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

REFERENCES

1 . C. Alexander, I. Giblin, and D. Newton, Symmetry groups on frac­

tals, The Mathematical lntelligencer 14 (1 992), no. 2, 32-38.

2. I. K. Argyros, D. Chen, and Q. Qian, The Jarratt method in Banach

space setting, J. Comput. Appl. Math. 51 (1 994), 1 03-1 06.

3. I. K. Argyros and F. Szidarovszky, The Theory and Applications of

Iteration Methods, CRC Press, Boca Raton, FL, 1 993.

4. W. Bergweiler, Iteration of meromorphic functions, Bull. Am. Math.

Soc. (N. S.) 29 (1 993), 1 51 -1 88.

5 . P. Blanchard, Complex analytic dynamics on the Riemann sphere,

Bull. Am. Math. Soc. (N. S.) 11 (1 984), 85-1 41 .

6. J. A. Ezquerro, J. M. Gutierrez, M. A. Hernandez, and M. A.

Salanova, The application of an inverse-free Jarratt-type approxi­

mation to nonlinear integral equations of Harnrnerstein-type, Com­

put. Math. Appl. 36 (1 998), 9-20.

7. J. A. Ezquerro and M. A. Hernandez, On a convex acceleration of

Newton's method, J. Optim. Theory Appl. 100 ( 1 999), 31 1 -326.

8 . W. Gander, On Halley's iteration method, Am. Math. Monthly 92 (1 985), 1 31 -1 34.

9. W. Gautschi, Numerical Analysis: An Introduction, Birkhii.user,

Boston, 1 997.

1 0. J. M . Gutierrez and M. A. Hernandez, A family of Chebyshev­

Halley type methods in Banach spaces, Bull. Austral. Math. Soc.

55 (1 997), 1 1 3-1 30.

1 1 . M. A. Hernandez, An acceleration procedure of the Whittaker

---- -

method by means of convexity, Zb. Rad. Prirod. -Mat. Fak. Ser.

Mat. 20 (1 990), 27-38.

1 2 . P. Jarratt, Some fourth order multipoint iterative methods for solv­

ing equations, Math. Camp. 20 (1 966), 434-437.

1 3. D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Sci­

entific Computing, 2nd ed., Brooks/Cole, Pacific Grove, CA, 1 996.

1 4. R. F. King, A family of fourth order methods for nonlinear equa­

tions, SIAM J. Numer. Anal. 10 (1 973), 876-879.

1 5. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear

Equations in Several Variables, Monographs Textbooks Comput.

Sci. Appl. Math. , Academic Press, New York, 1 970.

1 6. H. 0. Peitgen and P. H. Richter, The Beauty of Fractals, Springer­

Verlag, New York, 1 986.

1 7 . M. Shub, Mysteries of mathematics and computation, The Math­

ematical lntelligencer 16 (1 994), no. 2, 1 0-1 5.

1 8. J . F. Traub, Iterative Methods for the Solution of Equations, Pren­

tice-Hall, Englewood Cliffs, NJ, 1 964.

1 9. W. Werner, Some improvements of classical iterative methods for the

solution of nonlinear equations, in Numerical Solution of Nonlinear

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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the cenrur)', but i also full of remarkable insights inro its future development

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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2001/1236 PP., 253 ILLUS./HARDCOVER/$44.95/ISBN 3-540-66913-2

' Springer . www.springer-ny.com Promotion 1152560

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

· --- .. 1 2 • • 1 2

• • 1 2

3 • • 3 4

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

· --- . 1 2 1

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,

A Characterization of Affine Kac-Moody Lie Algebras, Communi­

cations in Mathematical Physics, 185 (1 997), 671 -688.

[2] Georgia Benkart, A Kac-Moody Bibliography and Some Related

References, Canadian Mathematical Society Conference Pro­

ceedings, vol. 5 (1 986), 1 1 1-135.

[3] Stephen Berman, Isomorphisms and Automorphisms of Universal

Heffalump Lie Algebras, Proceedings of the American Mathemat­

ical Society 65 (1 977), 29-34.

[4] Armand Borel, Some Recollections ofHarish-Chandra, Current Sci­

ence 65 (1 993), 91 9-921 .

