No title

Letters to the Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.

know that their female stu­

Girls and Boys in Moscow

professors

If a wild goose came across Konrad

dents are as good as and often better

Lorenz's wonderful books on ethology,

than their male students; why isn't this

it would read with great interest and

obvious to our Russian counterparts?

probably would like to add something.

[emphasis hers]" What is "this"? That

I have a similar feeling reading about a

some female students are better than

"country from which ... reliable data

most male students? This is indeed ob­

is not obtainable" in "Impoverishment,

vious.(Nor have I seen any indication

Feminization,

that girls have special difficulties in

and

Glass

Ceilings:

Women in Mathematics in Russia" by

"time-critical competitions," as Johns­

Karin Johnsgard

(Mathematical Intel­ ligencer, vol. 22 (2000), no. 4, 20-32).

gard suggests. Several girls from the

Let me first thank her for her sincere

the Moscow Mathematics Olympiad.I

interest and

was sorry, by the way, that one of these

sympathy

for

class mentioned above were winners in

Russia's

(certainly difficult) situation; but let

told us later that she does not want to

me add a few comments.

continue mathematics studies.) On the

I am a teacher in a specialized math

other hand, we do find that more boys

school which selects students from the

than girls are interested in mathemat­

whole Moscow region by running a se­

ics and perform well. Thus the graph

ries of problem-solving sessions. (Oc­

in the accompanying figure shows re­

casionally physics problems are in­

sults in a mathematics contest where

100-200 students aged

simple math problems were sent to

participate in these sessions

schools with an open invitation to stu­

cluded.) Usually

13

and

14

(each student comes 2-4 times), and

dents to write down their solutions and

the 20-25 students with the best results

send them in by mail.

are selected and invited to the school.

I am not sure that profound insights

Typically most students that come

can be gained by measuring correla­

to the problem session are boys. Writ­

tions between gender (or race) and

ing this, I have looked in our files. In

scientific achievements. But I believe

1996 there were 270 applicants;

girls among

that, whatever statistics are gathered,

the disproportion is

one should set aside one's preconcep­

about

60

similar among the students with the best results, with

6

girls among the

tions and deal with the facts as one

25

finds them.

students selected. In some years the disproportion was even greater, and

Alexander Shen

we decided to lower the threshold

Institute for Problems of Information

somewhat for girls (which has evident

Transmission

drawbacks). Similarly in departments

Ermolovoi 19

of mathematics, most applicants are

K-51 Moscow GSP-4, 1 01 447

male and most students are male.

Russia

Karin Johnsgard writes, "American

"

0

e-mail: [email protected]

•'

I

.1'

..... ,

•,

'

....

.

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. G RAVER, 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

where the observer's eye is located on the axis of revolu­

pair of compasses.

tion, which we take to be the y-axis of a standard euclid­

Introduction

above the xz-plane, which meets the mirror in a circle of

During the early part of the Industrial Revolution, engineers

radius

ean coordinate system in

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 con­ verts circular motion to mathematically perfect linear mo­ tion. Cf. [ 1 ; Ch.

4]

r0 s

1

R3.

We further suppose that

E is

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

outside C(6 would be seen by the observer at

located inside C(6 at the point

the plane

E as if it were

D on the segment between the D* for which IOD*I I O DI = 1. To achieve this, our mirror must reflect a ray from D* to E at an interme­ origin 0 and

·

diate point

M in such a way that the reflected ray appears D, as indicated in Figure 1. (From geometric optics, the tangent line to the mirror surface at M in the to come from

plane containing the incident ray and the reflected ray

and [2].

Because reflection in a line can be effected with a flat mirror, while controlled optical distortions can be pro­

makes equal angles with these rays.) The mirror images of lines outside C(6 would then appear as circles inside 'i6.

duced through reflection (in the optical sense) in curved

It will suffice to restrict our attention to a tangent line

mirrors, it is natural to wonder whether inversion in a cir­

to the cross section of the mirror in the xy-plane, as de­

cle can be achieved through reflection in a suitable mir­

picted in Figure 2. In this figure, Y is the y-coordinate of

rored surface. In this note we give some positive answers

the point

to this question, including equations for constructing such

of the point on the x-axis whose reflection is being viewed

E (the observer's eye),

mirrors. Specifically, we show how to design a mirror in

by the observer, and

which the viewer sees the exterior of a disk as though it

image.

w*

is the x-coordinate

w is is the x-coordinate of its virtual

had been geometrically inverted to the interior of the disk. The Differential Equation The Mirror

Let y

If such a mirror exists, it is a surface of revolution some­

pothesized mirror for x

what similar in shape to a cone. (In fact, it more closely re­

the mirror, let

sembles a bell.) Its exact shape depends upon the point

the graph off at (x,y) makes with the line of sight from the

4

THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK

E

= f(x) be the equation of the cross section of the hy­ 2: 0. If (x,y) represents a point on a

denote the angle that the tangent line to

E

(2)-(5)

also, by

y-aXIS

( Eye)

tan

(

2y -

;) = tan(y- ) =tan ((a+ y)- (a+ u)) w*-x Y- y y -x) (Y�y) (w* 1+ y u

_

=

-

x(w*- x)- y(Y- y) xy + (w* x)(Y- y)

(1- Yy)( Y- y)- x 2 Y

_

x-axis D*

.r

-x

x2Yy + (Y- y)(Y- y - x2Y)

so that

(6) The first expression for observer at (O,Y) to this point. Let

u

denote the angle the

tangent line makes with the horizontal and makes with the vertical. We note that conclude that tan(y)

=

rLr dy

y the angle it

= -tan( y)

-,

.

y

(1)

(2)

xY '

(3)

)

tan(a+

y

(5)

.

(

2'Y-

7T)

(6) is used to replace

u,

-1 when x

= 0.

=

Before working with this general equation, we consider

The View from Infinity

When Y---> when

we see from

x,

u = xy/(1- x2),

(6) that u---> xy/(1- x2); and, (7) has the partial

the right side of

( �) uy = ( �)

- 1 +

- 1+

1

� x 2·

Since this partial derivative is bounded on each x-interval

< b < 1, it follows from a standard theorem

1

1- tan2(y)

2 tan(y)

=

1- (y')2

2y'

lution 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), (y')Z +

E

(O,Y )

the quadratic equation for y' is

2 y x �?y' - 1

1- :L-

With the substitutions equation

(8) can be written 1 2y = -p-1- s sp '

= 0, (0 s; x < 1).

s = x2

dy, d.'

and

where p

By differentiating with respect to

=

(8)

p = -y'lx

(>0),

y -2d .

(9)

ds

s and eliminating

y and

we get the first-order equation

p dp ds- s(s- 1)(sp2 +

l§tijil;ifW

we get a first-order dif­

the more tractable limiting case as the viewer moves to­

y-axis

-1

(7)

(e.g., [3; p. 550]) that the limiting equation has a unique so­

2 = tan(2y) =

(Eye)

= -u- VUT+l;

ferential equation for the meridian curve. Note that y'

[O,b] where 0

From (1) we get

. u = -tan

y'

(4)

X

w*- x

gives the quadratic equation

derivative with respect to y given by

Y- y

= --;

y) =

u

Noting that y' is never positive, we

ward positive infinity.

Y- y w* = --·

u

= 0.

see that

and when

-1

2:

tan(a+

2uy' - 1

and

There are four other relations that we can easily see from Figure

(y')Z +

1)

(0

<

s < 1)

(10)

which, although not standard, admits integration. Indeed, with the

p = v + q,

and

q=

successive substitutions

1/s

= 1 + pq,

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

(1 - s) r eV212 dt = speV21 2 (for appropriate c) (11) v

v =p +

where

s-1 sp

--

(= 2y + sp) .

(12)

(11) and (12) determine p in terms y = 112(v - sp) can be ob­ tained as functions of x.] [In principle, equations

of

s

=

x2, so that

v

and hence

We can derive qualitative information about our implic­ itly determined solution. First, note that the integration constant

c

=

v(O)

=

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) de­ creases to a limit p 1 2:: 0. In fact, p1 = 0 since otherwise v = p + (s - 1)/sp has the positive limit v 1 = p 1 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 y 1 , say. (y 1 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

� ��

Proof: Observe that y"

y'' = =

(0 :S X :S 1).

(13)

= 2Vs(d/ds) ( -Vsp) so that, for 0 <

s gn

s

s

ds

-

-

where we have used

(9)

tivity of

and

p, s, sp2 + 1,

(10), 1 - s.

and

cline associated with slope m < with min

This approach

- 1 obtained by replacing y'

(8). We can put the resulting equation in the form:

2y/(1 - s) = 1/y - y!s so that y2

=

and we see that the isocline is a hyperbola having as asymp­ totes they-axis and the line

(

p

s(l - s) = xz 1 - x2 1 + x2 1+s

x=

�� �'

But these coordinates satisfy

-1 � y = -:;; :� � (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 con­

y = Y( 1 - x) gives the only decreasing linear u = P!Q, where P = x[(Y - y)(yY - 1) + x 2Y] and Q = (Y - y)Z(l - x 2) + x 2y2, 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 to verify that

solution. Now,

Py au Qy - = uy = - u-. Q Q ay

___

=

1/y

y-axis

or when

)

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 be­ as claimed. (Inflection must occur because near

cause it locates the boundary of the physical mirror. Con­

x0 as near 1 as pos­ 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 versely, it is clearly desirable to have

sible and to know how large we must take

(14)

THE MATHEMATICAL INTELLIGENCER

y = ("':,� 1) x. Moreover, the ver­

tex of the relevant branch of the hyperbola has coordinates

together with the posi­

We see that inflection occurs when

6

(8), we recover (13).

also leads to an interesting geometrical fact. Consider the iso­

siderably more complicated. However, it is straightforward

sp2 + 1 1 1 (Vsdp + 2vs . r p) = - gn ( s-1 + 2 ) 1 - gn( + 1 + ysp ) = sgn(l- yp), s-1 1 -s -

Upon substituting this in

2y(O) = 2y0,

since

s gn

-x y'=-. y

is given by

c

s < 1,

Yo--> + oo as x0 )" 1. (14) i s evaluated numerically, w e 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 which implies that

I f the integral i n

+prijii;JIM

We only outline the arguments supporting the remain­ ing assertions in this proposition. Note that along a solu­ tion curve

y" =

-(1

u(l + u 2)-ll2)u'

= !,u(x,y(x)), sgn y" = -sgn

u'(x)

in general,

=

+

(7) we have

= y'u'(1

+ u 2) -ll2

u'

= Ux + uyy' . Hence, u ' ; and at an inflection point, 0 with u.xUy 2: 0 (since y' < 0). Now, when (6) is used

where

u'

y(x) of

so that

for fixed Y, then formally

( 0, f)

u'

= R(x, y, y'),

where R is a rational function of its variables that is linear

-u - YT+U'2. By direct computation, we can u = xY and u'(x) * 0 at points on the horizon­ tal open segment M of height m = (Y2 + 1)/2Y between L and the y-axis. Moreover, since u(O) = 0, it is easy to ver­

in

y' =

show that

From the argument used at the beginning of the earlier sec­ tion titled "The View From Infinity," we see that, for each

y0 < Y,

y = y(x) of our equation on [0,1) with the initial value y(O) =Yo· More­ over, 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

there is a unique decreasing solution

(7), that

y'(xo) = -(V(xo!Y )2

+ 1-

ify that sgn y"(O) that, with sgn

y'"(x) = sgn ((y - m)[2x(Y - y - x 2Y)

x0 = 1, the situation is less clear. In fact, when Y > 4) that the point (1,0) ends the hyper­ bolic arc H defined by (Y - y)(y- l!Y) + x 2 = 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 avoids Hand L by having another inflection point. For y0 (1,Y), the solution curve must cross the circular arc

tion curves have no inflection points. We can extend this

argument to the case Yo

= l!Y where y"(O) = 0 but Y111(0) > 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 inflec­

0, since then

E

(m, Y), then Yo > m and y"(O) < 0; hence, y" cannot x with y(x) > m since there y"'(x) < 0. It

vanish at a "first"

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

C, de­

x < X£, YL < y :o:; 1), where YL = -Y(xL- 1), as shown. At the crossing point, (xc, Yc), say,

curve has slope

Q2]J,

x < 1 < Y, the second factor is not positive y = Y(1 ± x). When y0 E (0, l!Y), y"(O) > 0 and it follows that y" cannot vanish at a 111 "first" x value since there y (x) > 0; the associated solu­

E

:o:;

it can be easily verified from

p- ylp2 +

where, for 0 <

either crosses H with an intervening inflection point or it

1, (0

+

tion point, so that there cannot be more than one. Finally,

1, we note (see Fig.

y2 =

P and Q as before,

and it is strictly negative unless

if Yo

xo!Y) >-1.

But if

fined by x2 +

= sgn(l!Y- y0) when Yo < Y. If we fur­ y" = u ' = 0, we find (eventually)

ther differentiate and set

(6) and (7) that the solution

1

-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 prin­ cipal result: Proposition 2.

Suppose Y > 1. Then, ifYo 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 flection point.)

= Y the solution segment L has no in­

l@tijii;IIW VOLUME 24, NUMBER 1, 2002

7

AUTHORS

GERALD T. CARGO

JACK E. GRAVER

JOHN L. TROUTMAN

Department of Mathematics

Department of Mathematics

Department of Mathematics

Syracuse University

Syracuse University Syracuse,

NY 13244-1150

Syracuse,

USA

NY

Syracuse University Syracuse,

1 3244-1150

USA

NY 13244-1150 USA

e-mail: [email protected]

After earn i n g a Master's de g ree

in mathe-

matical statistics from the University of Michigan, Gerald Cargo served in the U.S. Army, where he worked with the world's first largescale computer, the ENIAC.

He returned

to

Mich igan and got a doctorate in 1959. Most

of hi s research publications have dealt w ith

Jack Graver, whose doctorate is frorn lndi-

John L. Troutman studied app lied mathe­

ana University, has been on

matics at Virginia Polytechnic Institute and at

the faculty of

Syracuse University for 35 years. His re-

search has been on desig n theory, intege r

and li nea r programming, and graph theory.

Among his books is an undergraduate exposition of ri g idity theory,

MAA, 2001 . He

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

inequalities or the boundary behavior of an-

gets particular satisfaction from teaching

ford and Dartmouth, and has recently

alytic functions. He also worked with h igh -

summer workshops for high-school teach-

after 30 years on t he mathematics faculty at

retired

for college credit. As Professor Emeritus he

ers, which he has done over the years in In-

Syracuse University. He has published arti­

d iana, New York, the Virg i n Islands, and Eng-

cles on real and complex analysis, and is the

has had time to cultivate his many interests,

land.

author of textbooks on variational calculus

school teachers who taught calculus courses

i nclud i n g math, travel, and swimm i ng

.

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 in­ verts 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 Y o = 1.245) gives the profile of a mirror that should faith­ fully 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 com­ puter-directed lathe1. REFERENCES

1 . Davis, P. 2.

=

��1 Patent

8

���� -

pending.

---------

THE MATHEMATICAL INTELLIGENCER

J. The Thread: A Mathematical Yarn.

Birkhauser, Boston, Kempe, A. B.

How to Draw a Straight Line.

Teachers of Mathematics, Reston, VA, 3.

Simmons, G. F.

The Harvester Press,

1983.

National Council of

1977.

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

This column is a placefor 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.

Michael Kleber and Ravi Vakil, Editors

The Best Card Trick Michael Kleber

Y

ou, my friend, are about to wit­ ness 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 ran­ domly, whichever you prefer-but do not show them to me! Show them in­ stead 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 il­ licitly? No, I assure you: the only in­ formation I have is a sequence of four of the cards you chose, and I can name the fifth one. The Story

Please send all submissions to Mathematical Entertainments Editor, Ravi Vakil,

Stanford University,

If you haven't seen this trick before, the effect really is remarkable; reading it in print does not do it justice. (I am for­ ever 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 re­ late some history and delay giving away the answer for a page or two. Fully appreciating the trick will involve

a little information theory and applica­ tions of the Birkhoff-von Neumann theorem as well as Hall's Marriage theorem. One caveat, though: fully ap­ preciating this article involves taking its title as a bit of showmanship, per­ haps a personal opinion, but certainly not a pronouncement of fact! The trick appeared in print in Wal­ lace 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 re­ ceiving his B.A and M.A. from the Uni­ versity 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 su­ pervised by C.L.E. Moore and titled "In­ finitesimal deformation of surfaces in Riemannian space." Fitch was an in­ structor and assistant professor then at the University of Hartford (Hillyer Col­ lege before 1957) until his retirement in 1971; he remained an aQjunct until his death in 1974. For a look at his extra-mathemati­ cal 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 Fran­ cisco in the early 1900s. He devoted countless hours to learning sleight­ of-hand skills and other "pocket magic" effects with which to enter­ tain friends and family. From the time of his initial teaching assign­ ments at Tufts College in the 1920s, he enjoyed introducing magic ef­ fects into the classroom, both to il-

Department of Mathematics, Bldg. 380, Stanford, CA 94305-2 1 25, USA e-mail: [email protected]

'Published by Seeman Printery, Durham, N.C., Hades International, Calgary, 1976.

1950:

Wallace Lee's Magic Studio, Durham, N.C.,

1960;

Mickey

© 2002 SPRINGER·VERLAG NEW YORK, VOLUME 24, NUMBER 1, 2002

9

lustrate points and to assure his

rums; on the

students'

once heard that it was posed to a can­

attentiveness.

He

also

tially by rank,

rec.puzzles newsgroup, I

A23 ... JQK, and break

ties by ordering the suits as in bridge

(i. e., alphabetical) order, 4- 0 \? •·

trained himself to be ambidextrous

didate at a job interview. It made a re­

(although

left-handed),

cent appearance in print in the "Problem

Then the three cards can be thought of

and amazed his classes with his abil­

Comer" section of the January 2001 Emissary, the newsletter of the Mathe­

the six permutations can be ordered,

naturally

ity to write equations simultane­ ously with both hands, meeting in

matical Sciences Research Institute. As

the center at the "equals" sign.

a result of writing this column, I am

Each month the magazine M-U-M, official publication of the Society of

e.g. , lexicographically.4

Now go out and amaze (and illumi­

learning about a slew of papers in prepa­ ration that discuss it as well. It is a card

nate5) your friends. But, please: just

make sure that you and your assistant

trick whose time has come.

agree on conventions and can name the hidden card flawlessly, say

American Magicians, includes a sec­ tion of new effects created by society

as smallest, middle, and largest, and

20 times in

a row, before you try this in public. As

The Workings

members, and "Fitch Cheney" was a

Now to business. Our "proof' of im­

we saw above, it's not hard to name the

regular by-line.

possibility ignored the other choice my

hidden card half the time-and it's

A number of his con­

tributions have a mathematical feel.

lovely assistant gets to make: which of

tough to win back your audience if you

His series of seven "Mental Dice Ef­

the five cards remains hidden. We can

happen to get the first one wrong. (I

fects" (beginning Dec.

