rameau and d'Alembert; do not fall into
the facile conclusion that the interaction
between music theorists and mathe
maticians was always friendly. Still the
interaction was there.
D E'M 0 N S T R A T I 0 N. ':lllCfS a1p:�l
a.P piO�V
. our tronnr Irs raifons de
. l' .4tu�tl lit 1/J foptilm• - fupnflNI •
I ll .fauc mplc:r lc:s nt>mbus de cellc·c:y , ou \4o. donnc:!a le Son grave de: cc: dcrnicr
'"· Ac,ord amfi 5 <40: 6o, 75• 90• IoB.l ( :R.c , La� Ud E 1 Mi� Sol J
I. Accord fondamcmat
de Ia fcpticmc In this report I have concen
trated on examples, starting with
Pythagoras and ending with Rameau. To
support the choice of Pythagoras and
Rameau, let me conclude by citing
Jacques Chailley [4]1
I ��� 1 Son gran de l' Aecord j de la 'Quintc-fupc:dlui!,
Figure 5. A diagram in Rameau's Traite, discussing ratios of frequencies of a dissonant En 2500 ans d 'histoire ecrite, la chord (containing a minor seventh). musique n'a peut-etre connu que deux
derived from the existence and the properties of the har-
monic sequence. For instance, in his analysis of chords,
the root of a triad is treated as a unit, in a mathematical
sense, and this point of view makes things simple and ev
ident. The theory of triads (consisting of three notes, like
C, E, G) had already been derived from the harmonic se
quence by Zarlino and Descartes, but Rameau worked on
a complete theory of dissonant chords. The diagram in
Figure 5 is one of Rameau's pictures in the Traite de l 'harmonie reduite a ses principes naturels, in which he
represents the dissonant chord La, Do#, Mi, Sol (that is,
A, C# , E, G), with four other derived chords. The num
bers below the notes are the corresponding elements of
the harmonic sequence.
veritables theoriciens, dont les autres n'ont guere fait qu'amenager ou rapetasser les propositions. L 'un, au VIe siecle avant notre ere, jut le jabuleux Pythagore. L 'autre mourut a Paris en 1 764: cejut JeanPhilippe Rameau.
In 2500 years of written history, music has perhaps known
only two genuine theoreticians, and what the others did
was only to repackage or patch up their propositions. The
first one, in the Vlth century before our era, was the fabu
lous Pythagoras. The other one died in Paris in 1764: this
was Jean-Philippe Rameau.
REFERENCES
[1 ] Benno Artmann, The liberal arts, Math. lntelligencer 20 (1 988), no.
3, 40-4 1 .
7J. Chailley was a famous musicologist, professor at the Conservatoire National Superieur de Musique de Paris and at the University o f Paris. I borrowed this quota
tion from the Introduction to the collected works of Rameau [10].
72 THE MATHEMATICAL INTELLIGENCER
[2] Milton Babbitt, Past and present concepts of the nature and lim�
its of music, International Musical Society Congress Reports 8
(1 96 1 ) , no. 1 , 399.
[3] J. M. Barbour, Music and ternary continued fractions, Amer. Math.
Monthly 55 (1 948), 545-555.
[4] Jacques Chailley, "Rameau et Ia theorie musicale", La Revue Mu�
sicale , Nurnero special 260, 1 964.
[5] Galileo Galilei, Discorsi e dimostrazioni matematiche intorno a due
nuove scienze, in Vol XII of the Complete Works, Societa Editrice
Fiorentina, 1 855.
[6] G . D. Hasley and Edwin Hewitt, More on the superparticular ratios
in music, Amer. Math. Monthly 79 (1 972), 1 096-1 1 00.
[7] Jamblichus (ca. 240 AD), The Life of Pythagoras, English transla�
tion by Thomas Taylor, London, John M. Watkins, 1 965.
[8] Johannes Kepler, Harmonicas Mundi. (I have used the French
translation with comments by J. Peyroux, Librairie A. Blanchard, 9
rue de Medicis, Paris, 1 977 . )
[9] A. L. Le1gh Silver, Musimatics or the nun 's fiddle, Amer. Math.
Monthly 78 ( 1 9 7 1 ) , 351 -357.