[5] Armand Borel, The Work of Cheval/ey in Lie Groups and Algebraic

Groups, Proceedings of the Hyderabad Conference on Algebraic

Groups, ed. S. Ramanan, Madras: Manoj Prakashan, 1 991 ,

1 -23.

[6] Elie Cartan, Les groupes projectifs qui ne laissent invariante au­

cune multiplicite plane, Bulletin de Ia Societe mathematique de

France 41 ( 1913) , 53-96.

[7] Elie Cartan, Les groupes reels simples, finis et continus, Annales

scientifiques de I 'Ecole normale superieure 31 (1 91 4) , 263-355.

[8] Elie Cartan, Premiere these: Sur Ia structure des groupes de trans­

formations finis et continus, Paris: Nony, 1 894.

[9] Arthur Cayley, On Jacobi's Elliptic Functions, in Reply to the Rev.

B. Bronwin; and on Quaternions, Philosophical Magazine 26 (1 845), 208-2 1 1 .

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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ing Algebra of a Semisimple Lie Algebra, Transactions of the Amer­

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metric Spaces, New York: Academic Press, Inc., 1 978.

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bridge: University Press, 1 990.

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Functional Analysis and Its Applications 1 (1 967), 328-329.

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Russian), Trudy MIFM (1 969}, 48-60.

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60 THE MATHEMATICAL INTELLIGENCER

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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.

Vn flu/ homme n'efl pas capable d'epuiflr une matiere auffi profonde que ctlle-cy ; il e_(l p1·;jque impojji�lc qu'il n'y ou�/ic to�jours que/que chofl , malgre tous fes foms; mats dte mom.r , /es nou"Ptlles decou�ertes qu'il pcut joindrc a cc qui a deja paru for /e mime fojet, font autant de routes ftayees piJur ccux qrei peu�ent a/ler plus loin.

lA Mujique eft une ftimc� qui doit a�oir des rcgl�s certaines ; ces regles doirvmt ltre tirdes d'un principe evident ' � ce principe nc pcut gutres nous ltre connufons le flcours des Mathematiques: Aujft dois-je a�ouer que , nonobflant toute l'exp�rience que ;c pou­'flois m'ltre a'juifl dans Ia Mujique , pour l'a'lloir pratiquee pendant une affe' longue foite de temps , ce n'efl cependant que p11.r lc flcours des Math�matiques que mts idees fl font debrouil. Lees , @- que Ia lumie1e y a foccede a tme certaine obfiurite , dol# je ne m'apperct"Pois pas aupara"Pant. Si je ne Jfa'llois pas foire Ia difference du principe 4 Ia regie , bien tdt ce principe s' eft o/Jert a moi ii."PeC autant de jimplicite que d'i�idence ; Its con­fiquei'Jces qu'il m'a fournies enfoite , m'ont fait cvnnottre en el/es autant de rtgles J qui de"Poient fl rapporter par confl'luent a ce principe ; le 'Veritable fins de ces regles , /u�r jufle application , leur rapport , @}- l'ordre qu'ellts doi'lltnt tenirtntr'el/es (/a plus ftmple y firrval# toiJjours d'introdMIJio• a Ia moins jimple , @j­ain.fi par degre") enfi• le choix des ttrmts; tot# cela , dis -je , que ;' ignorois aupar�VtJa11t, s' eft de'lltioppi dans mon t}prit a'lle& tant de nettetl (.$ de precifion, que j e _. ai p/J m' tmplcher de con"Penir qu' il flroit 4 foubaitel' ( commeon mt le difoit, un jour que j'applaN­dijfois J I• pe'.foiJion de ndm Mujique modtme ) qut les connoi.f­fonces des Mujicitns de ct jitc/t repotlliijfent II.NX heii.Ntt'{ de leurs Compojitiot�s. IL nt fojfit done pas de fl#ir les :elfets tf unt Science ou d' un Nt , il fat tit plus les concnoir de fafo" qu' on puiJ!e Its rendrt inttlligib/es'; (.i c'eft J quoi je me fois pl"iilcip•lemtnt applique dAns It corps de Cit OtVW11.ge , que j'11.i tiijlrib� '" quatre Iivres.

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

Mathematical Intelligencer? You are

welcome to submit an unsolicited

review of a book of your choice; or, if

you would welcome being assigned

a book to review, please write us,

telling us your expertise and your

predilections.

Column Editor's address: Department

of Mathematics, Drexel University,

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