1963) will ap­

put that choice to good use. With five

speak, sadly, from experience.)

peal to anyone who thinks it important

cards in your hand, there are certainly

to remember whether the numbers

1,

two of the same suit; we adopt the

The Big Time

2, 3 are oriented clockwise or counter­

strategy that the first card my assistant

Our scheme works beautifully with a

clockwise about their common vertex

shows me is of the same suit as the

standard deck, almost as if four suits

on a standard die. "Card Sense" (Oct.

card that stays hidden. Once I see the

of thirteen cards each were chosen just for this reason. While this satisfied

1961) encodes the rank of a card (pos­

first card, there are only twelve choices

sibly a joker) using the fourteen equiv­

for the hidden card. But a bit more

Wallace Lee, we would like to know

alence classes of permutations of

cleverness is required: by permuting

more. Can we do this with a larger deck

abed

which remain distinct if you declare

ac

=

ca and bd

=

db as substrings: the

the three remaining cards my assistant

of cards? And if we replace the hand

can send me one of only

size of five with

3!

=

6 mes­

n,

what happens?

sages, and again we are one bit short.

First we need a better analysis of the

whose four edges are folded over (by

The remaining choice my assistant

information-passing. My assistant is

the magician) to cover it, and examin­

makes is which card from the same­

card is placed on a piece of paper

sending me a message consisting of an

ing the creases gives precisely that

suit pair is displayed and which is hid­

much information about the order in

den. Consider the ranks of these cards

52 X 51 X 50 X 49

which they were folded. 2

to be two of the numbers from

1 to 13,

Since I see four of your cards and name

ble to add a number between

1 and 6

extract is an unordered set of five 5 cards, of which there are ( ), which

While Fitch was a mathematician, the five-card trick was passed down via Wal­

arranged in a circle. It is always possi­

ordered set of four cards; there are such

messages.

the fifth, the information I ultimately

l

lace Lee's book and the magic commu­

to one card (modulo

nity (1 don't know whether it appeared

other; this amounts to going around the

earlier in M-U-M or not.) The trick seems

circle "the short way." In summary, my

52 X 51 X 50 X 49 X 48/5!. So there is

to be making the rounds of the current

assistant can show me one card and

plenty of extra space: the set of mes­

math community and beyond, thanks to

13) and obtain the

transmit a number from

1 to 6; I incre­

for comparison we should write as

sages is

1:� = 2.5 times as large as the

mathematician and magician Art Ben­

jamin, who ran across a copy of Lee's

ment the rank of the card by the num­

set of situations. Indeed, we can see

ber, and leave the suit unchanged, to

some of that slop space in our algorithm:

book at a magic show, then taught the

identify the hidden card.

trick at the Hampshire College Summer

some hands are encoded by more than

It remains only for me and my as­

one message (any hand with more than

Studies in Mathematics program3 in

sistant to pick a convention for repre­

two cards of the same suit), and some

1986. Since then it has turned up regu­

senting the numbers from

messages never get used

larly in "brain teaser" puzzle-friendly fo-

totally order a deck of cards: say ini-

1 to 6. First,

(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 [email protected].

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 bound of n! + n- 1, this is a square cards, from which you draw a hand of matrix, and has exactly n! 1's in each n. Calculating as above, there are row and column. We conclude that d(d - 1) (d- n + 2) possible mes­ some subset of these 1's forms a per­ sages, and possible hands. The mutation matrix. But this is precisely a trick really is impossible (without sub­ strategy for me and my lovely assis­ terfuge) if there are more hands than tant-a bijection between hands and messages, i. e. , unless d :::; n! + n - 1. messages which can be used to repre­ The remarkable theorem is that this sent them. Indeed, by the above para­ upper bound on d is always attainable. graph, there is not just one strategy, While we calculated that there are but at least n!. enough messages to encode all the hands, it is far from obvious that we Perfection can match them up so each hand is en­ Technically the above proof is con­ coded by a message using only the n structive, in that the proof of Hall's cards available! But we can; the n = 5 Marriage theorem is itself a construc­ trick, which we can do with 52 cards, tion. But with n = 5 the above matrix can be done with a deck of 124. I will has 225,150,024 rows and columns, so give an algorithm in a moment, but first there is room for improvement. More­ an interesting nonconstructive proof. over, we would like a workable strat­ The Birkhoff-von Neumann theorem egy, one that we have a chance at per­ states that the convex hull of the per­ forming without consulting a cheat mutation matrices is precisely the set of sheet or scribbling on scrap paper. The doubly stochastic matrices: matrices perfect strategy below I learned from with entries in [0,1] with each row and Elwyn Berlekamp, and I've been told column summing to 1. We will use the that Stein Kulseth and Gadiel Seroussi equivalent discrete statement that any came up with essentially the same one matrix of nonnegative integers with independently; likely others have done constant row and column sums can be so too. Sadly, I have no information on written as a sum of permutation matri­ whether Fitch Cheney thought about ces.6 To prove this by induction (on the this generalization at all. constant sum) one need only show that Suppose for simplicity of exposition any such matrix is entrywise greater that n = 5. Number the cards in the deck than some permutation matrix. This is 0 through 123. Given a hand of five cards an application of Hall's Marriage theo­ co < c 1 < c2 < c < c4, my assistant will 3 rem, which states that it is possible to choose ci to remain hidden, where i = arrange suitable marriages between n co + c1 + c2 + c + c4 mod 5. 3 men and n women as long as any col­ To see how this works, suppose the lection of k women can concoct a list of message consists of four cards which at least k men that someone among sum to s mod 5. Then the hidden card them considers an eligible bachelor. Ap­ is congruent to -s + i mod 5 if it is ci. plying this to our nonnegative integer This is precisely the same as saying matrix, we can marry a row to a column that we renumber the cards from 0 only if their common entry is nonzero. to 119 by deleting the four cards used The constant row and column sums en­ in the message, the hidden card's new sure that any k rows have at least k number is congruent to -s mod 5. Now columns they consider eligible. it is clear that there are exactly 24 pos­ Now consider the (very large) 0-1 sibilities, and the permutation of the matrix with rows indexed by the four displayed cards communicates a hands, columns indexed by the number p from 0 to 23, in "base facto­ d!l(d - n + 1)! messages, and entries rial:" p = d 11! + d22! + d33! , where for equal to 1 indicating that the cards lex order, di :::; i counts how many used in the message all appear in the cards to the right of the (n- ith) are hand. When we take d to be our upper smaller than it. 7 Decoding the hidden ·

·

·

(�)

if

(�)

card is straightforward: take 5p + ( -s mod 5) and add 0, 1, 2, 3, or 4 to ac­ count for skipping the cards that ap­ pear in the message.8 Having performed the 124-card ver­ sion, 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 see­ ing four cards the performer both names the fifth and states whether the coin came up heads or tails. Encoding and de­ coding 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 en­ codes 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, respec­ tively. Thanks also to my lovely assis­ tants: 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.

6Exercise: Do so for your favorite magic square. 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

11

FEDERICA LA NAVE AND BARRY MAZUR

Reading Bombelli 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

d 'Histoire des Mathematiques:

Elements

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 equa­ tions and considers (to resort to modem language) cubic

Bombelli ...takes care to give explicitly the rules for

polynomials with "three real roots ."2 He produces the for­

calculation of complex numbers in a manner very close

mula (a sum of cube roots of conjugate quadratic imaginary

to modem expositions.

expressions) which yields ("formally," as we would say) a so­ lution to the cubic polynomial under examination.

This may be true, but is of limited help in understanding

Complex numbers, when they occur in Gerolamo Car­

the issues that the text is grappling with: if you open

dana's earlier treatise

Bombelli's treatise you discover nothing resembling com­

ties like 2 +

V-15.

Ars Magna,

occur neatly as quanti­

But they appear initially in Bombelli's

133,1 at which point certain math­

treatise as cubic radicals of the type of quantities discussed

ematical objects (that might be regarded by a modern as

by Cardano; a somewhat complicated way for them to arise

"complex numbers") burst onto the scene, in full battle ar­

in a treatise that is thought of as an organized exposition

plex numbers until page

ray, in the middle of an on-going discussion.Here is how

of the formal properties of complex numbers! Why doesn't

Bombelli introduces these mathematical objects.He writes,

Bombelli cite Cardano here? Why does he not mention his

''I have found another sort of cubic radical which behaves in a very different way from the others. "

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

Ho trovato un'altra sorte di R.c.Iegate molto differenti dall'altre. ...

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 cu­

The cubic radicals that Bombelli is contemplating here are the radicals that occur in the general solution of cubic

bic 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

problem of "using" the general solution by cubic radicals

number? Of course one can go in the opposite direction

to help you find, or even approximate, any of the three real

z and z3 with known cube root, and one might be lucky in guessing z, given y. Bombelli, for ex­ ample 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 = with ease: that is, one can take a complex number

cube it to get a number

y=

4 to the cubic equation x3

-

= 15x + 4. But the general prob­

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.

lem of extracting cube roots is of a different order. For how

Bombelli also develops a method (as he says, "in the plane")

would you go about solving the equation

for finding a real number solution to a cubic polynomial

(X +

iY)3 = A

+

equation. His method involves making a construction in

iB,

plane geometry dependent on a parameter (the parameter

or equivalently, the simultaneous (cubic, of course) equa­

being the angle that two specific lines in the construction

tions

subtend) and then "rotating" one of those lines (this "rota­ tion" effects other changes in his construction) until the lengths of two line segments in the construction are equal;

without having various eighteenth-century insights at your

these equal lengths then provide the answer he seeks. Later

disposal? There is surely the smell of circularity here, de­

in this discussion, we refer to this type of construction as

spite the fact that a "modem" can derive some simple plea­

a

sure in analyzing the 0-cycle of degree

sions-trisection of angle and

9 in complex pro­

jective 2-space given by the intersection of those two cubics. To Bombelli, his cubic radicals were indeed

new

kinds of radicals.

neusis

construction. To what extent do these discus­

neusis construction-play a ex­

role in providing a "demonstration" to Bombelli of the

istence of his yoked cubic radicals? We

discuss this in de­

tail in the latter part of this article.

Can we be content with this answer? A few paragraphs later Bombelli makes it clear that he

Tempering any answer that we might offer to any of these questions is the fact that the incubation period for

was quite dubious, at first, about the legitimacy of his dis­

Bombelli's text, and its writing, spanned more than two

covery and only slowly accustomed himself to it; he writes:

decades. Bombelli's treatise records the evolution of his

[This radical] will seem to most people more sophistic

of these questions change with time. Reading him may per­

thought, and the answers that Bombelli entertains for some than real. That was the opinion even I held, until I found

haps give us a portrait of an early father of algebra grap­

demonstration [of its existence] . . .

pling with what it means for a concept to

3

demonstration? existence? As we shall see, Bombelli only ascribes existence, whatever this means, to the yoked What, then, does Bombelli mean by

What does he mean by

sum of two cubic radicals (the radicands being, in effect,

conjugate complex numbers). As he puts it,

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-jour­ ney 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

It has never happened to me to find one of these kinds

drafts.

of cubic root without its conjugate.4 Bombelli's Writing Let us add a further element to this stew of questions:

Bombelli wrote in Italian (which, according to Dante, is the

In the "irreducible case," i.e., the case where the cubic poly­

language of the people). To our knowledge, his is the first

nomial has three real roots, does Bombelli believe that the

long treatise on mathematics written in Italian. He was

solution given by his "new kind of cubic radicals" corre­ sponds to any, or all, of the three solutions? (He seems to.)

faced, therefore, with something of a Dante-esque project: to choose words for existing terms (generally from Latin)

In what sense does Bombelli's general solution lead to a

and to invent Italian words for the various concepts that

numerical determination of one, or more, of the three roots

came along. That his book is in Italian has a mild disad­

of the polynomial? If you do not have Abraham de Moivre's

vantage, and a great advantage for a reader. On the one

insight, or anything equivalent, you may be stymied by the

hand many of Bombelli's neologisms never caught on, and

3Bombelli (1966), p. 1 33. 4Bombelli (1 966), p. 1 34. 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

ered a monstrous absurdity (Cardano called the expression

therefore must be carefully deciphered (we give a partial

containing square roots of negative numbers "sophistic and

glossary in Appendix B). On the other hand his style is quite

far from the nature of numbers" and also "wild").

personal (putting aside the lengthy computations about cu­

Bombelli gives a definition of

variable and notation for

bic irrationalities that are spelled out in prose !). At times

exponents. He studies monomials, polynomials, and rules

the text reads as if it were a private journal. To get a sense

for calculating with them. He treats the equations from the

of this, see Appendix A for a translation of his introductory

first to the fourth degree, and solves, among other things,

remarks. What we know of Bombelli's life comes, it seems,

all "42" possible cases of quartic equations (improving on

entirely from this treatise. More importantly, as already

the work of Ferraro and Cardano ). Following the practice

solid geometric demonstration

mentioned, Bombelli's informality allowed him to keep in

of the time, he also gives a

the text some of his early attitudes, as well as the changes

of the solution of cubic equations in terms of how a cube

in his outlook over the twenty-year period during which he

can be decomposed into two cubes and six parallelepipeds.

worked on

L 'Algebra.

Moreover, noticing the analogy between this problem and the classic problem of the insertion of two middle propor­

Bombelli and His Algebra

tionals, he also offers his

plane geometrical

construction

In

of the root of a cubic equation, which we discuss below.

calls himself "citizen of Bologna." Bombelli

This construction is perhaps superfluous for a cubic equa­

was a member of a noble family from the countryside

tion with only one real root, but it is necessary in the irre­

We do not know precisely where Bombelli was born.

L 'Algebra he

around Bologna. They came to Bologna at the beginning of

ducible case where the decomposition of the cube is im­

the 13th century. At the end of the same century they, be­

possible. In doing this Bombelli developed a geometric

ing "ghibellini," were forced to leave the city, and only re­

algebra (he refers to this as algebra linearia, that is to say linear algebra) which has a distinctly cartesian flavor. For

turned in the sixteenth century.

L 'Algebra he men­

at times Bombelli seems to be making the claim that geom­

tions his involvement in the project of draining the Chiana

etry is not necessarily the only way to prove things: rather,

Bombelli was a civil engineer, and in

swamp in Tuscany. He recounts that during periods of in­

certain geometric constructions are grounded in the un­

terruption of this project he wrote his book. The treatise

derlying

L 'Algebra as edited in a complete

algebra that represents

these

constructions.

edition in 1966 consists

Bombelli addresses the question of the relationship be­

of two "parts"6 which were, it seems, initially written in

tween the problem of the trisection of the angle and that

1550.7 After this first manuscript, Bombelli came to know Diophantus's

Arithmetic which was in a codex of the Vat­

of the solution of the cubic equation in the irreducible case. In his published treatise he expresses his intention to

use

ican Library. 8 Bombelli then made a general revision of his

the solution of the cubic equation in the irreducible case

manuscript and, among other things, included Diophantus's

to solve the angle-trisection problem.9 This represents a

problems in his text. He published none of it until 1572. At

change of viewpoint from the earlier version of his manu­

that time Bombelli published only the first part. He apolo­

script, in which Bombelli simply maintained that angle-tri- . section leads to cubic equations that cannot be solved. 10

gized, saying that he could not publish the other part be­ cause it had not yet been "brought to the level of perfec­

His treatise contains a collection of problems that in­

tion required by mathematics." However, it was surely

clude all the problems of the first four books of Diophan­

circulating among scholars, for in Bologna's libraries we

tus.

still find two copies of the manuscript. The second part of

damental text of advanced algebra. It was studied, for

L 'Algebra remained for more

than a century the fun­

the book was not published and was believed lost until the

example, by Christian Huygens and Gottfried Wilhelm

1920s when Bortolotti found the complete manuscript (not

Leibniz.

just the last part, but also the frrst in an unrevised version) in codex B 1560 of the "Biblioteca dell'Archiginnasio di

"Ho trovato un'altra sorte di R.c.legate molto

Bologna. "

differenti dall'altre . . . . "

Here is a run-down of the contents of Bombelli's five

books. As already mentioned, his great innovation was to

Here is how the text 1 1 continues. (We have shortened it a bit by putting the algebraic formulae in modem notation.)

have "solved" the "irreducible case" of the general cubic polynomial; i.e., the case when the root of Dal Ferro's for­

. . . I have found another kind of cubic root of a polyno­

mula for solving cubic equations involves the square root

mial which is very different from the others. This [cubic

of a negative number, a thing that at the time was consid-

root] arises in the chapter dealing with the equation of

6Part I consists of three "books"; Part II, of two. 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

14

THE MATHEMATICAL INTELLIGENCER

the division of a trinomial made by cubic roots of polynomials and number).

the kind :il = px + q, when p3/27 > q2/4, as we will show in that chapter. kind of square root has in its calcu­ lation [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 x 4 + ax 3 + b or x 4 + ax + b or x 4 + ax 3 + 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 above­ mentioned chapter on the plane). I will first treat multi­ plication, giving the law of plus and minus:12

This

bers as coefficients, so will treat separately (in different chapters) equations of the form x 3 + px q, etc. , terms be­ ing 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, treat­ ment of the general cubic equation =

x 3 = px + q. If we formally factor the polynomial

x 3 - 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 "for­ mula" for the real solution(s) to this polynomial is given by

( + )( +i) = +i (-)( +i) = -i (+)(-i) = -i (-)(-i) = +i ( + i)( +i) = ­ (+i)(-i) = + ( -i)( +i) = + (-i)(- i) = -

x=

if

Notice that this kind of root of polynomials cannot be obtained if not together with its conjugate. For in­ stance, 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 num­ ber and not a root (as we will see in solving equations). Yet, [even if the second quantity is a number] , an ex­ pression like 2 + 2i cannot be reduced to only one monomial, despite the fact that both 2 and 2i are num­ bers.