[1 0] J . Ph. Rameau, Complete Theoretical Writings, edited by R. Ja�
cobi, a facsimile of anginal editions, published by the American In�
stitute of Musicology, 1 967 .
[1 1 ] C. St0rmer, Sur une inequation indeterrninee, C. R. Acad. Sci. Paris
1 27 (1 898), 752-754.
[1 2] Theon of Smyrna (beginning of the second c. AD), Exposition of
the mathematical knowledge useful for the reading of Plato. A bil in�
gual (Greek�French) edition due to J. Dupuis (Paris 1 892) is
reprinted by Culture et Civilisation, 1 1 5 Av. Gabriel Lebon, Brus�
sels, 1 966.
A U T H O R
ATHANASE PAPADOPOULOS
Institute de Mathematiques, CNRS
7 rue Rene Descartes
67084 Strasbourg Cedex
France
e-mail: papadopoulos@math.u-strasbg.tr
Athanase Papadopoulos graduated as an engineer from the
Ecole Centrale de Paris in 1 981 , and got his doctorate in math�
ematics at Universite de Paris�Sud in 1 983. Since 1 984 he
has been a researcher at CNRS, specializing in low�dimen�
sional topology, geometry, and dynamical systems. In addi
tion, he teaches a course on mathematics and music at the
Universite Louis Pasteur, Strasbourg. He was choir director of
the Russian Orthodox Church of Strasbourg from 1 989 to
1 999.
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VOLUME 24. NUMBER 1 . 2002 73
I ;J§Ih§'.lfj .J et Wim p , Editor I
Feel like writing a review for The
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Column Editor's address: Department
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Philadelphia, PA 1 9 1 04 USA.
The Math Gene: How Mathematical Th inking Evolved and Why Numbers Are Like Gossip by Keith J. Devlin
NEW YORK: BASIC BOOKS, 2000
328 pp. US $25.00; ISBN 0-4650-161 8-9
REVIEWED BY REUBEN HERSH
Devlin is one of the most prolific
popularizers of mathematics, not
only in print but also on the electronic
media. He was the first recipient of a
newly established prize of the Mathe
matical Association of America, for
popularizing math.
The title The Math Gene is a mis
nomer, commercially workable but not
quite honest. Devlin informs the reader
early on that there is no math gene.
The book only superficially appears
to be a popularization. Actually, it's a
daring presentation of a complex, origi
nal theory of the origin and nature of
mathematical thinking. Rarely does one
book combine such super-easy read
ability with such radical theoretical
speculation-speculation that brings in
sociobiology, linguistics, neurology, an
thropology, philosophy of mathematics,
and, of course, mathematics itself.
Devlin doesn't claim expertise in all
these fields: he necessarily depends on
reading and consultation with appro
priate experts. Unfortunately, espe
cially regarding linguistics and socio
biology, there is no consensus among
experts on some major questions De
vlin confronts. He doesn't tell the
reader that some opinions he quotes
are controversial, not to be taken on
authority. If he was unaware of this, his
consultants are to be blamed more
than he. If he was aware of it, he owed
it to himself and his readers to consider
the other views on these questions.
74 THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YOCIK
The main conclusion Devlin draws
from his excursions into linguistics,
brain functioning, and human evolu
tion are actually not that surprising or
controversial; in fact, they are reassur
ing. He says that mathematical think
ing is a normal part of human thinking
in general, not a rare gift confined to
an elite. He says that the main thing
that makes some people good at math
is that they are deeply interested in it;
they really like it, and think about it a
lot.
He gives a more-than-welcome,
badly needed challenge to some fash
ionable catchwords. For instance,
we're warned ad nauseam that U.S.
children's math test scores ranking
only 19th internationally is a grave
threat to our economic viability. But
anyone with open eyes and ears knows
that few jobs in our production or ser
vice sectors need even "intermediate
algebra," let alone "rigorous proof" a la
Euclid, or calculus or the infamous
"pre-calculus." These offerings are com
pulsory by tradition. They maintain em
ployment for their teachers and the
teachers of their teachers, and they are
claimed to be vestiges of intellectual in
tegrity in the mush and applesauce of
contemporary American schooling.