-\/

-\/

-\1

Commentary

The cube equal to a coefficient times the unknown plus a number refers to the equation which in modern dress is x 3 = px + q. Here, p is the coefficient and q is the number. Bombelli prefers to think of his equations having only positive num-

j

q 1,� - + - v - M3 + 2

6

j

q

1,�

- - - v - M3 2

6

'

where Ll is negative (and we are looking for the unique real solution) the above formula has an unambiguous in­ terpretation as a real number and gives the solution. If, however, Ll is positive (which is what Bombelli is en­ countering 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 negative and

J! - �

F

� �

is imaginary), the above solution, i.e.,

' %+

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 , you get the three real solutions. -

i v=LV3

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.

VOLUME 24. NUMBER

1 , 2002

15

q p

Figure

r

1

t and the u that worked; but ig­

Geometrical "Demonstration"

have to arrange to find the

Bombelli knows that any cubic polynomial has a root. The

nore this, and let us proceed. Substituting

(post-cartesian) argument (that a cubic polynomial

p(x) takes

on positive and negative values, is a continuous function of

p u=3t

x, and therefore, as x ranges through all real numbers, it must

traverse the value 0 at least once) is not in Bombelli's vo­ cabulary, but as the reader will see, there remains a shade of

in the equation

t3

-

u3 = q, we get

this argument in Bombelli's geometrical "demonstration." Bombelli convinces himself that cubic polynomials have roots by

two distinct methods-the first by consideration of not work in the irre­

volumes in space, a method that does

ducible case; and the second by consideration of areas in the plane, a method that does work in the irreducible case.

or

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

which we think of as a quadratic equation in

t3:

nesting in opposite comers of the big cube, these being of linear dimensions, say,

u

and

t - u,

and three parallelop­

ipeds, following the algebraic formula:

(t - u)3 + 3tu(t - u)

=

and applying the quadratic formula (available, of course, in

t3 - u3.

Bombelli's time) we get

s/27 2 ts = q ± Yq + 4p ' 2

Stripping the rest of Bombelli's demonstration of its geo­

p : = 3tu and t - u is a so­

metric language, here is how it proceeds. Put

q : = t3 - u3 ,

and note that the quantity

x :=

lution of the cubic equation

x3

+

px = q.

Of course, if we had such an equation with given con­ stants

p, q > 0

x = t - ft of the cu­ q. All this is per­ formable geometrically to produce the x only if t3 is a real i.e., Cardano's formula for the solution

which we wished to solve, we would first

bic equations of the form

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. 1 4This type of "decomposition of the cube" argument had already been used by Cardano in the Ars one can derive his formula; Cardano never considered the irreducible case.

16

THE MATHEMATICAL INTELLIGENCER

x3 + px =

Magna

to explain how, for a particular equation (x6 + 6x = 20),

The method in the plane

Bombelli's second method resembles some of the

neusis­

constructions used in questions of angle-trisection in an­ cient Greek geometry (see below), and indeed does work

features of this particular member of the family to the prob­ lem one wishes to solve. In the

Book of Lemmas Archimedes (3rd century BCE) neusis construction. (We

trisects a general angle using a

in the irreducible case. Bombelli promotes this method (in­

do not have the original Greek of this work; we have an

voking the august authority of the ancient authors, who

Arabic translation that does not seem to be completely

used similar methods) because, he claims, it provides a

faithful to the original Archimedean text.) Hippias (end of

"geometric demonstration" that his cubic radicals "exist."

By a gnomon let us mean an "L-shaped" figure; i.e., two

the 5th century BCE), instead, used a curve that he invented,

the so-called Quadratix of Hippias. By means of this curve

closed line segments joined at a 90 degree angle at their

it is possible to divide a general angle into any number of

common point, the

equal parts. Nicomedes (2nd century

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 man­ uscript) shown as Figure 1 . cubic equation x 3 bers

p

= px + q, i.e., from the pair of real num­

con­

choid curve by means of a neusis construction and he used

the conchoid to solve the problem of trisection. Apollonius (late 3rd to 2nd century

H e will construct such a diagram from the data o n his

BCE) made his

BCE) achieved angle-trisection us­

ing conics (the two cases we have, transmitted to us by Pappus in his

Mathematical Collection, use a hyperbola).

and q; from dimension considerations, we can ex­

pect p to appear as an area, and q/p as a linear measure­

Suggestions

ment. Let us calibrate the diagram by putting

We feel that there are two distinct elements that contribute to Bombelli's "faith" in cubic radicals.

lm = unity.

First, Bombelli deals with the "inverse problem," and he

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 -

and the area of the rectangle as the length

li.

on occasion, what the cube root of a specific number is + 1 1 v=I is 2 + v=l) and thereby ex­

(the cube root of 2

plicitly solves an equation (e.g.,

q

la = ­ ' p

for such a dia�ram, the root

does this in two ways: As mentioned, he explicitly tells us,

1 5x

x = 4 is a solution of x3 =

+ 4) saying that if one follows his geometrical method

for the solution of this problem one obtains that same so­

abfl equal to p, and moreover, x of his equation will appear

lution. But he also may simply start with a sum of two yoked cubic radicals,

Va + iVb + Va - iVb,

Neusis-Constructions and the Trisection of Angles

The problem of trisecting a general angle with the aid of

and discover the cubic equation of which this is a root. 1 6

no more than an unmarked straightedge and compass, as

Since he has proven by his geometric method that the cu­

posed by the ancient Greek mathematicians, is impossible.

bic equation has a real solution (in fact "three" of them), it

The fact that (the general solution of) this problem is im­

follows that this sum of two yoked cubic radicals in some

represents such a solution (and, thus, in some sense,

possible was established only in 1 837 by Pierre Laurant

sense

Wantzel, who also made explicit the connection between

represents a number). But whether it represents one, or all

trisection and solutions of cubic equations. But ancient

three, of the solutions is not dealt with. It would be diffi­

mathematicians had an assortment of methods of angle-tri­

cult, in any case, for us to say what it meant for Bombelli's

section that made use of "equipment" more powerful than

yoked cubic radicals to

mere compass and straightedge. One such method (re­

they don't lead to the determination or approximation of

ferred to as

the number that they represent.

neusis: verging,

inclination) useful for solving

represent numbers for him,

since

certain problems involves making (as in the gnomon con­

We have put quotation-marks around "three" when we

struction of Bombelli's that we have just sketched) a plane

discussed the "three" solutions to the cubic equation in the

geometric construction or, more precisely, a "family of con­

structions" dependent upon a single parameter

irreducible case because Bombelli does not consider neg­

of varia­

ative solutions. Nevertheless, by appropriately transform­

the construction" one can arrange it so that two designated

of an equation into positive solutions of the transformed

tion. 15 In general, the strategy is to show that by "varying

points on a specific line (of the construction) switch their

ing the equation, Bombelli is able to tum negative solutions equation. See page 230 where Bombelli transforms the

orem, that there is a member of the family where the two

x3 + 2 = 3x into the equation y3 = 3y + 2, where y = -x, and pp. 230-231 where Bombelli divides x 3 - 3x + 2 by x + 2 (y = 2). In his discussion of reducible cases of

designated points actually coincide. One then applies the

cubic polynomials, however, Bombelli talked of their (sin-

order on the line, under the variation. This then allows one to argue, in the spirit of the modem intermediate-value the­

equation

1 5For neusis see, for instance, Fowler (1 987), 8.2; Heath (1 92 1 ) , 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. 1 6Cf. 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 rele­

Rose

P. L. The Italian Renaissance of Mathematics.

Geneva: Librairie

Droz, 1 975.

vant "yoked cubic radical" so as to provide the two com­ plex roots of the cubic polynomial.

On the relation between angle trisection and cubic equations in Bombelli

Second, it seems to us that Bombelli gains confidence in the "existence" of his yoked cubic radicals through his abil­

see: Bortolotti, E. "La trisezione dell'angolo ed il caso irreducible dell'e­

ity to perform algebraic operations with them, and thirdly,

quazione cubica neii'Aigebra di Raffaele Bombelli,"

by his increased understanding of the relationship between

Bologna

Rend. Ace. di

(1 923), 1 25-1 39.

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.

school of mathematics see: Bortolotti, E. "I contributi del Tartaglia, del Cardano, del Ferrari,

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 Bambelli da Bologna. Libri IV e V

comprendenti "La parte geometrica " inedita tratta dal manoscritto B.

1 569, [della] Biblioteca deii'Archiginnasio di Bologna. Pubblicata a

cura di Ettore Bortolotti Bologna: Zanichelli, 1 929.

matematica nella Universita di Bologna .

Bologna: Zanichelli, 1 94 7. Bortolotti, E. "L'Aigebra nella scuola matematica bolognese del sec. Periodico di matematica ,

Cossali , Pietro. arricchita.

cubiche,"

Studi e mem. deii'Univ. di Bologna

9 (1 926).

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),

Smirnova G. S. "Geometric solution of cubic equations in Raffaele !star. Metoda!. Estestv. Nauk.

36 (1 989),

1 23-129. (Russian) On Bombelli and imaginary numbers see: Hofmann, J. E. "R. Bombelli- Erstentdecker des lmaginaren II,"

series IV (5) (1 925).

Origine, trasporto in ltalia, primi progressi in essa del­

l'a!gebra; storia critica di nuove disquisizioni analitiche e metafisiche

Math. ---

Praxis

1 4 ( 1 0) (1 972), 25 1 -254. . "R. Bombelli- Erstentdecker des lmaginaren,"

Praxis Math.

1 4 (9) (1 972), 225-230.

Parma: Reale Tipografia, 1 797-1 799. 2 vols.

Libri, Guillaume.

della

Bortolotti, E. "Sulla scoperta della risoluzione algebrica delle equazioni

Bombelli's 'Algebra,' "

and particularly in Bologna, see: Bortolotti, Ettore. La storia della

e

Scuola Matematica Bolognese alia teoria algebrica delle equazioni

1 38-1 46. (Russian)

On the mathematical environment at Bombelli's time in Italy in general

XVI,"

On cubic and quartic equations in Cardano, Bombelli, and the Bologna

Histoire des sciences mathematiques en ltalie, depuis

Ia reinaissance des lettres jusqu'a Ia fin du dix-septieme siecle.

Vols.

Wieleitner, H. "Zur Frugeschichte des l maginaren," Deutschen Mathematiker- Vereinigung

Jahresbericht der

36 (1 927), 74-88.

2 and 3. 2nd ed. Halle: Schmidt, 1 865. On Bombelli's For information about Bombelli's life see: Gillispie, Charles Coulston, editor in chief. raphy.

L 'Aigebra

and its influence on Leibniz see:

Hofmann, J. E. "Bombelli's Algebra. Eine genialische Einzelleistung und Dictionary

of Scientific

Biog­

ihre Einwirkung auf Leibniz,"

Studia Leibnitiana

4 (3-4) (1 972),

1 96-252.

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

Born belli e Ia sua famiglia." Atti Accad. Rend.

54 (1 963), 391 -395.

. "Documenti inediti degli archivi di Bologna intorno a Raffaele Sci. !st. Bologna C!. Sci. Fis.

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.

1 0 (2) (1 962/1 963), 235-247. On Bombelli as engineer see:

For the history of algebra during Bornbelli's age see: Giusti, E. "Algebra and Geometry in Bombelli and Viete," Sci. Mat.

Jayawardene, S. A. "Rafael Bombelli, Engineer-Architect: Some Un­ Boll. Storia

1 2 (2) (1 992), 303-328.

Maracchia, Silvio. Oa Cardano

published Documents of the Apostolic Camera,"

Isis

56 ( 1 965),

298-306.

a Galois: momenti di storia dell'algebra.

Milano: Feltrinelli, 1 979.

--- . "The influence of practical arithmetics on the Algebra of Rafael Bombelli,''

Isis

64 (224) (1 973), 51 0-523.

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 Bombelli,"

Arch. Hist. Exact Sci.

52 (2)

Une introduction a l'histoire de l'algebre.

Ancient World

Ball Rouse W. W., and H . S. M. Coxeter.

(1 998), 1 61 -1 93. (Italian) Sesiano, Jacques.

Books on Mathematical Problems in the

Lausanne:

Presses polytechniques et universitaires romandes, 1 999.

and Essays.

Bold,

Mathematical Recreations

New York: Dover, 1 987.

B. Famous Problems of Geometry and How to Solve Them.

New

York: Dover, 1 982. On the relationship between mathematicians and humanists in the re­ vival of Greek mathematics:

18

THE MATHEMATICAL INTELLIGENCER

Boyer, Carl B., and U. C. Merzbach. York: John Wiley & Sons, 1 989.

A History of Mathematics.

New

Bunt, Lucas N. H . , P. S. Jones, and J. D. Bedient. of Elementary Mathematics .

Englewood

The Historical Roots

Cliffs, NJ: Prentice-Hall,

Courant, R., and H. Robbins.

What Is Mathematics? An Elementary Ap­

proach to Ideas and Methods. New

York: Oxford University Press,

Great Problems of Elementary Mathematics: Their His­

tory and Solutions.

Trans. David Antin.

New

York: Dover, 1 965.

The Mathematics of Plato's Academy.

Fowler, D. H.

A

New

Recon­

struction. Oxford: Clarendon Press, 1 987. Grattan-Guinness, ences. New

[of their lack of interest in algebra] is the weakness or roughness of their own minds. In fact, given that all math­ ematics is concerned with speculation, one who is not

1 996.

Dorrie, H. 100

lvor. The Norton History of the Mathematical Sci­

York: W.W. Norton & Company, 1 997.

A Short History of Greek Mathematics .

Stechert & Co. , 1 923.

Heath, Thomas.

A History of Greek Mathematics.

speculative works hard, and in vain, to learn mathematics. I do not deny that for students of algebra a cause of enor­ mous suffering and an obstacle to understanding is the con­ fusion created by writers and by the lack of order that there is in this discipline. Thus, to remove every obstacle to those who are spec­

New York:

G.E.

ulative and who are in love with this science, and to take every excuse away from the cowardly and inept, I turned

Oxford: Clarendon

my mind to try to bring perfect order to algebra and to dis­ cuss everything about the subject not mentioned by others.

Press, 1 92 1 . Klein, Jacob.

tect themselves by making such excuses. If they were will­ ing to tell the truth they should rather say that the real cause

1 976.

Gow, James.

use. But I think rather that these people want only to pro­

Greek Mathematical Thought and the Origin of Algebra.

Trans. Eva Brann. Cambridge, MA: The M . I.T. Press, 1 968.

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 ex­

Appendix A.

Bombelli's Preface

To the reader

amining what most of the other authors had already writ­ ten on the subject. My aim was to compensate for what

I know that I would be wasting my time if I tried to use

they missed. There are many such authors, the Arab

mere finite words to explain the infinite excellence of the

Muhammad ibn Musa being considered the first. Muham­

mathematical disciplines. To be sure, the excellence of

mad ibn Musa is the author of a minor work, not of great

mathematics has been celebrated by many rare minds and

value. I believe that the name "algebra" came from him. For

honored authors. Nevertheless, despite my shortcomings,

the friar Luca Pacioli of Bargo del San Sepolcro from the

I feel obliged to speak of the supremacy, among all the

Minorite order, writing about algebra in both Latin and Ital­

mathematical disciplines, of the subject that is nowadays

ian, said that the name "algebra" came from the Arabic, that

called algebra by the common people.

its translation in our language was "position" and that this

All the other mathematical disciplines must use algebra. In fact the arithmetician and the geometer could not solve

science came from the Arabs. This, likewise, had been be­ lieved and said by those who wrote after him.

their problems and establish their demonstrations without

Yet, in these past years, a Greek work on this discipline

algebra; nor could the astronomer measure the heavens,

was found in the library of our Lord in the Vatican. The au­

and the degrees, and, together with the cosmographer, find

thor of this work is a certain Diophantus Alexandrine, a

the intersection of circles and straight lines without having

Greek who lived in the time of Antoninus Pius. Antonio

been compelled to rely on tables made by others. These ta­

Maria Pazzi, from Reggio, public lector of mathematics in

bles, having been printed over and over again, and fur­

Rome, showed Diophantus's work to me. To enrich the

thermore by people with little knowledge of mathematics,

world with such a work, we began to translate it. For we

are extremely corrupted. Thus, anyone using them for cal­

bothjudged Diophantus to be an author who was extremely

culation is certain to make an infinite number of errors.

intelligent with numbers (he does not deal with irrational

The musician, without algebra, can have little or no un­

numbers, but only in his calculations does one truly see

derstanding of musical proportion. And what about archi­

perfect order). We translated five books of the seven that

tecture? Only algebra gives us the way (by means of lines of

constitute his work We could not finish the books that re­

force) to build fortresses, war machines, and everything that

mained due to commitments we both had. In this work we

can be measured: solid, and proportions, as occurs when

found that Diophantus often cites Indian authors. Thus, I

dealing with perspective and other aspects of architecture.

came to know that this discipline was known to the Indi­

Algebra also allows us to understand the errors that can occur in architecture. Setting all these (self-evident) things aside, I will say

ans before the Arabs. A good deal after this, Leonardo Fi­ bonacci wrote about algebra in Latin. After him and up to the above mentioned Luca Pacioli there was no one who

only this: either because of the inherent difficulty of alge­

said anything of value. The friar Luca Pacioli, although he

bra, or because of the confused way that people write about

was a careless writer and therefore made some mistakes,

it, the more algebra is perfected the less I see people work­

nevertheless was the first to enlighten this science. This is

ing on it. I have thought about this situation for a long time

so, despite the fact that there are those who pretend to be

and have not been able to figure out why. Many have said

originators,

that their hesitations with algebra stemmed from the dis­

wickedly accusing the few errors of the friar, and saying

and ascribe to themselves all the honor,

trust they had of it, not being able to learn it, and from the

nothing about the parts of his work that are good. Coming

ignorance that people generally have of algebra and of its

to our time, both foreigners and Italians wrote about alge-

VOLUME 24, NUMBER 1 , 2002

19

A U T H O R S

BARRY MAZUR

FEDERICA LA NAVE

Department of Mathematics

Department of History of Science

Harvard University

Harvard University

Cambridge, MA 02138

Cambridge, MA 02 1 38 USA

USA

e-mail: [email protected]

e-mail: [email protected]

Federica La Nave is a graduate stu dent in history of science.

Her interests in clude classical philosophy, medieval log ic, and medieval

m us ic She works on Aristotle, Abelard, .

Duns

Sco­

tus, William of Ockham, and philosophical issues in mathe­

Banry Mazur is well known to lntelligencer readers for his math­ ematical contributions, especially to number theory and alge­ braic geometry.

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 cer­ tain 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 intelli­ gent 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.

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. Ac­ cept, 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 arith­ metic (or algebra), which may God like, it being in his praise and for the benefit of living beings.

1 7 Bortolotti, 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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21

11'1Ffii•i§rrF'h£119·1rr1rriil•iht¥J

Remem bering A. S. Kronrod E. M . Landis and I . M . Yaglom

Translation by Viola Brudno Edited by Walter Gautschi

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A

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lexander Semenovich Kronrod was born on October 22, 192 1, in Moscow. Sasha Kronrod discovered mathe­ matics when he was a participant in the now legendary study group for school­ children that was affiliated with Moscow State University. His teacher, D. 0. Shklyarskii, was a talented young scientist and an outstanding peda­ gogue. His general method was to en­ courage 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 para­ doxical statements-even by his ap­ pearance-he was tall and had a beau­ tiful 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 su­ pervised 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 dis­ continuity of a function that is differ­ entiable at the points of continuity. In 1939, Kronrod's first scientific article, in which this problem was solved, ap­ peared 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; stu­ dents at the graduate level were ex-

I

empt from conscription. In subsequent years, they were sent to military acad­ emies. In the early days of the war they were mobilized to build trenches around Moscow. On his return, he re­ newed his application for enlistment, was accepted, and was sent to the front. His military career was not easy. During the winter offensive of the So­ viet army near Moscow, his bravery re­ sulted 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 mathe­ matics, 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 nat­ ural 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 diver­ gent one? Kronrod greatly extended the scope of the problem. He managed to prove that, with respect to their action on (conditionally convergent) series, per­ mutations fall into several categories. There are permutations mapping some convergent series into divergent ones-Kronrod called these "left." Per­ mutations transforming some diver­ gent 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

�

Mathematical Communities Editor, Marjorie Senechal,

Department

of Mathematics, Smith College, Northampton, MA 01 063 USA e-mail: [email protected]

22

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.

THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK

A. S. Kronrod.

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 =

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 Matem­ aticheskii 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

au ax

=

au A(x y) .E!:'.. = -B(x y) � ' 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.

from the one-variable theory to the two­

the original function, a metric can be de­

It was not in his character to deal only

variable theory, features depending on

fmed and, on the tree, a function. The

with a particular problem; we will speak

variation dichotomize, so that for func­

linear variation then turns out to be

tions of two variables it is natural to in­

equal to the usual variation of the func­

life as well as in science). Dealing with

troduce two variations. One of them he

tion defined on a one-dimensional tree.

functions of two variables, Kronrod dis­

called

planar, the other linear. The

The boundedness of both the planar and

covered that, while the theory of con­

boundedness of the planar variation

linear variation guarantees the exis­

tinuous functions of one (real) variable

guarantees the existence almost every­

tence almost everywhere of the usual

had achieved some degree of complete­

where of an asymptotic total differential.

ness at that time, a theory of functions

For a smooth function, this variation

total differential.

of two (and more) variables simply did

turns out to be equal to the integral of

functions, but the concepts he intro­

not exist. Only the most elementary

the absolute value of its gradient, ex­

duced can easily be carried over to the

facts from the theory of functions of one

tended over the domain of definition.

case of discontinuous functions. He

below about Kronrod's maximalism (in

Kronrod

considered

continuous

variable had been extended, and they

The linear variation was basically a

also outlined a program for investigat­

did not contain anything "essentially

new object. Kronrod introduced the

ing functions of many variables, which

two-dimensional." If the theory does not

concept of a monotone function of two

was later carried out by his students. At that time, an active group

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 problem area Over four years, Kronrod de­ veloped an orderly theory, con­

Kronrod i ntrod uced the

of students congregated around

concept of a monotone

pedagogical

Kronrod.

(More of Kronrod's activity

is

dis­

cussed below.) Among them

function of two variab l es .

were A. G. Vitushkin, who de­ veloped a theory for variations

taining properties of functions of two

variables, a natural generalization of the

of functions and sets of many variables,

real variables and their connections

corresponding concept for a function of

and A. Ya. Dubovitskii, who studied in

with the concept of variation; it paved

a single variable. He proved that the

detail the set of clitical points for func­

the way for studying functions of many

boundedness of the linear variation per­

tions of many variables and smooth

variables.

mits the function to be represented as

mappings. In particular, he reproved A.

a difference of two monotone functions.

Sard's theorem, at the time not known

From the beginning, he avoided us­ ing definitions that depend on the choice

For the linear variation itself, he gave a

in Moscow, and he also obtained a se­

of a given orthogonal coordinate system

number of equivalent definitions, one of

lies of more refmed theorems on the

(e.g., Tonelli variation), and he intro­

which is of particular interest. It turns

duced concepts that are invariant with

out that with a continuous function of

From the modem perspective, half

respect to orthogonal mappings. Varia­

two variables one can associate a one­

a century later, it is not A. S. Kronrod's

tions of functions of two variables are

dimensional tree, the elements of which

results themselves that are of the

fundamental concepts for his theory.

are the components of the level-sets of

greatest interest. (They represent an

Kronrod showed that in the transition

the function. On them, with the help of

stl1lcture of the clitical points.

important but closed phase of devel­ opment.) The main value lies in the apparatus he created for obtaining the

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

Gauss points and adding 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

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

timally to a (2n + I)-point nlle by retaining the n n +

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.

Gc G � IR a smooth function, E1 = {x E Glf(x) = t} the level-sets of the function !, and ds the

following theorem of Kronrod: Let

!Rn be a domain andf:

(n

-

1 )-dimensional surface element

on E1• Then meas

Walter Gautschi

G

=

raxf f mmf

Et

(I:!: I) dt. v

f

of Computer Sciences

This theorem, for example, lies at the

West Lafayette, IN 47907-1 398

theory of partial differential equations.

Department

Purdue University USA

e-mail: [email protected]

basis of many modern proofs in the Kronrod's work on the theory of functions of two variables constituted the contents of his Masters thesis,

24

THE MATHEMATICAL INTELLIGENCER

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 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 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

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 used the "cascade method" (a kind of parallelism, a topical modern problem) and the Shannon counter, which was then largely unknown in 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

Kro n rod and Bessonov conceived

the idea of a u n iversal prog ram ­ control led dig ital com puter. various problems (e.g., independently of some other authors, he discovered the sweep method 1 ). 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-

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 2 4 , NUMBER 1 . 2002

25

pace. The machine was cheap, and un­ crease the speed, but in fact brought fortunately this created an attitude of down the speed to a very low level. Yet, low interest toward it. Quite competent the relay machine still remained his fa­ and well-meaning people gave Kronrod vorite accomplishment, bringing tears wise advice on how to speed up the when it was dismantled. During the period 1950-1955, Kron­ construction. For example, one could make contacts out of gold, which rod's main activity was finding numer­ would somewhat improve the quality ical solutions to physical problems. He of the machine, and would make it con­ collaborated much with physicists, in theoretical physicists, siderably more expensive. This would particular radically change the attitude toward among whom, with respect to work, he the machine. Kronrod could only laugh was closest to I. Ya. Pomeranchuk, at this kind of advice. His honesty and, on a purely personal level, L. D. would never allow him to use such Landau. For his work on problems of tricks. By the time the machine was importance to the state he was completed, a project to build the first awarded the Stalin Prize and an Order electronic computer had already been of the Red Banner. Only in 1955 did a real opportunity started. Thanks to the many rich ideas incorporated into the design of the arise for A S. Kronrod to work with an RVM, it would have operated at the electronic computer. It was the M-2 high speed of the EVM, but, of 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. We are talking about this RVM in computer constructed by I. S. Bruk, M. such detail in order to underscore one A Kartsev, and N. Ya. Matyukhin in the of A S. Kronrod's leading principles: an laboratory of the Institute of Energy idea is nothing; its implementation, named after Krzhizhanovski'i and di­ everything. Even though rich with bril­ rected by I. S. Bruk. This laboratory liant ideas, he did not value them. He later became the Institute for Elec­ gracefully gave them away left and tronic Control Machines. The mathe­ right, quite honestly convinced that the matics/machine interface was devel­ authorship belongs to the one who im­ oped by A L. Brudno, a great personal plements them. In this respect, he was and like-minded friend of Kronrod. At this point, a new period started quite the opposite of his teacher, Luzin. With regard to the RVM, he resolutely in the life of A S. Kronrod. We will declared Bessonov (definitely a tal­ speak about this later, but to preserve the chronological order, we will men­ ented person) to be its sole inventor. Having had a clear and deep insight, tion yet another aspect of his activity. Kronrod quickly realized the advan­ During the years 1946-1953, he led a tages of electronic computers over re­ seminar, called the Kronrod circle. At lay computers. He actively participated that time, it was probably not less in discussions on building the first known among young mathematicians EVM. He was a member of many and than the Luzin seminar. An atmosphere diverse committees planning to build of enthusiasm always surrounded the such a machine at that time. One must seminars he led. Its participants were say, though, that, his ideas often being convinced that mathematics was the ahead of their time, he was often left most important science and that in the minority in these discussions. A S. Kronrod was one of its prophets. For example, he unsuccessfully in­ At the same time, he was not the mas­ sisted on hardware support for float­ ter, but simply Sasha, and it so contin­ ing-point numbers. However, our first ued to the end of his days. His seminar machines used fixed-point numbers; studied the theory of functions of a real operations with floating-point numbers variable, set theory, and set-theoretical were implemented by means of soft­ topology. Work continued with the ware. This, theoretically, would in- same fervor, even after he left pure

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 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, 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

An idea is noth ing ; its i m ­ plementation , everyth i ng .

26

THE MATHEMATICAL INTELLIGENCER

chosen and programmed was the card

colleagues treated heuristic program­

game "crazy eights." This choice (in

ming and anything not connected with

him maximum ease and liberate him

spite of the smiles it provoked) was not

their immediate needs as mere enter­

from all tasks not requiring his qualifi­

accidental and not meant to be frivo­

tainment.

sary efforts, however, one must provide

cations. The mathematician would use a

lous. It is a complex game with no es­

He organized a chess match be­

tablished theory. Considering the low

tween the institute's program and the

guage, write on a form printed on high­

capabilities of the computer and its lim­

best (at that time) American program,

quality paper, using a pencil that allowed erasing an unlimited number of times.

language that is close to common lan­

ited memory, the game's simple de­

developed at Stanford University un­

scription of positions was very impor­

der the guidance of J. McCarthy. Over

There was a rich library of standard pro­

tant. The program was written and

the telegraph a match of four games

grams which were easily accessible. A

played. It worked fine as long as there

was played, ending with a score of

program (or any piece of it) would be

were enough cards remaining and in

3 to 1 in favor of the institute's program.

conditions of "incomplete information. "

However, the Mathematical Depart­

ing the code, punching cards, checking

After the game became open and every­

ment, of course, existed as a service

the cards-all this did not require the

sent to the coding center. Coding, check­

thing was reduced to an enumeration of

medium for physical problems, and the

programmer's attention. The next day he

all possible strategies, the computer's

time has come to say how this work

would receive two copies of the pro­

capacity was too limited to handle the

was organized by A S. Kronrod. This

gram without any coding or punching

extremely large size of the game's tree.

may be instructive, for in all scientific

mistakes. The debugging was done in

(The game was abandoned,

front of the control panel, and

never again to be resumed.

The prog ra m m i n g

It is not clear whether even

modem

computers

have

enough capacity for this

there was no time problem. A programmer was given as much access to the control panel as he

m ust be done by the

needed, and he did not need

mathematician .

into small blocks, each of which

game.) In the process of creat­

much. Programs were partitioned could be debugged separately and

ing the program, general heuristic programming principles were

institutes with a need for mathematical

usually ran the very first time. A correc­

formulated

service, work is organized differently.

tion could be introduced into a program

for the first time.

They pro­

Kronrod believed that a mathemati­

by pushing a key on the control panel,

grammed search (a priori it is not clear

cian solving the mathematical aspect

just as an editor does now. A woman re­

included

a

length-independent

whether this is possible or not), algo­

of a physical problem should under­

sponsible

rithms for organizing information, etc.

stand this problem, beginning with its

next to the programmer and could im­

for card-punching

worked

Since the "crazy eights" game clearly did

formulation, and should understand

not qualify as a standard text problem

how the results obtained are going to

because it was a strictly regional (or na­

be used. Moreover, the mathematician

next day, a corrected white card took its

tional) game, Kronrod proposed as a

must work out the algorithm, usually

place in the deck.

standard another game-chess. Chess is

according to the physical formulation,

mediately change the respective card.

For this, colored cards were used. The

Each program was required to un­

the

write the program, and run it. The pro­

USA, people had already started to

gramming must be done by the mathe­

general rule, strictly followed, was that

create chess-playing programs. Such

matician, because only in this way can

a program which worked and pro­

programs were already developed on

the optimal variant of the solution be

duced reasonable answers is not nec­

special-purpose machines: in the Math­

chosen. For this, one needs mathe­

essarily correct, even if the result is ac­

ematics Division of the ITEF a first, and

maticians with sufficiently high quali­

curate in special cases.

then a second M-20 machine was in­

fications, and Kronrod attracted many

It turned out that the work of the

stalled. The chess program was written

good graduates from the Faculty of Me­

coding and card-punching groups was extremely important in the course of

played throughout the world.

In

dergo a check by hand computation. A

by a group of mathematicians (Adel'son­

chanics and Mathematics to ITEF, also

Vel'skii, V. I. Arlazarov, A R. Bitman, and

those who specialized in abstract ar­

writing a program. These groups con­

A V. Uskov) which did not include Kro­

eas. Why precisely people from the

sisted of women, since they were be­ lieved to be more accurate in this kind

nrod himself. Nevertheless, when a dif­

Faculty of Mechanics and Mathemat­

ficulty was encountered regarding the

ics? He liked to quote I. M. Gel'fand:

of work; on each form for writing a pro­

development of a general recursive

"The objective of the Faculty of Me­

gram which was prepared for Kron­

search scheme, he entered the group

chanics and Mathematics is to make

rod's department, on the bottom was

and invented an improvement which

people capable," meaning that for a

written "program written by (a male

helped to overcome the difficulties. He

mathematician it suffices to formulate

name)," "coded by (a female name),"

assumed the role of an organizer. It was

the definitions and the rules operating

"coding checked by (a female name),"

necessary, but not easy, to create ap­

on them.

propriate working conditions for the

For a mathematician to be able to

chess group at the institute. Most of his

program, without expending unneces-

"punched by (a female name)," "punch­ ing 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 comput­

It must be said that Kronrod's per­

erized differential diagnostics for some

sonality attracted many talented peo­

diseases.

used in hopeless cases for patients who

were doomed to die. Milil became well

In the Cancer Institute named

ple from quite different fields. And

known and accepted to some degree:

after Gertsen, a laboratory was created,

while some of them were attracted by

A A Vishnevskll set aside a ward at his

Kunin, a

his professional competence (e.g., for

institute to treat patients according to

physicist by training and one of Kron­

the prominent oil researcher Lapuk he

the method of A S. Kronrod. Kronrod was promised a laboratory for animal

which was headed by P. E.

rod's students in heuristic programming.

had to compute the optimal regime for

The laboratory conducted research, in

exploiting oil and gas deposits), com­

particular on the differential diagnostics

municating with others involved quite

promise, and he did all the experiments

of lung cancer and central pneumonia.

different interests. You could meet at

on himself.

(The results were considered crucial for

his home with the actor Evstigneev, the

No longer a novice in medicine, Kron­

deciding whether surgery was needed.)

screenwriter Nusinov, and others. Kron­

rod replaced mathematics books with medical books, many of which he ob­

experimentation, but this remained a

Kronrod supervised the research. Quite

rod could be seen with academician

encouraging results were obtained. The

tained from physicians he knew. He al­

sudden death of

Kunin cut short this

I. G. Petrovski! at the Burdel sculpture exhibition, not discussing mathemati­

ready had considerable clinical experi­

cal problems, but questions of fine art.

ence. He kept a large card file on the

During this time, Kronrod organized

Among his friends also were prominent

history of patients' diseases. And he

work mathematics courses for high schools

physicians: the surgeon Simonyan, the

had

and developed teaching methods for

pediatrician Pobedinskaya, the radiol­

physicians: he could do a correct sta­

ogist-oncologist

and

tistical inference from the thousands of

of philan­

apist I. G. Barenblat (the father of the

them. After signing a petition in 1968 in support of the prominent dissident and

Marmorshtem,

others.

an

important

advantage

over

cards in his file. The well-known ther­

Having a keen

sense

logician Alexander Esenin-Volpin, the

thropy, with a strong desire to imme­

mechanical engineer G. I. Barenblat)

son of the famous poet Esenin, Kron­

diately help people, he was captivated

was struck, after a conversation with

rod was summarily fired from his po­

by the professional stories of physi­

Kronrod, by his medical erudition. And

sition at ITEF. He later became head of

cians, sharing their successes and fail­

is it surprising? If a very talented per­

the mathematical laboratory of the

ures. Gradually he understood that sav­

son works hard in a medical field, and

Central Scientific Research Institute of

ing the

if he is helped by good specialists, he

Patent Information (CNIIPI). Setting

important thing that can and must be

is likely to become proficient in it, at

up the mathematical and informational

done. At that time, he became ac­

least as much as an average, or even a

terminally ill

is the

most

part (and for this, among other things,

quainted with a Bulgarian doctor, Bog­

good student in a medical school. But

he

danov, developer of a medicine called

he did not have a medical degree, and

needed

to create software

for

In

Kronrod conditions for the machine

anabol, based on a Bulgarian sour milk

milil was not an approved medicine.

"Razdan" located at the CNIIPI and to

extract. This medicine often caused

the medical field, this could not be tol­

assemble a cohesive group of mathe­

remission in cancer patients. Inciden­

erated. Recall, for example, the story

maticians), Kronrod became interested

tally, Bogdanov treated i. N. Vekua and

of artificial pneumothorax.

in matters strictly related to patents

S. A Lebedev with anabol.

other hand, Kronrod did not treat pa­

and discovered that, here also, radical reforms were needed that would stim­ ulate inventions.