We math teachers have accepted
the embarrassing, shameful role of
"gate-keepers." We're custodians of a
narrow opening, through which
squeeze aspirants to the "degrees" that
are the virtual sine qua non of re
spectability and affluence. "It's a dirty
job, but somebody has to do it." It
keeps us supported by state legisla
tures.
This seldom-acknowledged reality
contradicts the role some, at least,
would rather play-to provide more
students intellectual challenge and
pleasure that they could have enjoyed,
and will never know they missed.
Devlin's forthright explanation of
these important facts makes his book
worthy of the largest possible reader
ship.
Most of the book develops a highly
speculative theory. The language abil
ity, says Devlin, is the same as the
ability to think "off-line," as he calls
it. That is, to think about stuff that's
not in front of you at the time. In other
words, abstract thinking. Math think
ing is just abstract thinking at a level
one step higher, where we think
about our thoughts. He relies heavily
on work of Derek Bickerton, an Eng
lish specialist in "creoles. " These are
new languages resulting when two
populations speaking radically differ
ent languages are forced to become
mutually comprehensible. Bickerton
claims that the jump from a mere mix
ture of alien vocabularies to a genuine new language, complete with its own
grammar and syntax, happens in a
great leap, in one generation. By anal
ogy, Devlin speculates that the human
race's jump from proto-language (vo
cabulary without syntax and gram
mar) to genuine language, capable of
abstraction and therefore of mathe
matics, also took place in one sudden
leap, possibly by an actual genetic
mutation. (Bickerton's claims are
carefully examined and found want
ing in an important paper, "Perspec
tives on an Emerging Language," by
Judy A Kegl and John McWhorter, in
E. V. Clark (Ed.) (1997), the Proceedings of the 28th annual Child Language Research Forum at Stanford
CSLI.)
All this is in the context of Chom
skyite nativistic linguistic theory. My
information is that Chomsky-Pinker
linguistics has not justified its claims
or fulfilled its promises. Study of actual acquisition of language by actual living children is giving rise to new
perspectives. The interaction between
the innate and the learned is a wide
open problem, not a settled matter.
Devlin's grandiose speculations ignore
this difficulty, rendering them signifi
cantly less convincing.
Nevertheless, The Math Gene is a
great read, and a great contribution
against the self-serving Philistinism,
hypocrisy, and politicization so ram
pant in the fight about math education.
Thanks to Dan Slobin and Vera
John-Steiner for invaluable consulta
tion.
Department of Mathematics
University of New Mexico
Albuquerque, NM 871 31
USA
e-mail: rhersh@math unm.edu
Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being by George Lakoff and
Rafael E. Nunez
NEW YORK: BASIC BOOKS 2000
489 pp. US $30; ISBN:0-465·03770-4
REVIEWED BY DAVID W. HENDERSON
This book is an attempt by cognitive
scientists to launch a new disci
pline: cognitive science of mathematics. This discipline would include the
subdiscipline of mathematical idea analysis.
What prompted me to read this book
were the endorsements on the back
cover by four well-known mathemati
cians: Reuben Hersh, Felix Browder,
Bill Thurston, and Keith Devlin. I was
excited by the authors' purpose stated
in their Preface and Introduction:
Mathematical idea analysis, as we
seek to develop it, asks what theo
rems mean and why they are true
on the basis of what they mean. We
believe it is important to reorient
mathematics teaching toward un
derstanding mathematical ideas and
understanding why theorems are
true. (page xv) We will be asking how normal
human cognitive mechanisms are
employed in the creation and un
derstanding of mathematical ideas.
(page 2)
It was with enthusiasm that I read
the book together with the members of
a mathematics department seminar at
Cornell University. However, there
were major disappointments:
• There are numerous errors in math
ematical fact. Only some of these are
corrected on the book's web page:
http://www. unifr. chlpersolnunezrl welcome.html. There are so many er
rors that it seems inconceivable that
the four mathematicians who have
endorsements on the back cover
could have read the book without
noticing them. On the web page the
authors blame the publisher for most
of the errors. They report that the
second printing has even more er
rors and has been recalled!
• The authors assert, "The cognitive sci
ence of mathematics asks questions
that mathematics does not, and cannot, ask about itself." (page 7) [my
emphasis] . I will show below that this
statement is false. Most of the book
after the third chapter provides a pow
erful argument that a mathematics
that asks these questions is precisely
what is needed.