Kronrod

started

advertising

On the

this

tients without physicians, and was not

medicine. The medicine was not easy

paid for the treatment. In fact, he spent

to obtain, as it was produced in Bul­

his fortune on the treatment. (At the

Kronrod proposed a number of mea­

garia in limited quantities. Kronrod or­

sures that would help improve the pre­

ganized the delivery of this medicine

vailing situation, and entered the high

. for terminally ill patients. But this was

echelons, where he found understand­

not the solution; anabol was rare and

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 Kro­

ing. The director of the CNIIPI, who

expensive. It had to be produced in

nrod and, more seriously, his card files

was supportive of Kronrod, departed,

large quantities and by a simple proce­

were confiscated. The story had a tragi­

and the new director wanted to free

dure. Thus, a new medicine appeared,

comic ending. Either the mother or the

himself of such a worrisome colleague.

which was sour-clotted milk, based on

wife of the prosecutor who brought the

A S. Kronrod left the CNIIPI.

a Bulgarian milk extract. He gave this

case had cancer. And he needed milil.

medicine the name milil (in honor of

Naturally, the case was dismissed, and

stitute called the Central Geophysical

Mechnikov Il'ya Il'ich). He developed a

the card file was returned. But for A

Expedition. Here Kronrod headed a

simple technology for its production

S. Kronrod himself, the story turned

laboratory

and ways of using the medicine.

into a tragedy. He had a stroke, and he

His last employment was at an in­

processing

exploration­

drilling data. He implemented a series

Kronrod did not treat patients with­

out a physician. Physicians used milil ac­

completely lost his speech and his abil­

of new computational ideas, but this work, of course, did not match the

cording to his instructions (there were

very slow, but he learned how to speak,

level of his talent, and so he set new

more and more who came to believe in

read, and write once again. He left his

goals for himself.

Kronrod's medicine). The medicine was

position at the Central Geophysical Ex-

ities to read and write. Recovery was

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.

E. M. LANDIS

I. M. YAGLOM

E. M. Landis was born in 1 92 1 in Kharkov

Isaak Moiseevich Yaglom was born in

Bibliography: Publications of

and was raised in Moscow. He was ad­

Ukraine but raised in Moscow. His "can­

A.S. Kronrod

mitted to Mathematics and Mechanics at

didate's" and doctoral theses were on

Sur Ia structure de /'ensemble

Moscow State University in 1 939, b ut im­

extensions of some very classical geo­

des points de discontinuite d'une fonction

mediately had to leave for six years of mil­

metric ideas. Throughout his life he con­

itary service. Only after the war could he

tributed to mathematics of this sort and

get back to his studies.

championed it. Twice subjected to grossly

1 . A. Kronrod, derivable

en

ses

points

de

continuite

(Russian), Bull. Acad. Sci. URSS, Ser.

his early research he fol­

unfair dismissals from university posts

lowed the interest in real analysis of his

(1 949 and 1 968), he never lost heart, and

S. Kronrod. Later, his pri­

remained a sing ularly humorous and gen­

In much of

Math. [Izvestia Akad. Nauk SSSR] 1 939, 569-578.

first teacher, A.

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.

mary area was partial differential equa­

erous human being. Among his many

tions, and he had many results and many

books and articles, some of the most ad­

His achievements in

mired and widely read are historical es­

students in this area.

3. G.M. Adel'son-Vel'skiy and AS. Kronrod,

programming and algorithms

were widely

says and expository texts. He died unex­

On the level set of continuous functions

influential as well. He worked

at Moscow

pectedly in 1 988 of complications following

possessing partial derivatives,

State University from 1 953 until his death

an

in 1 987.

lived, he might feel today that his strug­

Dokl. Akad.

Nauk SSSR (N.S.) 50 (1 945), 239-241 .

4. G.M. Adel'son-Vel'skiy and A.S. Kronrod,

He was a music lover and could often

On the maximum principle for an elliptic

be found at the Moscow Conservatory.

system,

His paintings appeared in a faculty exhi­

Dokl. Akad. Nauk SSSR (N .S.) 50 ,

6. AS. Kronrod and E . M . Landis,

On level

allowance saturation,

5 (1 960), 5 1 3-51 4.

sets of a function of several variables

1 2. AS. Kronrod,

(Russian), Dokl. Akad. Nauk SSSR (N.S.)

accuracy,

58 (1 947), 1 269-1 272.

7. AS. Kronrod,

1 7-1 9.

On lin ear and planar varia­

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),

1 04 1 -1 044.

9. AS. Kronrod,

On surfaces of bounded

(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

Uspehi Matern.

Nauk (N.S.) 5

(1 950), no. 1 (35), 24- 1 34. 1 1 . AS. Kronrod,

Numerical solution to the

equation of the magnetic field in iron with

Integration with control of

Soviet Physics Dokl. 9 (1 964),

1 3. AS. Kronrod, Nodes and weights of quad­ rature formulas. Sixteen-place tables.

Au­

thorized translation from the Russian, Con­ 1 4. V.D. Belkin, A.S. Kronrod, U.A. Nazarov, The rational price calcula­ . tion based on co ntemporary economic in­

and V.Y. Pan, formation,

i

THE MATHEMATICAL INTELUGENCEA

Akad. Nauk SSSR, Ekonomika

Maternaticeskie Metodi (1 965) 1 , no. 5,

699-7 1 7.

gle to rehabilitate classical geometry was emerging victorious.

rod,

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.

1 9 . AS. Kronrod,

A nonmajorizable prescrip­

tion for the choice of a confidence region

1 5. V.L. Arlazarov, AS. Kronrod, and V.A. Kron­ 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,

30

Soviet Physics Dokl.

sultants Bureau, New York, 1 965.

Dokl. Akad. Nauk SSSR (N.S.) 66 (1 949),

ables,

he

(Russian), Rec. Math. [Mat.

Sbornik] N.S. 18 (60) (1 946), 237-280.

area

Had

On permutation of terms of nu­

merical series

tions

operation.

bition at the university.

(1 945), 559-561 . 5. A. Kronrod

uncomplicated

The choice of the step in the

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 Seq uences s a route to ever more exact knowledge, successive approximation has been a major theme in the development of science. Many algorithms to find approximations 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

aries have been a favorite showpiece in popularizing frac­

a dictionary, the first step is to open the dictionary close

tals (see for instance [DS]).

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 approx­

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.

imation. In the search for zeros of functions, you need to

How do these sequences behave? I will examine here the

know that a zero exists and how the map behaves in the

particular case of the quadratic family x

neighborhood of that zero. Newton formulated a general and simple method to fmd

c,

where

sion to

c is a real positive parameter.

E fR � fc(x) = :i2 +

The natural exten­

C of each map of the family has the real line fR X

{0 I

approximations of zeros of functions. For a real (or com­

as the boundary of the basins of attraction of its two (com­

plex) function f with a zero at {, and an initial choice x0,

plex) roots, so its geometry is trivial. However, the sequences

Newton suggested the following recurrence formula to ob­ tain better approximations of {: _

Xn + l - Xn -

f(xn)

f' (Xn)

After a clever change of variable, analysis of the se­

,

which is defined if the derivative off vanishes at no Xn, and

which, if convergent, will surely pick up a zero off as its

x1

(xn)n E No show irregular and unpredictable behavior, which nevertheless has an underlying order that I will describe. (xn)n E No will be straightforward by appealing to some easy techniques and results from dynamical systems quences

and elementary number theory. The main result is that ra­ tional initial conditions produce finite or infinite periodic

is obtained by considering the

sequences, whereas the irrational ones yield infinite but not

tangent line at (x0, j(x0)) to the graph ofj and intersecting

periodic sequences. This recalls what happens with deci­

it with the real axis; to get the whole sequence, just iterate

mal or binary expansions (luckily, even the terminology is

this process. Sufficient conditions for the method to work

the same), and the sensitivity with respect to the initial

limit. Given x0, the term

are easy to state, but a major problem arises: the competi­

choice x0 is evinced at once. Moreover, the dynamics as­

tion among the several zeros of the function. As a conse­

sociated with these sequences is modeled by a left shift on

quence, the basin of attraction of each zero (that is, the set

the binary representation of x0 in the new variable.

of initial conditions x0 such that the corresponding se­ quence

(xn)n E N o

converges to the specified zero) may

have a very complicated boundary, and the dynamics as­

sociated to the sequences (Xn)n E N o may be highly sensi­ tive to perturbations on initial conditions. These bound-

Let me start by taking a brief tour of discrete dynamical systems. Given a map

G : X � X,

I may compose

G with it­

self as many times as I please (the n-fold composition of G with itself is denoted by sequence

(Gn(x))n E

Gn).

Therefore for each x in X the

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 isomor­ phism between two dynamical systems

f: X � X

aN(b)N - l

and

+

·

· · +

a 1b

+ ao + c (i) + . . . + ck (it + . . . . . .

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 con­

information only about the digits ck. Rational numbers have

jugacy, the asymptotic behaviors of each orbit and how they

finite or infinite periodic representations in any base, in

vary with x. The fiXed points are the orbits easier to detect

general not unique; irrationals appear as unique non-peri­

and the ones to look for first; more generally, an orbit is

if it is a fiXed point of GP; if

periodic with period p E N nothing is said to the con­ trary, p is understood to be

the smallest period. An or­

pre-periodic with pre-period n E N0 and pe­ riod p E N if Gn(x) is a bit

is

It will be found useful to discard the integer part and keep

odic infinite representations. To simplify the notation, a pe­

Rational n u m bers have

· · · , ap , a 1 , a2 , · · · , ap, · · · · · · ), will be de­ noted by a 1 a2 · · · ap , and

fi n ite or i nfi n ite period ic

binary

riodic sequence of

I,

For maps G defined on subsets of �' the composition of

abbreviated to

G with itself may be pictured on the graph of G, and this is a good way of guessing how the orbits behave. For in­ stance, consider G : [0, 1 ] � [0, 1 ] given by G(x)

= x if and

only if x

= t,

=

1

- x.

for this is the only in­

tersection 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

dinality of their range of values. There are dynamical sys­ tems that contain essentially all the kinds of orbits that non­

=

(I )

w) =

i�

·

··, · ·

ap,

·

· · · · ·

)

(II )

E N,

=

, ap, a1, a2, · · · , ap, · · · · ·

·

for more details). I will consider each element of I as a bi­

nary expansion of a number in [0 , 1 ] ; in this process, the fi­

nite binary representation (of each dyadic rational) is

I, from 001 1 1 1 1 1

·

(III )

The denominator is even but not a power of 2-that is, r0 = t12nQ , an irreduciblefraction where Q is odd and n is a positive integer-if and only if the bi­ nary representation is infinite pre-periodic with a pre-period n.

ck •

• · · • · (b) with

aj, ck in {0,

1, · · ·

the number is given by the sum

32

THE MATHEMATICAL INTELLIGENCER

,b-

1}, meaning that

1.

= 0.000 1lc2)

has period

4 as 1/5

Cases (II) and (III) merit closer inspection: (N) If an

irreducible fraction of positive integers tJq E an odd denominator, it may be expressed in the form s/(2P - 1) where s and p are positive in­ tegers and are minimal. Once this is achieved, p gives the length of the period of its binary representation. ]0,

In expansion of the real numbers in a given base

b, each a0 · c 1 c2c3 · · ·

1/(2 · 5)

For example, and pre-period

· · · , although they are expansions of the

number is replaced by a sequence aN · · ·

=

=

12

0.01c2) is , and is distinct, in

same number.

4 = ¢(5) and 1/13 ¢( 13).)

riod

thought of as having an infinite tail of zeros: thus,

01000000 · ·

115 = 0.0011c2) 0.00 0100 1 1 1011c2) has pe­

dependent of the numerator. (For instance, has period

);

0 or 1 ordered by their length-see [D]

E

¢(q) is the number of positive integers less than q and

and it has dense orbits (e.g., that of the element of I that

the element of I given by

rational r0 E ]0, 1 [ has infinite binary represen­ tation with a period that starts just after the deci­ mal point if and only if it is an irreducible frac­ tion tJq where q is odd.

A

co-prime to q); in fact, it divides ¢(denominator) and is in­

is obtained by writing down consecutively all possible fi­ nite blocks of digits

rational ro E ]0, 1[ has finite binary representa­ tion 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.

A

In this case the (finite) representation of r0 has precisely

N,

odic points of all periods, because, for each p

ap, ab a2 , · (a1 , a2,

it is known

¢(q), where ¢ is the Euler totient function (for each q

is continuous with respect to the above metric; it has peri­

· · ·,

b=2

Furthermore, the length of the period does not exceed

l aj - bjl , 2j

= (al , a2, · · · , an, · · · ) and w = (b 1 , b2 , · · · , bn, · · · ). Acting on I, the one-sided full shift map u takes each se­ quence (a , a2 , · · · , an, · · · ) to (a2, · · · , an, · · · ). This map 1 for z

an +pan + l . . . . will be

an+ 2 . . . an+p . . · an an+ l an +2 · · · an +p·

n digits.

{(ab a2 , · · · , am · · · ) : aj E

oo

an + 2 · · ·

that (see [RT]):

is based on the space of sequences constructed with the

0 and 1, say I = {0, 1jf'' {0, 1 } }, with the metric

pre-periodic

information on its expansion in a given base can be read

injective maps may be expected to have. One such system digits

O.a 1a2 · ·

a

representation

from the denominator only. In the case

2: I suggest you check this on the graph of G.

to their topological properties, asymptotic behavior, or car­

uP(a1 , a2,

similarly

When a rational number is written in irreducible form,

The orbits may present many differences with respect

D(z,

(ai. a2 ,

representations in any base .

fiXed point for GP.

Then G(x)

say

1[

has

For example,

1

5

=

24

3

_

1

-

1

5 X 63

= O.OOllc2); 1:3 = 12 2

_

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, mini­ mal, 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 -

= 0.000 1 c2)·

1))

1

1

2(23

14

_

(IV) if n is also allowed to be zero; to prove (V), con­

nary expansion:

X

d1 d2

=QX =Q X

r1

As the remainders

+

+ r2

cause, by (II), the binary representation of 1/Q has a period that starts just after the decimal point. Therefore there ex­

ists a positive index p such that rp = 1 , and so the last of

the above equations, before they start repeating, is equation by

21' - 1,

them all to get

21' =

Q [ 21' -

Therefore [ 21'

1 Q

- 1 d1

dp +

1

.

2

X

Multiply the second the third one by 21'- 2 and so on, and add

1 d1

X

+ 21'- 2 d2 + . . .

+ 21'- 2 d2 + . . . +

21' -

1 ) = 0.00001 c2)·

_

r0 E iQ =>

N

r0 = k/2n => has a finite binary representation that terminates at 0 after n 3k,

nE

digits

3k, n E

N

:

3p E No : r0 = k/(21'(2n -

1))

=>

It is time to go back to Newton's method and the map

x0 E �. then the cor­ (xn)n ENo' if well defined, is

If I start with an initial condition

responding Newton's sequence

real and thus cannot converge: if it did, the recurrence for­ mula

Xn + 1

=

(x� - 1)/2xn

- 1.

would imply that the limit

� verifies the impossible equation 2L2

=

L2

+ 2dp - 1 + dp]

+ 1.

+ dp]

A

2dp - 1

1

LE

The dy­

namical system associated with this recurrence formula may be described by the iterates of the map C§ : � - �.

C§(t i= 0) = quence

(t2 - 1)/2t,

C§(O) = 0. If well defined, the se­

(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 New­ ton'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§:

21' -

( 1) Consider x0 1

'

so

At

t = --- ' 21' - 1

Q

= 0. 1010c2);

1)

r0 tE iQ => r0 has a unique representation, infinite, non­

f1 .

r1 are positive integers less than and co­

dp + rp = Q

_

Let me summarize for later use:

0

most. The first remainder to reappear is precisely 1 be­

= Q X

9

2(2 3

p and period n

< r1 < Q 0 < r2 < Q

r1

prime to Q, they repeat themselves after cjJ(Q) steps, at the

rp - 1

=

unique binary representation with pre-period

= Q X 0 + 1 X 1

9

14

periodic

sider the fraction 1/Q and the equations that produce its bi­

1 2 2

= o . oo0 1 c2);

1 1 = 2 28 2 (23

Let me sketch a proof of these two properties. (V) im­ plies

1)

1;

Xn

= 1; then C§(x0) = 0, so C§n(x0) = 0 for n :::::

is not defined for n ::::: 2. I describe this by saying

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 ::::: that the orbit of

3. This orbit is also finite and terminates at 0 after two iterates.

s

At

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

(3) Take Further, the type of the binary representation of sf(� (2P is the same as that of

1/(2n (21'

-

1))

- 1)), and the latter may be

of degree

obtained from the following calculation:

1

2n (21' -

1

1)

2n

1 - 1121' 1

I �

2 j� 1

(-)j 1

21'

So

= o.o . . . ooo . . . mc2),

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. No­ tice that if the denominator is even but not a power of then p must be bigger than or equal to

2.

2,

The effect of the

in the denominator is to push the period to the

right, creating a pre-period of length this on some examples, such as

x0 is a pre-periodic orbit of period two and pre-pe­

riod one.

where the first block of zeros has size

2n

.

(4) If Xo = V3, then C§(x0) = llv'3 and C§2 (C§(xo)) = C§(xo).

1/21'

= ----;;:;:

power

-

4 with only even exponents; it has no solu­ llv'3 and 11V3

tions other than

n. I suggest you check

More sophisticated tools are needed to detect other kinds of orbit. The recurrence formula Xn + 1 =

(2xn)

((xn? - 1)/

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 ]0,

1T [

r0 E

]0, 1 [: this is permissible since cotan:

- � 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 cotan( 1T2nr0) for each n, provided that 2n 11ro is

C§n (x0) =

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 dyadic rationals; more precisely:

n are just the

r0 k/2n, with k, n E N and k an odd in­ teger, if and only if the orbit by <§ terminates at 0 after n iterates. 1st Conclusion:

=

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. Con­ versely, 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 or­ bits 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 ra­ tionals 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 deter­ mined by the denominator of r0. In particular, if ro is ir­ rational, then Xn is defined for all n E N . Let me review in this new setting some of the above ex­ amples.

(a) ro =

-2

113 11(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. =

=

=

=

-

P 2, and cotan(?T/3) llv'3. The orbit of x0 is pre-periodic with pre-period N 1 and period P 2.

(b) r0 = 1/6 112(22 - 1) 0.001c2J: N 1, x0 = cotan(?TI6) = V3, xi = cotan(27r/6) =

=

=

=

=

=

=

=

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.

(c) ro =

=

=

=

=

=

=

I suggest you now compare the following diagram with the similar one above.

I

l

r0 $. OJ ==> its orbit by <§ is infinite non-periodic ::lk, n E N : r0 k/2n ==> its orbit by <§ ==> terminates at 0 after n iterations ro E Q ::lk, n E N 3p E N 0 : 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 suc­ cessive values of cotan( 'lT2nr0), what matters is the frac­ tional part of 2nro (denoted by {2nr0}). If the irrational ro is written in base 2 as ro = 0 . aia2a3 · ak · · (2), this rep­ resentation is unique, and 2ro a I . a2a3 ak · (2) · Dis­ missing the integer part, we are left with {2r0} = O.a2a3 · · ak (2) and, by (2), ·

·

·

·

·

=

·

·

·

·

·

·

·

(cotan({ 'lT2nro}))n E N0 = (COtan( 'lT . 0 . an + I an + 2 . . . (2J))n

(cotan('lT2nro))n E No

=

E

N0,

which corresponds, up to the action of cotan o ('lT X ), ex­ actly to the iteration a n of the shift on the sequence . More precisely, the map aia2a3 · · ak ·

·

·

·

·

]0, 1[ I {dyadic numbers} � ]0, 1[ I {dyadic numbers} 0 . aia ak (2) � 0 . a2a3 ·. ak · · · · · C2J 2 ·

·

·

·

·

·

·

�,

if � :=:::

(that is, '!f(t) = 2t if 0 :=::: t < '!f(t) 2t - 1 t < 1) is conjugated by z � cotan( '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 . a i a2

·

·

·

ak

·

·

·

C2J)

=

a i a2a3

=

·

· ·

ak

·

·

·

· · ·

is a conjugacy between the chosen restrictions of '!f and a. Let me illustrate the use of these observations in two examples:

(i)

c2J, where each digit If r0 0.10100100010000 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 for all n. But, since {2nr0} gets arbitrarily close to 0, this orbit is not bounded from above. ·

=

·

·

·

·

·

=

·

-i4

·

·

=

(ii) If r0 is an irrational number whose binary representa­ tion 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 rep­ resentation with ending zeros (e.g., writing 112 0. 10000 c2J instead of 0.0111 c2J), then the corre­ sponding extension of h is not continuous. However, I I let 'JC(x) = h((li?T)cotan - x) , then the equation a o 'JC(x) = 'JC o <&(x) is still valid for all x =I= 0. This yields the following: =

·

·

·

·

·

·

if

The dynamics of the Newton's sequences initial conditions Xo, is deter­ mined by the binary representations of the initial con­ ditions in the new variable r0. 3rd Conclusion:

(xn)n E N0, for allowed real

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 uninterest­ ing: for all Xo E IR, the sequence ((C£10)n (xo))n EN has limit 0, the unique fixed-point of C£10. If a > 0, then for x of=. 0 we have

A U T H O R

=

=

=

=

MARIA PIRES DE CARVALHO Centro de Matematica Prac;:a Gomes

do Porto

Teixeira

4099·002 Porto

Portugal e·mail: mpca!Val@fc,up.pt

that is,

Maria Carvalho and her twin sister were born in Africa. She

completed her first degree in mathematics at the U niversity of Porto, where she is now an associate professor. Her post­

graduate studies were completed at l nstituto de Matematica

Pura e Aplicada, in Rio de Janeiro, where she specialized in Ergodic Theory and completed her Ph.D. under the guidance

This suggests the change of variable

of Ricardo Mane, Maria shares a cat with her husband and is

xo to - -, a _

enthusiastic about l i te rature and jazz music,

which leads to

t1

=

x1 a

=

the quadratic family Cfc)c extend easily to all quadratic poly­ nomials. Given a polynomial p(x) d2x2 + d 1x + 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 be­ comes

(to)2 - 1 2t0

=

=

and, in general, to

=

=

=

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 imag­ inary 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 . . - 1, �Pn + l 2 �Pn 2Pn .

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 =

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 0 1 267 USA e-mail: [email protected]

36

Colin Ada m s , Editor

Fields Medalist Stripped Colin Adams

M

arch 3, 2005: The International Congress of Mathematics an­ nounced today that Wendell Holcomb will be stripped of his Fields Medal af­ ter testing positive for intelligence-en­ hancing drugs. Holcomb has denied the charges. "Just because I never finished high school, and then solved the three­ dimensional 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 di­ mension up, which I called the 3sphere. But could a different 3-dimen­ sional 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 ob­ ject 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 math­ ematicians. 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.

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 co­ operation of the administration. But Holcomb had been hoofing it over to the Seven Eleven and using the facili­ ties there to avoid detection. Ulti­ mately 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 conjec­ tures 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 intel­ lectual alertness. It is acknowledged that without coffee, mathematical pro­ ductivity would have been half of what it was. But the new class of beta-en­ hancers 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 Car­ olyn 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 degrada­ tion 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 dimin­ ished that the drug brings one back up to a functional level only, and then not even that." Holcomb plans to appeal the deci­ sion. "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 i c and N u merical Co m parison Between Iterative Method s Dedicated to the memory of Jose J. Guadalupe ("Chicho''), my Ph.D. Advisor

l

et 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

Xn+l

=

f(xn) Xn - ----;--- n = 0, f (Xn) '

1, 2, . . .

generates a sequence {xn J:= 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 New­ ton-Raphson method. (It has also been described as the tan­ gent 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

Zn+l

=

fCzn) Zn - ----;--- n = 0, f (Zn) '

1, 2, . . . ,

( 1)

he looked for conditions under which the sequence {zn }�= o converges to a root. In particular, if we denominate the at­ traction basin of a root ? as the set of all z0 E C such that the method converges to ?, he was interested in identify­ ing the attraction basin for any root. He solved the prob­ lem whenf is a quadratic polynomial. For cubic polynomi­ als, 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 New­ ton's method fails to converge-has fractional dimension, and it coincides with the frontier of the attraction basins k of the three complex roots e2 7Ti13, k = 0, 1, 2. With the aid of computer-generated graphics, we can show the com­ plexity 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 itera­ tive 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 numeri­ cal 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.

Figure 2. Newton's method for multiple roots.

Newton'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 z n - 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 (z 2 - 1)(z 2 + 0. 16) (due to S. Sutherland, see the cover illustration of [ 17]).

2

-

Figure

3.

Convex acceleration of Whittaker's

method.

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 {zn l�=O be a complex sequence. We say that is the order of convergence of the sequence if

l zn+ l - � . �co I Zn - �ri a nhm

=

C,

-1

4. Double

convex

acceleration

Whittaker's method.

38

THE MATHEMATICAL INTELUGENCER

of

Figure

)

oo

(2)

� ::::::

2 -2

E [1,

where � is a complex number and C a nonzero constant; here, if a = 1 , we assume an extra condition l e i < 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 -

2

Figure

a

5. Halley's method.

Figure

6.

Chebyshev's method.

Figure

7.

Convex acceleration of Newton's

Figure

8.

Figure 9. Steffensen's method.

(Shifted) Stirling'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 sume that they have, respectively, orders of con­ vergence a and a', where ' a > a . Then, it is clear that, asymptotically, the sequence {znl�=o con­ verges to its limit more 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 in­ dex (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 a 11d,

where a is the order of convergence. For the methods that I am dealing with here, it is easy to derive both the infor­ mational efficiency and the efficiency index from the or­ der. 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 instance, let us suppose that we have an iterative process Zn+ 1 =
The order of convergence is

=

used to compare the speed of

convergence of sequences .

�

Figure

10.

Midpoint method.

Figure

11.

=

2

Traub-Ostrowski's method and

Figure

12.

Inverse-free Jarratt's method.

Jarratt's method.

VOLUME 24, NUMBER 1 . 2002

39

count the computational work involved in computing f, 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 to f is E(cfJ,f) = a 11!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 nec­ essary to fmd the root with a required precision. Finally, note that, to ensure the convergence of an iter­ ative 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 guar­ antee the convergence (I will give references for each method) are hard to check; and, moreover, are too de­ manding. So, if we want to solve f(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 differ­ ent functions (simple and hard to evaluate) that allow com­ parisons 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 con­ ditions on the initial point.

f,�l has a simple root at { Then, we only need to apply the ordinary Newton's method to the equation g(z) = 0.

•

with L1 (z) =

=

•

•

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

Double convex acceleration of Whittaker's method [ 1 1 ] :

This is a new convex acceleration for the previous iter­ ative process. It has order 3. •

Halley's method (see [ 18, p. 91], [3, p. 247], [9, p. 257], [8]): Zn+ 1 = Zn -

f(zn)

2

J' (Zn) 2 - Lj(Zn)

= Zn -

1 f'Czn) f(z.)

j"(zn) 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 fre­ quently rediscovered iterative functions in the literature. From its geometric interpretation for real functions, it is also known as the method of tangent hyperbolas. Alter­ natively, it can be interpreted as applying Newton's method to the equation g(z) = 0 with g(z) = f(z)l . Its order of convergence is 3.

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 refer­ ences. 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:

j(z)f"(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 ac­ celeration is an order 2 method.

The Numerical Methods

•

Convex acceleration of Whittaker's method [ 1 1 ] :

v7'(Z)

•

Chebyshev's method (see [ 18, p. 76 and p. 81] or [3, p. 246]): Zn+ 1 = Zn -

ftzn) f' (zn)

(

)

Lj (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 super­ Halley method [7]:

Zn+ 1 = Zn -

1 - LJ (Zn) j(Zn) = Zn - ' f (zn)

(

1

1 +

2LJ (Zn)

)

point methods can be found in [ 18, Ch. 8 and 9]. Let us look at some of them. •

1 - Lj(Zn) .

Zn + 1 = 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+ 1 = Zn -

Zn - F(zn) 1 - F' (F(zn))

(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 tech­ niques for solving nonlinear equations that require the evaluation of f or its derivatives at more than one point in each step. These iterative methods are known as mul­ tipoint methods. They are usually employed to increase the order of convergence without computing more deriv­ atives of the function involved. A general study of multi-

f(zn)

-

g (Zn)

j(z +j(z)) - f(z) · one of the simplest mul. Th"IS IS WI"th g(z) = f(z) 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+h f(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). ·

t

=

•

Midpoint method (see [ 18, p. 164] or [3, p. 197]): Zn+ 1 = Zn -

.

If we want to solve an equation fl:z) = 0, we can trans­ form 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 non­ vanishing 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 solve j(z) = 0 by means of an iterative method Zn+ 1 = ¢(zn), like the ones shown above, and {zn l�=O converges to �. it is clear that � is a fixed point for ¢ (upon requiring that ¢ be a continuous func­ tion and taking limits in Zn + 1 = ¢(zn)). So, without notic­ ing, 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 hypothe­ sis. In this way, we have •

Steffensen's method (see [15, p. 198] or [ 18, p. 178]):

(

)

fl: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) 2j(Zn - U(Zn)) - j(zn)

Zn + 1 = Zn - U (Zn)

J,�;l.

Its order of convergence is 4 , the high­ with u(z) = est for the methods that we are studying. •

Jarratt's method [ 12, 2] (for different expressions, see also [3, p. 230 and p. 234]): 1

Zn+ 1 = Zn - -;;_u(Zn) +

where, again, u(z) = •

f(zn)

------

f

j' (Zn) - 3j' (Zn - u(zn))

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)

j(z) . With u(z) = f'(z) and h(z) der 4 method.

=

(

f

)

1 - h(Zn) ,

j'(z - �u(z)) - f'(z) f'(z)

. Also an or-

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 f(z) = z3 - 1 and .f'(z) = exp

(100) sin(z)

(z 3 - 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 gen­ eral 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.

Table

Function f and rectangle Rb Ord

Eff

Nw

2

1 .41

1/P

NC

T

P/S

3.

1/S

Function

and rectangle Rb

f*

Ord

Eff

NC

1/P

2

1 .4 1

3.06

8.17

T

P/S

1/S

0.00267

7.52

0.00381

7.93

1 .17

0.857

0.904

NwM

2

1 .26

8.2

1 .4 7

0.681

0.683

1 8.9

3.23

0.309

0.778

CaWh

2

1 .4 1

33.2

1 9.9

3.58

0.279

0.679

1 8. 1

11

0. 71 4

Nw

NwM

2

1 .26

CaWh

2

1 .4 1

DcaWh

3

1 .44

0. 1 25

6.5

1 .41

0. 71 1

0.6 1 5

DcaWh

3

1 .44

Ha

3

1 .44

0

4.38

0.901

1.11

0.646

Ha

3

1 .44

Ch

3

1 .44

0.0492

6.27

1 .11

0.902

0. 752

Ch

3

1 .44

CaN/sH

3

1 .44

0

3.82

0.81 5

1 .23

0.623

CaN/sH

3

1 .44

86.6

36.4

4.71

0.212

1 .03

Stir

2

1 .4 1

87.7

36.5

4.04

0.248

1.10

85

35.7

5.79

0.1 73

0.820

Steff

2

1 .41

84.5

35.6

3.39

0.295

1 .28

Stir

2

1 .41

Steff

2

1 .41

24.5

2.86

0.321 1 1 .5 1 .92

1 .88

0.532

4.48

0.91 8

1 .09

0.597

9.1 1

1 .56

0.641

0.7 1 4

4.59

0.907

1 .10

0.61 9

Mid

3

1 .44

4.62

6.32

1.1

0.91 1

0. 766

Mid

3

1 .44

5.61

6.57

1 .2 1

0.824

0.662

Tr-Os

4

1 .59

0

3.69

0.696

1 .44

0. 705

Tr-Os

4

1 .59

1.10

4.03

0.677

1 .48

0. 729

Ja

4

1 .59

0

3.69

0.699

1 .43

0. 702

Ja

4

1 .59

3.99

0. 777

1 .29

0.628

lfJa

4

1 .59

1 .62

7.45

1 .41

0.71 1

0.705

lfJa

4

1 .59

1 . 71

0.584

0.797

eral. (Note that many of these iterative methods are also adapted to solve systems of equations or equations in Ba­ nach spaces. Here, to evaluate Frechet derivatives is, usu­ ally, very difficult.) I take a rectangle D c IC and I apply the iterative meth­ ods 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 dif­ ferent regions: the rectangle Rb = [ - 2.5, 2.5] X [ - 2.5, 2.5] and a small rectangle near the root e2 7Ti13 ( - 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 max­ imum of 40 iterations. The three roots are denoted by �k = e2k 7T'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 Nw

2.

Function f and rectangle Rs Ord

Eff

NC

1/P

2

1 . 41

0

2.97

P/S

1 1 .2

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 (mid­ point), Tr-Os (Traub-Ostrowski), Ja (Jarratt), IfJa (inverse­ free 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 T

0.965 19

1/S

4.

Function

f*

and rectangle R5

Ord

Eff

NC

1/P

Nw

2

1 .4 1

0

2.97

T

P/S

1/S

0.666

NwM

2

1 .26

0

2.97

1.1

0.91 0

0.910

NwM

2

1 .26

0

2.97

1 .50

0.666

CaWh

2

1 . 41

0

3.23

1 .39

0.71 9

0.781

CaWh

2

1 .41

0

3.22

1 .67

0.599

0.649

DcaWh

3

1 .44

0

2

1.1

0.91 1

0.6 1 3

DcaWh

3

1 .44

0

2

1 .1 3

0.883

0.594

Ha

3

1 .44

0

2

1 .03

0.974

0.656

Ha

3

1 .44

0

2

1 .1 0

0.906

0.61

Ch

3

1 .44

0

2

0.914

1 .09

0.737

Ch

3

1 .44

0

2

1 .06

0.944

0.636

CaN/sH

3

1 . 44

0

2

1 .06

0.946

0.636

CaN/sH

3

1 .44

0

2

1 .12

0.895

0.602

Stir

2

1 .4 1

0

4.15

1 .36

0.733

1 .02

Stir

2

1 .4 1

0

4.13

1 .38

0.724

1 .01

Steff

2

1 .41

0

3.44

1 .42

0.706

0.82

Steff

2

1 .41

0

3.43

1 .06

0.945

1 .09

Mid

3

1 . 44

0

2

0.898

1 .1 1

0.749

Mid

3

1 .44

0

2

1 .02

0.979

0.659

Tr-Os

4

1 .59

0

1 .96

0.925

1 .08

0.714

Tr-Os

4

1 .59

0

1 .96

0.909

1 .1

0.727

Ja

4

1 . 59

0

1 .96

0.928

1 .08

0.712

Ja

4

1 .59

0

1 .96

1 .04

0.959

0.634

lfJa

4

1 .59

0

1 .99

0.969

1 .03

0.690

lfJa

4

1 .59

0

1 .99

1 .05

0.955

0.639

42

THE MATHEMATICAL INTELLIGENCER

To construct the tables, I used a Power Macintosh

82001120

C + + program in a

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 per­ mits 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 · · J*CzJ · do not s1mp lify any factor m Cf*J'CzJ . Also, 1·f a subexpresswn 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 func­

c:f>(z)

1 + 12z3+54z6+ 14z9

6zz + 42zs + sszs

=

of them is the same (Figure the data of Tables

c:f>(zn) for J, adapting and simplifying its expression.

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 func­ tionj{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 1, e2 7Ti13, and e47T'i13, lighter or darker according

three roots

to the number of iterations needed to reach the root with

cumstances. This is good entertainment. Stirling's and Steffensen's methods are a case apart. First, they are the most demanding with respect to the ini­ tial point (in the tables, see the percentage of nonconver­ gent points; in the figures, see the black areas). And, sec­ ond, in their graphics, the symmetry of angle

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 pos­

sible 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 eas­

ier if we employ a computer package with graphics facili­ ties, such us Mathematica, Maple, or Matlab. The graphics that I show here were generated with Mathematica 3.0 (see

[20]);

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

pro­

[20]

grams used. First, we need to define function f and its derivatives. This

can

be

done

d f [ z_ ] : = 3 * z " 2

by

using

f [ z_ ] : = z " 3 - 1 1

and d2 f [ z_ ] : = 6 * z , but it is faster

if we use the compiled versions

f = Comp i l e [ { { z I _Comp l e x } }

I

z"3-1 ] ;

df = Comp i l e [ { { z i _Comp l e x } }

I

3 * z"2 ] ;

d2 f = Comp i l e [ { { z I _Comp l ex } }

I

6*z ] ;

Of

such

as

course,

exp

( sin�l )

any

(z3 -

1),

other

function,

f*(z)

can be used.

=

The three complex roots off are Do [ root [ k ] = N [ Exp [ 2 * ( k- 1 ) * P i * I / 3 l l � {kl 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 _Comp l ex } } Whi c h [ Abs [ z - ro o t [ 1 ] ] <

10 . 0" ( -3 ) 1

Abs ( z - ro o t f [ 2 ] ] < 1 0 . 0 " ( - 3 ) ,

E Rb for which the corresponding iterative method start­

ing in z0 does not reach any root with tolerance w-3 in a

and the same happens for

havior and suitability of any method depending on the cir­

the fixed precision required. I mark with black the points z0

1 1),

2.