• The authors seem to be working
from a common misconception
about what mathematicians do.
This book is nevertheless a serious
attempt to understand the meaning of
mathematics. I hope it will encourage
cognitive scientists and mathemati
cians to talk to one another. Perhaps
together we can develop a clearer un
derstanding of the meanings of mathe
matical concepts, a deeper under
standing of mathematical intuition. As expressed by David Hilbert-
If we now begin to construct math
ematics, we shall first set our sights
upon elementary number theory; we
recognize that we can obtain and
prove its truths through contentual
intuitive considerations. ([5), page
469)
Cognitive Science- Cognitive
Metaphor
The authors start in Chapter One by
surveying discoveries by cognitive sci
ence of an innate arithmetic of the
numbers 1 through 4 in most humans
(and some animals). The problem is to
connect this innate arithmetic to the
arithmetic of all numbers and to the
rest of mathematics. According to the
authors, "One of the principal results
of cognitive science is that abstract
concepts are typically understood, via
VOLUME 24, NUMBER 1 , 2002 75
metaphor, in terms of more concrete concepts. This phenomenon has been studied scientifically for more than two decades and is in general as well established as any result in cognitive science." (page 39, 41)
For the authors, "metaphor" has a much more complex (and technical) meaning than it does for most of us. They describe a cognitive metaphor as an "inference-preserving cross-domain mapping-a neural mechanism that allows us to use the inferential structure of one conceptual domain (say, geometry) to reason about another (say, arithmetic)." For some cognitive metaphors, cognitive scientists have detected actual neural connections in the brain.
To illustrate the authors' notion of cognitive metaphor, let us look at the "Arithmetic Is Object Collection" metaphor. This metaphor, as with all cognitive metaphors, consists of two domains and a mapping:
• a source domain: "collections of objects of the same size (based on our commonest experiences with grouping objects)";
• a target domain: natural numbers with addition and subtraction (which the authors call "arithmetic");
• a cross-domain mapping as described in the accompanying table (Arithmetic Is Object Collection):
It is basic to the authors' arguments that the notions in the left-hand column have literal meaning, while the notions in the right-hand column do not. The notions in the right-hand column gain their meanings from the notions in the left-hand column via the metaphor.
Each conceptual metaphor has entailments, which for this metaphor the authors describe as follows:
Take the basic truths about collections of physical objects. Map them onto statements about numbers, using the metaphorical mapping. The result is a set of "truths" about the natural numbers under the operations of addition and subtraction. (page 56)
They list 17 such entailments for arithmetic that seem to me to be part of the what Hilbert called the "the truths" (involving only addition and subtraction) of elementary number theory "that we can obtain and prove through contentual intuitive considerations."
Mathematicians Are Needed
As far as I can tell it is at this point (in Chapter 3 out of 16) that the authors leave results established by cognitive science research and move into the realm they describe as "hypothetical" and "plausible." The remainder of the book deals with plausible cognitive metaphors, which the authors hypoth
esize account for our understanding of the meanings of real numbers, set theory, infinity (in varied fonns), continuity, space-filling curves, infinitesimals, and the Euler equation em + 1 = 0. It is also at this point that I think the authors' arguments and discussions need input from mathematicians and teachers of mathematics. I will illustrate by describing some of the authors' metaphors and the improvements that I think mathematicians can make.
Arithmetic Is Object Collection
Source Domain Object Collection
Collections of objects of the same size
The size of the collection Bigger Smaller The smallest collection Putting collections together Taking a smaller collection
(from a larger collection)
76 THE MATHEMATICAL INTELLIGENCER
---->
---->
---->
---->
---->
---->
---->
Target Doma'in Arithmetic
Natural numbers
The size of the number Greater Less The unit (One) Addition Subtraction
Actual Infinity: The authors "hypothesize" that the idea of infinity in mathematics is metaphorical.
Literally, there is no such thing as the result of an endless process: If a process has no end, there can be no "ultimate result." However, the mechanism of metaphor allows us to conceptualize the "result" of an infinite process in terms of a process that does have an end. (page 158)
The authors "hypothesize that all cases of actual infinity are special cases of" a single cognitive metaphor which they call the Basic Metaphor of Infinity or BMI. BMI is a mapping from the domain, Completed Iterative Processes, to the domain, Iterative Processes That Go On and On. A completed iterative process has four parts, all literal: the beginning state; the process that from an intermediate state produces the next state; an intermediate state; and the final resultant state that is unique and follows every non-final state. These are mapped onto four parts (with the same names and descriptions) of an Iterative Process That Goes On and On, wh�re the first three parts have literal meaning but the last part (the "final resultant state") has meaning only metaphorically from the cognitive mapping.
I illustrate with a special case from the book.
Parallel lines meet at infinity (using BMI): How do we conceptualize (or give meaning to) the notion in Projective Geometry that two parallel lines meet at infinity? If m and l are two parallel lines in the plane, then let the line segment AB be a common perpendicular between them, and consider the isosceles triangles on one side of AB. The authors call this the frame. They then construct the special case of BMI given in the table on page 77 (Parallel Lines Meet at Infinity).
They remind the reader that theirs is "not a mathematical analysis, is not meant to be one, and should not be confused with one." They state their "cognitive claim: The concept of 'point at infinity' in projective geometry is, from a cognitive perspective, a special case of the general notion of actual in-
Parallel Lines Meet at Infinity
Source Domain Target Domain Completed Iterative Processes Isosceles Triangles with Base AB The beginning state ---+ Isosceles triangle ABG0, where length of
AGo (=EGo) is Do The process that from ( n - 1 )th ---+ Form ABGn from ABGn-1 by making
"Dn arbitrarily larger than Dn- 1" state produces nth state
Intermediate state ---+ "Dn > Dn - 1 and (90° - an) < (90° - an -d"
The final resultant state, unique ---+ "ax = 90°, Dx is infinitely long", and the and following every non-final sides meet at a unique Gx, a point
state at x (because Dx > Dn- 1 , for all
finite n)
finity." They admit that they "have at
present no experimental evidence to
back up this claim. "
OK, let us look at this as mathe
maticians. There are several problems
with this metaphor as presented-the
three most important (from my per
spective) are as follows:
• As we teach in first-year calculus,
not every monotone increasing se
quence is unbounded. Thus we need
more than "Dn arbitrarily larger than
Dn-1" to insure that "Dx is infinitely
long."
• This metaphor entails a unique point
at infinity on both ends of a line,
which does not agree with the usage
in projective geometry-nor with
our intuitive fmite experience with
lines, as I will show.
• The metaphor only indirectly in
volves "lines" (the primary objects
under consideration) and does not
give meaning to the question: Why does a line have only one point at infinity? A Mathematician's Metaphor
point of intersection when m is paral
lel to l. To be more specific: Imagine
that m starts perpendicular to l and
then rotates at a constant rate so that
at time T it is parallel to l, and then
stops when it is again perpendicular to
l. We now define a cognitive metaphor
in the authors' sense with
• Source Domain: Continuous motion
of a particle along a curve through a
point P. (Let T be the time that the
particle is at P.)
• Target Domain: The rotating line frame described above.
• Gmss Domain mapping: (see accom
panying table-Projective Metaphor)
In a course presenting projective
geometry, I show how a projective trans
formation can give a way of actually see
ing (an image of) the point at oo.
I see no need in this description for
the authors' Basic Metaphor of Infinity.
I propose this metaphor as a coun
terexample to the authors' hypothesis
"that all cases of actual infinity are spe
cial cases of" the single cognitive
metaphor BMI.
Not always metaphors: In addition,
there are many cases (especially in
geometry, which the authors consider
only lightly) where our cognitive analy
sis does not produce cognitive meta
phors. For example, look at the notion
of "straightness." We say that "straight
lines" in spherical geometry are the
great circles on the sphere, but how do
we understand what is the meaning of
"straight" in this case? An answer
sometimes given in textbooks is that,
of course, great circles are not literally
straight, but we will (metaphorically)
call them straight. However, I have ar
gued in [2] and [3] that great circles on
a sphere are literally straight from an
intrinsic proper point-of-view. Ex
trinsically (our ordinary view of an ob
server imaging the sphere from a posi
tion in three-space outside the sphere)
the great circles are certainly not
straight. They are intrinsically (the
point of view of a 2-dimensional bug
whose universe is the sphere) straight;
that is, the 2-dimensional bug would
experience the great circles as straight
in its spherical universe. I would like
to see a cognitive scientist analyze this
situation which (at first sight) involves
more centrally imagination and pointof-view rather than metaphor, per se.