The tables and the figures provide empirical data. From

=

Now, let us go back to the other target of this paper: to

and

1

them, and the indications given here, we can guess the be­

tion! (or if we wanted a figure in the fastest way), it would be possible to modify the procedure that iterates Zn + 1

Hence the fractal figure for both

•

Abs [ z - ro o t f [ 3 ] ] < 1 0 . 0 " ( - 3 ) , True ,

I

3,

2,

1,

0] ,

{ { roo t f [ _ ] , _ Comp l ex } }

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 , _Comp l e x } } , z-f [ zJ

I df [ z J J

and, for Halley's method, i terHa l l ey = Comp i l e [ { { z , _Comp l e x } } , B l o c k [ { v = df [ z ] } , - ( d2 f [ z ] )

I

z - 1.0

( 2 . 0 *v) ) ]

I

( v/ f [ z ]

in the next section, I show the programs used to ob­ (observe that an extra variable

tain the figures. Note that both Traub-Ostrowski's method and Jarratt's method for j{z)

=

z3 -

1 lead to the iterative function

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 t e rAlgori thm [ i t erMethod_ , x_ , y_ , l im_ ] Block [ { z , c t , r } , r

=

I;

. ­

ct = O ;

O f f [ Comp i l edFunc t i on : : c c c x ) ;

roo t P o s i t i on [ z J ;

Whi l e [ ( r ++ct ;

l ;

z = x+y

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

==

&&

0)

O f f [ Comp i l edFunc t i on : : c fn ) ; ( c t < l im) ,

O f f [ Comp i l edFunc t i on : : c f c x J ;

z = i t erMethod [ z ) ;

O f f [ Comp i l edFunc tion : : c f ex ) ;

r = r o o t Po s i t i on [ z )

O f f [ Comp i l edFunc t i on : : c r c x ) ; O f f [ Comp i l edFunc t i on : : i l sm )

I f [ Head [ r )

== (*

Return

Wh i c h , " Wh i c h "

r = O) ; uneva luated

*)

[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 r o o t Po 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:

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

l imi t erat i ons = 2 5 ; xxMin = - 2 . 5 ;

xxMax = 2 . 5 ;

yyM i n = - 2 . 5 ;

yyMax = 2 . 5 ;

+ +c t ;

p l o t Frac tal [ i terMe thod_ , points_] : = De n s i t y P l o t [ i t e rA l g o r i thm [ i t e r M e t ho d , x , y , l imitera t i ons ) , {x,

xxMin ,

xxMax } ,

{y,

yyMi n ,

yyMax } ,

l ;

( c t < l im ) ,

r = r o o t Po s i t i on [ z J ==

Whi c h , " Wh i c h "

;

r = O) ; uneva luated

*)

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

0 . 4 * Frac t i onalPart [ 4 *p ] J

256)

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 t Frac t a l :

THE MATHEMATICAL INTELLIGENCER

&&

0)

c o l orLeve l = Comp i l e [ { { p , _Rea l } } ,

p l o tFractal [ i terNewt on ,

44

ct = O ;

P l o t P o i n t s � p o i nt s ,

Note that I I T iming at the end allows us to observe the time that Mathematica employs when p l otFra c t a l is used. Then a graphic is obtained in this way (the example is a black-and-white version of Figure 1 ):

O f f [ Genera l : : ovf l ) ;

==

(*

Timing

O f f [ In f i n i ty : : inde t )

I;

z = x + y

z = i terMethod [ z )

I f [ Head [ r )

Mesh � F a l s e

II

-

r = roo t P o s i t i on [ z ) ; Whi l e [ ( r

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 p o i n t s means that, to gener­ ate the picture, a p o i n t s X p o i n t s grid must be used.

P l o t Range � { 0 , 3 } ,

.

[ { z , ct , r} ,

O f f [ Genera l : : un f l ) ;

and f r a c t a lC o l o r [ p_ ] Bl ock [ { pp

=

.

-

c o l o rLevel [ p ) } ,

Swi t c h [ I n t egerPar t [ 4 * p ) , 3,

CMYKColor [ 0 . 6 +pp , 0 . , 0 . , 2 * pp ) ,

2,

CMYKC o l o r [ 0 . , 0 . 6 +pp , 0 . , 2 * pp ] ,

0,

CMYKC o l o r [ 0 . , 0 . , 0 .

1,

CMYKC o l o r [ 0 . , 0 . , 0 . 6 +pp , 2 * pp ) , ,

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 t er C o l orAlgori thm has a range of [0, 4]; this is the reason for using 4 * p in some places in

and frac t a l C o l or. Also, note that c o l ­ can b e changed to modify the intensity of the col­ ors; 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 pro­ cedure c o l orLevel

orLevel

p l otCol orFrac tal [ i terMethod_ , points_] Dens i ty P l o t [ i t e rC o l orAlgori thm [ i t erMethod , x , y , l im i t e ra t i on s ] , { x , xxMin , xxMax } , { y , yyMi n , yyMax } , P l o tRange � { 0 , 4 } , P l o t Po i n t s � po int s ,

Mesh � Fal s e ,

C o l o r Fun c t i on � frac t a l C o l or

II

T iming

For instance, p l o t C o l o rFrac ta l [ i terNewt on ,

is just Figure

2 56]

1.

Families o f 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) Zn+ l = Zn f'(zn)

(

)

1 LJ(Zn) 1+2 1 - f3L./._zn) '

with {3 a real parameter. These are order 3 methods for solv­ ing 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 in­ teresting to note that any iterative process given by the ex­ pression

f,��) ·

Traub­ where {3 is an arbitrary real number and u(z) = Ostrowski's method is the particular case {3 = 0. Finally, here is another order 4 multipoint family:

ft{;;

where {3 is a parameter and u, h denote u(z) = and rl. f,ucz)j -f'Cz) . z Here' for {3 = 0 we getJarratt's method h(z) = 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 evolu­ tion of the fractal images when the parameter varies. Un­ fortunately, it is not possible to show moving images on pa­ per. To generate them in a computer, one can use small modifications of the Mathematica programs from the pre­ vious section, using also the Mathematica commands An­ ima t e 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 re­ quires 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

f(zn) H(L./._zn)), Zn + l = Zn f'(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 taking H(x) = 1 + =

1 :r 2 1 - (3:1:•

JUAN L. VARONA

A multipoint family (see

[18, p. 178])

-

Departamento de Matematicas y Computaci6n

is

Universidad de La Rioja

26004 Logroiio

f(zn) Zn+l = zn g(Zn) f(z

+

[3f(z)) - f(z)

Spain •

g(z) = and {3 an arb1trary constant f3f(z) ({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]):

with

Zn+ 1

=

Zn - U(Zn) f(Zn - u(zn)) fCzn) + f3f(zn - u(zn)) f' (zn) f(zn) + ({3 - 2) j(Zn - u(zn)) '

e-mail: [email protected]

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 Spai n . 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

method by means of convexity,

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

Mat.

tals,

The Mathematical lntelligencer

14 (1 992), no. 2, 32-38.

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

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

Comput. Appl. Math.

3. I. K. Argyros and F. Szidarovszky, Iteration Methods,

51 (1 994), 1 03-1 06. FL,

Soc.

(N.

S.)

entific Computing,

29 (1 993), 1 51 -1 88.

tions,

Bull. Am. Math.

11 (1 984), 85-1 41 .

36 (1 998), 9-20.

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

Com­

ematical lntelligencer

1 8. J. F. Traub,

100 (1 999), 31 1 -326. 8. W. Gander, On Halley's iteration method, Am. Math. Monthly 92

The Math­

Iterative Methods for the Solution o f Equations,

solution of nonlinear equations, in Equations

Birkhii.user,

0.

Boston, 1 997.

Numerical Solution of Nonlinear

Peitgen, eds. ,

Lecture Notes in Math.

The Mathematica Book,

878 (1 981 ) , 427-440.

3rd ed. , Wolfram Media/Cam­

bridge University Press, 1 996.

Bull. Austral. Math. Soc.

21 . J. W. Neuberger,

The Mathematical lntelligencer,

21 (1 999), no. 3,

1 8-23.

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

-

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 Unlimited 200 1 and Beyond

Contents: Amman, S . : Nonlinear Continuum Physics.

the reader. It not only depicts the

I./Tinsley Oden, J.: Co mputational Mechanics: Where is it Going?

the cenrur)', but i also fu ll of

Developments and Future Outlook.

remarkable insights inro its fu tu re development

as we enrer a new millennium. True ro irs ti de, the book extends

•

other related sciences. You will enjoy reading the many stimulating

•

ematics and the pe rspectives for its future. One of the ediror , Bj orn Engquist, is a world-renowned researcher in computational science

and engineering. The second edito r, Wi lfried Schm id, is a d isringuished 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 sci­ entist and, in particular, for graduates still choosing their specialty.

Order Today!

• Fax: {201 }-348-4505

Visit: http://www.springer-ny.com

THE MATHEMATICAL INTELLIGENCER

• Darmon, H . : p-adic L-func­ Fai ri ngs, G.: Di ophan ti ne Equations. • Farin, G.: SHAP Jorgensen, J ./Lang, S.: The Hear Kernel All Over rhe Place . •

Kllippelberg, C.: Devel o pments in Insurance Mathematics. Koblitz, N.: Cryprography.

•

Marsden, j./Ccndra, H./Ratiu,

T.: Geometric Mechanics, Lagrangian Reduction and Systems. •

Babuska,

Bailey, D.H./Borwein J.M.: Experimental Mathematics: Recent

tions. •

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46

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beyond the spectrum of mathematics ro i n clude contributions from

Call: 1 -800-SPRINCER

•

This is a book guara nteed ro del ight state of mathe m atics at the end of

Pren­

(Proc. , Bremen, 1 980), E. L. Allgower, K. Glashoff and H .

20. S. Wolfram,

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

----

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

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

(1 985), 1 31 -1 34.

Halley type methods in Banach spaces,

Springer­

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

Optim. Theory Appl.

Numerical Analysis: An Introduction,

The Beauty of Fractals,

Verlag, New York, 1 986.

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

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

Iterative Solution of Nonlinear

Monographs Textbooks Comput.

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

mation to nonlinear integral equations of Harnrnerstein-type,

9. W. Gautschi,

10 (1 973), 876-879.

1 6. H. 0. Peitgen and P. H. Richter,

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

Newton's method, J.

SIAM J. Numer. Anal.

Equations in Several Variables,

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

put. Math. Appl.

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

1 5. J. M. Ortega and W. C. Rheinboldt,

5. P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Am. Math. Soc. (N. S.)

20 (1 966), 434-437.

Numerical Analysis: Mathematics of Sci­

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

1 993.

4. W. Bergweiler, Iteration of meromorphic functions,

Math. Camp.

1 3. D. Kincaid and W. Cheney,

The Theory and Applications of

CRC Press, Boca Raton,

Zb. Rad. Prirod. -Mat. Fak. Ser.

20 (1 990), 27-38.

•

onholonomic

Roy, M.-F. : Four Problems in Real Algebraic Geometry.

Serre, D . : Systems of Conservation Laws: A Chall e nge for the

• Spencer, J.: Discrete Probability. • van der Geer, G . : Error Correcting Codes and Curves Over Finite Fields. • von Storch, H./von Storch, J.-S., and M u ller, P.: oisc in Climate Models . . . And

XX !st Century.

many more. 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

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, 8400

Aartshertogstraat 42,

Oostende, Belgium

e-mail: [email protected]

48

D irk H uylebrouck, Editor

L

et us add a few words to Alice Sil­ verberg's informative article about the birthplace of Ernmy Noether [ 1]. We have long admired Emmy Noe­ 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."

THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK

I

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.

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:

1 882- 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 Life and Work.

Tribute to Her

J.W. Brewer, M . K. Smith,

eds., Marcel Dekker, New York, 1 981 , p. 1 4 . [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: [email protected] Magdolna Hargittai Ebtvbs University and Hungarian Academy of Sciences The stone in the pavement under which Emmy Noether's ashes rest (photograph by I. and M .

e-mail: [email protected]

Hargittai).

H-1 521 Budapest, Hungary

VOLUME 24, NUMBER 1 , 2002

49

STEPHEN BERMAN AND KAREN H U N GE R PARSHALL

Victor Kac and Robert M oody : Thei 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

effect a decomposition that not only revealed the internal

trace zero give a model for the simple Lie algebra of type

the information so obtained.3 Their attack was, in broad

the great classes, for l

2:: 1, the (l + 1) X (l + 1) matrices of

structure of the Lie algebra but also efficiently synthesized

A1, while the orthogonal or symplectic Lie algebras simi­

terms, linearly algebraic; they used what would now be

larly supply a model for the others, namely, the algebras of

termed "generalized eigenspace

type B1

space decompositions in their setting-relative to the so­

(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 alge­ bra of derivations of the octonions.2 For the others, E6 , E7 ,

decompositions"-root

called Cartan subalgebra, a maximal abelian diagonalizable subalgebra of the Lie algebra. They then distilled, from the

E8, and F4, however, questions of existence and of finding

root system derived from this decomposition, the funda­

models are highly non-trivial and deeply influenced the de­

mental system of simple roots associated with the Lie al­

velopment of Lie theory.

gebra. They used the latter to define a "finite Cartan ma­

Killing and Cartan approached their analysis of the fi­ nite-dimensional simple Lie algebras over

1[: by considering

each Lie algebra as a decomposable entity. They aimed to

trix," 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

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 Claud e Chevalley and Harish-Chandra

The line of research from Lie through Killing and Cartan to

Ae Be

Victor Kac and Robert Moody.

of the Lie algebras, and they had succeeded in efficiently and completely encoding in the finite Cartan matrix the pertinent structural information about the Lie algebra. The classification then proceeded by enumerating these matrices. 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, following a number of conventions, recoverable from its associated Dynkin diagram.4 Thus, the nine types of simple algebras correspond to the nine types of finite Cartan rnatrices, 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 representations associated with the [mite-dimensional simple Lie algebras 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-toone correspondence with the so-called dominant highest weights. Moreover, just as the root spaces were the fundamental building blocks of the fmite-dimensional simple Lie algebras, the weight spaces played that key role in the associated theory of fmite-dimensional irreducible representations. 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

Ce De E6

E1

Es F4

G2

•

•

•

•

•

•

•===7•

•

•

------ �·

•

•

•

•

!

•

•

•

•

!

•

•

•

•

•

•

•

I

•

•

•===7•

•

!

•

•

·�

Figure 1 . Dynkin diagrams, the finite case.

4See [24, pp. 56-63] for the conventions and the exact associations. 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 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 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 1 948, 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 [ 1 0 ] . 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 o f 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

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 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 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-

The n ine types of s i m ple algebras correspon d to

the n in e types of fin ite Cartan m atrices .

-

6Again, Carlan was one of the pioneers. See [7]. 7Andre 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 rig­ orous," 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 re­ sult 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 alge­ bras, 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 Cheval­ ley, found the origins of the ideas in the work of Cartan and Weyl. "The representation theory of semisimple Lie al­ gebras 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 re­ quire 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 for­ mulation 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-di­ mensional 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 si­ multaneously is due to Chevalley [C] who obtained inde­ pendently a proof of the theorem. . . . I present here a mod­ ified version of my original proof so as to be able to consider the two questions together. But in this modifica­ tion I have adopted several of Chevalley's ideas" (20, p. 30]. In his paper, Harish-Chandra worked over an alge­ braically 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

(/3) det A =F 0; and

=

0;

( Y) the Weyl group associated with A (defmed immedi­ ately below) is a finite group. He then considered an [-dimensional vector space with ba­ sis 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 repre­ sentations.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 neverthe­ less essential" [20, p. 3 1 ] . More specifically, the construction involved taking a free associative algebra on the set {eiJi, hi II :=::: i :=::: l} and a nat­ ural 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 =

i

(ad firAj + 1 jj,

1 :=:::

i, j, :=::: z, i * j,

where (ad x)(y) = [x y] for x, y in the Lie algebra with prod­ uct [ · · ] . Here, the Cartan matrix associated with the Lie al­ gebra 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 con­ tained results about representations of both the groups and algebras acting on Hilbert spaces. Of broad scope, this pa­ per 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 Math­ ematical Society for it in 1954. The method developed independently by Chevalley and Harish-Chandra was ultimately presented, with simplifica­ tions 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. Ja­ cobson factored the free Lie algebra by the relations

(hi, hJ)

= 0, [k;,

ej)

=

AjieJ, l hi, .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 repre­ sentations. This resulted in the desired finite-dimensional simple Lie algebra with Cartan matrix A. Interestingly, Ja­ cobson 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 satisfy­ ing 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 presen­ tation in [ 46] for all of the finite-dimensional simple Lie al­ gebras 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 finite­ dimensional 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 in­ volving 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 pro­ gram 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 Ja­ cobson, Maria Wonenburger. In Coxeter's lectures on reg­ ular 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

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 differ­ ent 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 won­ derful 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 Amer­ ican 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 Jacob­ son had laid out in Chapter 7 of [25], Moody presented his algebras in terms of generators and relations that corre­ sponded to what he called the "generalized Cartan matri­ ces" 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 set­ ting 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-di­ mensional) which are constructed from matrices (called

1 °For an exposition of this work, see [24) and [46). 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 an undergraduate in 1960 and had begun attending 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 Weyl 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

and his advisor, E. B. Vinberg.

Growth," was originally communicated in Russian on 7 July 1967 to the journal Functional Analysis and Its Applica­ tions [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 d 1 , 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" i s a key technical condition that bounds the size o f the root spaces of the algebra. 1 5As 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

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THE MATHEMATICAL INTELLIGENCER

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ied by E lie 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

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Lie" by Gelfand and A. A. Kirillov [ 1 7], 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, 1 6 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 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 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. 1 7

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 Weyl 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 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 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,

Beg i n ni n g i n the m id - 1 960s ,

Moody i n Canada and Kac in R ussia worked s i m u lta­

neously and independently .

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 Weyl de­ nominator formula [37, p. 1 16]. 18 Whereas Weyl's formula equated a certain sum over the elements in the Weyl group

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

1 6 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 mean­

(finite-dimensional) complex semisimple Lie algebras to the

ing of [the factor] becomes obvious and the identities

Kac-Moody Lie algebras defined by symmetrizable Cartan

take on a simpler and even more beautiful appearance.

matrices" [ 16, p. 37) . This was the first use of the term "Kac­

In their new form, the identities give a marvellous re­

Moody Lie algebras" in the literature. Within a decade, it

lationship between the Weyl group, the root system, and

would not only be universally adopted, it would come to de­

the dimensions of the root spaces. It is not unreason­

fine a vibrant and burgeoning subfield of mathematics with

able to expect that similar identities may hold for all the

deep and surptising physical applications [ 14).