Misconceptions About
Mathematics
The above discussions of the pro
jective metaphor and of straightness
constitute a counterexample to the au
thors' assertion that
The cognitive science of mathemat
ics asks questions that mathematics does not, and cannot, ask about itself How do we understand such
basic concepts as infinity, zero, lines,
points, and sets using our everyday
conceptual apparatus? (page 7) [my
emphasis].
On the book's web page http://www. unifr.chlperso/nunezr!warning.html they explain further:
[O]ur goal is to characterize mathe
matics in terms of cognitive mecha
nisms, not in terms of mathematics
(without BMI): I propose a different
metaphor that I have used for years in
my geometry classes. This metaphor
more closely uses the main notions of
projective geometry: lines and their in
tersections. For the frame of the
metaphor we construct the mtating line frame: Take a line l in the Euclid
ean plane, a point A not on l, and a line
m in the plane that is conceived of as
free to rotate about A. As we rotate m about A, most positions of m result in
a literal unique intersection with l, and different positions result in different
intersection points. There is no (literal)
Source Domain
Proj ective Metaphor
Target Domain
Motion of the particle before T Motion of the particle after T At time T the particle is at a
unique point P
---+ Motion of the particle before T ---+ Motion of the particle after T ---+ At time T the particle is at a unique point
point (which we call the point at oo on l)
VOLUME 24, NUMBER 1 , 2002 77
itself, e.g., formal definitions, axioms, and so on. Indeed, part of our job is to characterize how such formal definitions and axioms are themselves understood in cognitive terms.
This quotation contains two related misconceptions about mathematics:
• Misconception 1: Mathematics is formal, consisting of formal definitions, axioms, theorems, and proof.
• Misconception 2: Mathematics does not (and cannot) ask what mathematical ideas mean, how they can be understood, and why they are true.
These misconceptions of mathematics are prevalent among non-mathematicians. The blame for this lies mostly with us, the mathematicians. Collectively, we have not done an effective job of communicating to the outside world the nature of our discipline. Nevertheless, many of us (including all four of the mathematicians whose endorsements are on the back cover) have in our writings attempted to dispel these misconceptions (see for example, [1 ] , [2] , [3], [4], [6] , [7]). In particular, let me quote David Hilbert from the preface of Geometry and the Imagination [6] . This book is important because Hilbert is considered to be the "Father of Formalism," and yet he writes:
In mathematics, as in any scientific research, we fmd two tendencies present. On the one hand, the tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material that is being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations. [Hilbert's emphasis]
On Hilbert's "one hand" is the tendency of formal mathematics that Lakoff and NUiiez are looking at. On Hilbert's "other hand" is the tendency of mathematics to consider much of what Lakoff and NUiiez say that it "does not, and cannot, ask about itself."
78 THE MATHEMATICAL INTELLIGENCER
Philosophy of Mathematics
I find the authors' discussion of their philosophy of embodied mathematics to be profound, and I think any mathematician who studies it will find her/his own understandings of mathematics stimulated and challenged in constructive ways. But first the mathematicians must overcome their reactions to being told (incorrectly) that mathematicians do not and cannot ask how they understand the meanings of mathematical ideas and results.
The authors summarize their view of the philosophy of mathematics with the following statement:
Mathematics as we know it is . . . a product of the human mind. . . . It comes from us! We create it, but it is not arbitrary [because] it uses the basic conceptual mechanisms of the embodied human mind as it has evolved in the real world. Mathematics is a product of the neural capacities of our brains, the nature of our bodies, our evolution, our environment, and our long social and cultural history. (page 9)
In Part V of the book the authors expand on this summary and proceed to dismiss (or "disconfirm") other philosophies of mathematics. I recommend that the mathematical reader skip over all arguments dismissing various other philosophies of mathematics, because for the most part these arguments are based on shallow summaries of what the various philosophies assert. Further, I do not think that the settling of these arguments is important or necessary for understanding the authors' main points. Regardless of one's philosophical beliefs, I think all mathematicians (and teachers of mathematics) would welcome
conceptual foundations [for mathematics that] would consist of a thorough mathematical ideas analysis that worked out in detail the conceptual structure of each mathematical domain, showing how those concepts are ultimately grounded in bodily experience and just what the network of ideas across mathematical disciplines looks like. (page 379)
We Need to Work Together
Cognitive scientists and mathematicians need to work together to develop mathematical idea analysis. I believe that most mathematicians and teachers of mathematics are concerned exactly with the things mentioned by these authors:
We believe that revealing the cognitive structure of mathematics makes mathematics more accessible and comprehensible . . . . [M]athematical ideas . . . can be understood for the most part in everyday tem1s. (page 7).
We, mathematicians and teachers would certainly be thankful to cognitive scientists if they could help us in our grappling "not just with what is true but with what mathematical ideas mean, how they can be understood, and why they are true." (page 8)
REFERENCES
[ 1 ] Keith Devl in, The Math Gene: How Mathe
matical Thinking Evolved and Why Numbers
Are Like Gossip , New York: Basic Books,
2000.
[2] David W. Henderson, Differential Geometry:
A Geometric Introduction, Upper Saddle
River, NJ : Prentice-Hall, 1 998
[3] David W. Henderson, Experiencing Geom
etry in Euclidean, Spherical, and Hyperbolic
Spaces, Upper Saddle River, NJ: Prentice
Hall, 2001 .
[4] Reuben Hersh, What Is Mathematics, Re
ally?, New York: Oxford University Press,
1 997.
[5] David Hilbert, "The foundations of mathe
matics, " translated in Jean Van Heijenoort,
From Frege to G6del; A Source Book in
Mathematical Logic, 1879-193 1 , Cam
bridge: Harvard University Press, 1 967.
[6] David Hilbert and S. Cohn-Vossen, Geom
etry and the Imagination, New York:
Chelsea Publishing Co. , 1 983.
[7] W. P. Thurston, "On Proof and Progress in
Mathematics," Bull. Amer. Math. Soc. 30,
1 61 -1 77, 1 994.
Department of Mathematics
Cornell University
Ithaca, NY 1 4853-7901
USA
e-mail: dwh2@cornell.edu
K-jfl .. i.C+J.JQ.t§i Robin Wilson]
The Deve lopment of Computing
Visual display unit
Integrated circuit
Please send all submissions to
the Stamp Corner Editor,
Robin Wilson, Faculty of Mathematics,
The Open University, Milton Keynes,
MK7 6AA, England
e-mail: r.j.wilson@open.ac.uk
Since the 1950s computers have developed at an ever-accelerating pace, with a massive increase in speed and power and a corresponding decrease in size and cost.
The 'first generation' of electronic digital computers spanned the 1950s. These computers stored their programs internally and initially used vacuum tubes as their switching technology. Because such tubes were bulky, hot and unreliable, they were gradually replaced in the 'second generation' of computers by transistors with thousands of interconnected simple circuits.
In the late 1960s, the 'third generation' of computers saw the development of printed circuit boards on which thin strips of copper were 'printed, ' connecting the transistors and other electronic components. This led to the all-important integrated circuit, an assembly of many transistors, resistors, capacitors, and other devices,
M AG YA R P O STA
Computer drawing
1 2 � 8
nederland Isometric projection
80 THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
interconnected electronically and packaged as a single functional item. In the 1970s the first personal computers became available, for use in the home and office.
Computer-aided design also developed rapidly, and in 1970 the Netherlands produced the first set of computer-generated stamp designs, such as an isometric projection in which cir
cles expand and are transformed into squares. The computer drawing of a head is a graphic from the 1972 computer-animated film Dilemma.
The invention of the World Wide Web by Tim Berners-Lee in the early 1990s has led to the information superhighway, whereby all types of information from around the world has become easily accessible. Communications have been transformed with the introduction of electronic mail; a stamp portrays King Bhumibol of Thailand checking his e-mail.
King Bhumibol at e-mail
Tim Berners-Lee
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