Lie algebras determined by arbitrary Cartan matrices

Kac and Moody independently discovered their new class of algebras during the course of their doctoral re­

[41, p. 43] .

search. Both drew on groundbreaking work of Chevalley B y the time Moody received the proof sheets of his paper,

and Harish-Chandra as filtered through Jacobson's influ­

he had seen the Russian version of Kac's note [29] and had

ential textbook. Both recognized the special nature of the

recognized that Kac had, in fact, proven the latter result

affine Kac-Moody Lie algebras and gave the standard real­

when the Cartan matrices were symmetrizable. Moody ac­

izations of them. Both appreciated the implications that

knowledged that Kac's "work establishes the Macdonald

their algebras held for Macdonald's results. Neither, how­

identities . . . by techniques which are intrinsically related

ever, would likely have predicted in the mid-1970s that the

to the corresponding Lie algebras" [41, p. 5 1 ] . Thus, Moody,

algebras they had isolated would so quickly define an area

like Kac, realized that here was the context-and a fasci­

of such spectacular growth and influence in both physics

nating one at that-that these Lie algebras had lacked. Now

and mathematics. Still less would they have suspected the

all they needed was a name, but Moody seemed to sense

kudos the field would elicit. Their algebraic work led to

that as well.

deep results in physics and won for them the prestigious

Although he used his former nomenclature "Euclidean

Wigner Medal in 1994; it also paved the way for the def­

Lie algebras" in the title of [41 ] , Moody coined a fanciful

inition and development of vertex operator algebras that

new term for the wider class of all Kac-Moody Lie algebras

won the highly prized Fields Medal for Richard Borcherds

in the same paper, "heffalump Lie algebras" [41 , p. 44).

in 1998. Kac and Moody sensed the importance of affine

Moody had been poring through the densely packed pages

Kac-Moody Lie algebras early on. The centrality of these al­

Linear Lie Groups, at

gebras in both mathematics and physics attests to the

of Freudenthal and de Vries's book,

the same time that he had been reading A. A. Milne's clas­ sic,

power of that intuition.

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

\3{2 (C) h, e, j [ l 5, p. 497]. For fixed i, the ei, ji, hi in (2)

the smallest finite-dimensional simple Lie algebra is denoted

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 con­ dition

(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 hef­ falump that had evaded Pooh and Piglet [38). Although Moody's terminology did not catch on,

1 9 the al­

gebras that Kac and Moody had discovered attracted in­ creasing attention following their linkage to Macdonald's re­ sults. 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

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[1] Bruce N. Allison, Stephen Berman, Yun Gao, and Arturo Pianzola, A Characterization of Affine Kac-Moody Lie Algebras,

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[2] Georgia Benkart, References,

ceedings, vol. 5 (1 986), 1 1 1 - 1 35. [3] Stephen Berman,

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[5] Armand Borel, Groups,

The Work of Cheval/ey in Lie Groups and Algebraic

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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,

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France 41 ( 1 9 1 3) , 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] E lie Cartan, Premiere these: Sur Ia structure des formations finis et continus,

[9] Arthur Cayley,

nilradicals of parabolic subalgebras in certain modules, from

Bronwin;

groupes de trans­

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(1 845), 208-2 1 1 .

THE MATHEMATICAL INTELLIGENCER

Isomorphisms and Automorphisms o f Universal

Heffalump Lie Algebras,

fundamental result . . . on the homology (or cohomology) of

58

A Kac-Moody Bibliography and Some Related

Canadian Mathematical Society Conference Pro­

B.

1 9See [3], however, for at least one paper that adopted it.

Communi­

and on

Quaternions,

Philosophical Magazine 26

A U T H O R

STEPHEN BEAMAN

KAREN HUNGER PARSHALL

Department of Mathematics and Statistics

Departments of History and Mathematics

University of Saskatchewan

University of Virginia

Saskatoon, SK S7N 5E6

Charlottesville, VA 22904-4 1 37

Canada

USA

e- mail : kh p3k@virgin ia. edu

e-mail: [email protected]

Stephen Berman got his undergraduate d egree from Worcester

Karen Parshall, alter receivin g a B.A. (1 977) in French and mathe­

Polytechnic Institute, and his Ph.D. in 1 97 1 from Indiana Univer­

matics and an M.S. (1 978) in mat hematics, both at the University

sity under the supervision of Maria J. Wo n enbu rger . Since that time

of Virgin ia , has worked in history of mathematics. Her doctoral su­

he has been on the faculty of the U n iversity of Saskatchewan. His

pervisors at the University of Chicago were Yitz Herstein in math­

mathematical interests are in infinite-dimensional Ue theory, rep­

ematics and Allen G. Debus in history of science. She has been

resentation theory, and vertex operator algebras. His hobbies in­

on the Virginia faculty since 1 988. She is currently working on a bi­

clude playing the guitar and T'ai Chi Ch'uan.

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 enj oy­ ing the northern lights, that they got the idea of writing the present paper.

[1 0] Claude Chevalley, Sur Ia classification des algebres de Lie simples et de leurs representations,

Cornptes rendus de I'Acadernie des

Sciences de Paris 227 ( 1 948), 1 1 36-1 1 38.

Comptes rendus de I'Academie des Sciences de Paris 227

(1 948), 1 1 97 .

[1 2] H. S. M. Coxeter and W . 0. J . Moser, for Discrete Groups,

Generators and Relations

2d ed. , Ergebnisse der Mathematik und ihrer

Grenzgebiete, vol. 1 4 , New York: Springer-Verlag, 1 965. [1 3] H. S. M. Coxeter,

Regular Polytopes,

2d ed. , New York: Macmil­

lan, 1 963. [ 1 4] Louise Dolan,

Notices of the American Mathematical Society 42 (Dec. 1 995), 1 489-1 495.

Linear Lie Groups,

New York

and London: Academic Press, Inc., 1 969. [1 6] Howard Garland and James Lepowsky, Lie Algebra Homology and the

[1 7]

Transitive Differential Geometry,

Macdonald-Kac

Formulas,

lnventiones Mathematicae 34

(1 976), 37-76.

I. M. Gelfand and A A Ki ri llov, Sur les corps lies aux algebres en­

veloppantes des algebres de Lie,

lnstitut des Hautes Etudes Sci­

entifiques, Publications mathematiques, 31 (1 966), 5- 1 9.

[1 8] Roe Goodman and Nolan R. Wal lach,

Representations and In-

An Algebraic Model of

Bulletin of the American Mathe­

matical Society 70 (1 964), 1 6-47.

[20) Harish-Chandra,

On Some Applications of the Universal Envelop­

ing Algebra of a Semisimple Lie Algebra,

Transactions of the Amer­

ican Mathematical Society 70 (1 951 ), 28-99.

[21 ] Thomas Hawkins, Algebras,

1 27-1 92. The Beacon of Kac-Moody Symmetry for Physics,

[1 5] Hans Freudenthal and H. de Vries,

Encyclopedia of Mathematics and

Its Applications, vol. 68, Cambridge: University Press, 1 998. [1 9] Victor Guillemin and Shlomo Sternberg,

[1 1 ] Claude Chevalley, Sur les representations des algebres de Lie sim­ ples,

variants of the Classical Groups ,

Wilhelm Killing

and the Structure

of Lie

Archive for History of Exact Sciences 23 ( 1 98 2) ,

[22] Sigurdur Helgason, metric Spaces,

Differential Geometry, Lie Groups, and Sym­

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

[23] Rebecca Herb, Harish-Chandra and His Work, Bulletin of the Amer­ ican Mathematical Society [2] 25 (1 991 ) , 1 -1 7.

[24] James E. Humphreys, sentation Theory,

Introduction to Lie Algebras and Repre­

New York: Springer-Verlag, 1 972.

[25] Nathan Jacobson,

Lie Algebras,

New York: John Wiley & Sons,

1 962. [26] Victor G. Kac, Automorphisms Algebras,

252-254.

Functional

of Finite Order of Semisimple Lie

Analysis and Its Applications 3 (1 969),

[27] Victor G. Kac, Correspondence to the authors 1 0 March, 2000.

VOLUME 24. NUMBER 1. 2002

59

Graduated Lie Algebras and Symmetric Spaces,

[28] Victor G. Kac,

Functional Analysis and Its Applications 2 (1 968), 1 82-1 83.

[29] Victor G. Kac, Function,

bras,

Infinite-dimensional Lie Algebras and Dedekind's '1]­

Functional Analysis and Its Applications 8 ( 1 97 4), 68-70.

[30] Victor G. Kac,

Infinite Dimensional Lie Algebras,

3d ed. , Cam­

Simple Graduated Lie Algebras of Finite Growth,

[32] Victor G. Kac,

Simple Irreducible Graded Lie Algebras of Finite

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

Growth,

Math. USSR-Izvestiya 2 (1 968), 1 27 1 - 1 3 1 1 .

(in

viet Mathematics- Doklady 9 (1 968), 409-4 1 2 .

Lie Groups: Beyond an Introduction,

So­

Basel/Berlin: Birkhauser, 1 996. Harish-Chandra,

tion,

Affine Root Systems and Dedekind's '1]-Func­

l nventiones Mathematicae 1 5 (1 972), 91-1 43.

[38] Robert V. Moody, Correspondence with the authors 1 5 February, 2000. [39] Robert V. Moody,

Euclidean Lie Algebras,

Mathematics 21 (1 969), 1 432-1 454.

[40] Robert V. Moody, Matrices,

2 1 7-22 1 .

60

Canadian Journal of

Lie Algebras Associated to Generalized Cartan

Bulletin of the American Mathematical Society, 73 (1 967),

THE MATHEMATICAL INTELLIGENCER

Wigner Medal Acceptance Speech,

1 994, un­

published. lar Decompositions,

Lie Algebras with Triangu­

New York: John Wiley and Sons, Inc . , 1 995. complexes,

New

York: W. A. Benjamin, Inc., 1 966. and Cartan,

On the Infinite Groups of Lie

Journal d'analyse mathematique 15 (1 965), 1 -1 1 4 .

[48] Ernst Steinitz, Biographical Memoirs of

Fellows of the Royal Society 31 (1 985), 1 97-225.

[37] Jan G. Macdonald,

Simple Quotients of Euclidean Lie Algebras,

[47] I. M. Singer and Shlomo Sternberg, Boston/

Journal of Alge­

Canadian Journal of Mathematics 22 (1 970), 839-846.

[46] Jean-Pierre Serre, Algebres de Lie semi-simples

Simple Graded Infinite-Dimensional Lie Algebras,

[36] Robert P. Langlands,

A New Class of Lie Algebras,

bra 10 (1 968), 2 1 1 -230.

[45] Robert V. Moody and Arturo Pianzola,

Some Properties of Contragredient Lie Algebras

[35] Anthony W . Knapp,

(1 975), 43-52. [42] Robert V. Moody,

[44] Robert V. Moody,

Russian), Trudy MIFM (1 969}, 48-60. [34] I. L. Kantor,

Macdonald Identities and Euclidean Lie Alge­

Proceedings of the American Mathematical Society 48

[43] Robert V. Moody,

bridge: University Press, 1 990. [31 ] Victor G. Kac,

[33] Victor G. Kac,

[41 ] Robert V. Moody,

Algebraische Theorie der K6rper,

Journal fUr die

reine und angewandte Mathematik 1 16 (1 9 1 0), 1 -1 32.

[49] Baertel L. van der Waerden, Lieschen Gruppen,

462. [50] V. S. Varadarajan,

Die Kassifikation der einfachen

Mathematische Zeitschrift 37 (1 933), 446-

Harish-Chandra and His Mathematical Work,

Current Science 65 (1 993), 91 8-9 1 9.

[51 ] Hermann Weyl,

Theorie der Darsteflung kontin uierlicher halb­

einfacher Gruppen durch lineare Transformationen, Tei/ 3,

matische Zeitschrift 24 (1 926), 377-395.

[52] Ernst Witt,

Mathe­

Spiegelungsgruppen und Aufzahlung halbeinfacher Li­

escher Ringe,

Hamburger Abhandlungen 14 (1 94 1 ) , 289-322.

l'iilfW·\·1·1

Dav i d E.

Rowe , E d i t o r

On the Myriad Mathematical Traditions of Ancient Greece David E. Rowe

T

I

o exert one's historical imagina­

bear on this problem. These were the

tion is to plunge into delicate de­

basis for the mature theories found in

Elements: Theaetetus's classi­

liberations that involve personal judg­

Euclid's

ments and tastes. Historians can and

fication scheme for ratios of lines ap­

do argue like lawyers, but their argu­

pears in Book X, the longest and most

ments are often made on behalf of an

technically demanding of the thirteen

image of the past, and these historical

books, whereas Book V presents Eu­

images obviously change over time.

doxus's general theory of proportions,

Why should the history of mathemat­

which elegantly skirts the problem of

ics be any different?

representing ratios of incommensu­

When we imagine the world of an­

rable magnitudes by providing a gen­

cient Greek mathematics, the works of

eral criterion for determining when

Euclid, Archimedes, and Apollonius

two ratios are equal (Definition V.5). A

easily spring to mind. Our dominant

standard picture of the activity that led

image of Greek mathematical tradi­

to this work has a group of mathemati­

tions stresses the rigor and creative

cians huddled over a diagram at Plato's

achievement that are found in texts by

Academy during the early fourth cen­

these three famous authors. Thanks to

tury. Some of these geometers have fa­

the efforts of Thomas Little Heath, the

miliar names, and a few even appear in

English-speaking world has long en­

Plato's

joyed easy access to this trio's major

eral vivid scenes and vital clues for his­

Dialogues,

which contain sev­

works and much else besides. Yet our

torians of mathematics. A few of its pas­

conventional picture of Greek mathe­

sages have provided some of the most

matics has drawn on little of this plen­

tantalizing tidbits of information that

tiful source material. Our image of

have come down to us.

Greek geometry, as conveyed in math­ ematical texts and most books on the

Particularly sage in Plato's

famous

is

Theaetetus

the

pas­

where the

history of mathematics, has tended

young mathematician recounts how

to stress the formal structure and

his teacher, Theodorus, had managed to

methodological sophistication found in

prove the irrationality of the sides of

a handful

of canonical

works-or,

squares with integral non-square areas,

more accurately, in selected portions

but only up to the square of area 17.

of them. Even the first two books of Euclid's

Elements,

which concern the

Given that Theaetetus is credited with having solved this problem on his way

congruence properties of rectilinear

to developing the massive theory of ir­

figures and culminate in theorem

rational lines that received its final form

II

14

Elements, the sig­

showing how to square such a figure,

in Book X of Euclid's

have often been trivialized. Many writ­

nificance of the historical events Plato

this passage has long been

ers have distilled their content down to

alludes to in

a few definitions, postulates, and ele­

clear. Little wonder that experts like the

mentary propositions, intended merely

late Wilbur Knorr were tempted to tease

to illustrate the axiomatic-deductive

out of it as much as they could, begin­

method in classical geometry.

ning with the obvious question: why did

Talk of the origins of Greek mathe­

Theodorus stop with the square of area

matics shows similar selectivity. The

1 7? Knorr and numerous others have of­

discovery of incommensurables, though

fered ingenious speculations about what

shrouded in mystery, presumably took

went wrong with Theodorus's proof.

place around the time of Plato's birth.

Needless to say, such efforts to recon­

Send submissions to David E. Rowe,

Two younger contemporaries, Theaete­

struct Theodorus's argument on the ba­

Fachbereich 1 7 - Mathematik,

tus and Eudoxus, both of whom had

sis of the meager remarks contained in

Johannes Gutenberg University,

ties with the Academy, are credited

the Platonic passage are driven by math­

055099 Mainz, Germany.

with having developed theories that

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 Acad­ emy, 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 cians will never tire of modernizing works that Pappus of Alexandria later older theories, we might still do well to attributed to Euclid? On the basis of in­ ask what consequences this activity has ternal evidence alone, it seems unlikely for historical understanding. The re­ that the Data and the Elements were flection is required most urgently for written by the same person. But what Euclid's Elements, a work that has gone about all the other mostly nameless through more shifts of meaning and scholars who surely must have mingled context than any other. Reading Euclid with Euclid in Alexandria shortly after (carefully) had profound consequences Alexander's death? Perhaps our Euclid for Isaac Newton, who soon thereafter was actually a gifted administrator who immersed himself in the lesser-known worked at the library and headed a re­ works of ancient Greek geometers. He search group to produce standard texts emerged a different mathematician, set of ancient mathematical works. Is it too on defending the Ancients against Mod­ farfetched to imagine Euclid as the an­ erns like Rene Descartes, who claimed cient Greek counterpart to the twentieth to have found a methodology superior century's Bourbaki? to Greek analysis. We need not puzzle But leaving these biographical spec­ over why Newton wrote his Principia ulations aside, we can easily agree that in the language of geometry, once we the Elements established a paradigm for understand his strong identification classical Greek geometry, or what came with what he understood by the prob­ to be known as ruler-and-compass lem-solving tradition of the ancient geometry. Indeed, synthetic geometry in Greeks. Nothing rankled him more than the style of Euclid's Elements continued Cartesian boasting about how this tra­ to serve as the centerpiece of the Eng­ dition had been supplanted by modem lish mathematical curriculum until well analysis. into the nineteenth century. For Anglo­ For ourselves, looking from a post­ American gentlemen steeped in the Hilbertian perspective, the question can classics, no formal education was com­ be posed like this: If we continue to view plete without a sprinkling of Euclidean Greek mathematics through the prism of geometry. This mainly meant mimicking Euclid's Elements, and to view the Ele­ an old-fashioned style of deductive rea­ ments mainly as a model of axiomatic soning that many believed disciplined rigor, what effect will this have on our the mind and prepared the soul to un­ conception of the more remote past in derstand and appreciate Reason and which Greek mathematics grew? One of Truth. With David Hilbert's Grundlagen the more obvious consequences has der Geometrie, published in 1899, the been the glorification of the ancient Euclidean style may be said to have Greeks at the expense of other ancient made its peace with mathematical cultures. This theme has been the sub­ modernity. Hilbert upgraded its struc­ ject of much bickering ever since the ture and redesigned its packaging, but publication of Martin Bernal's Black most of all he gave it a new modernized Athena. I will not enter this fracas here; system of axioms. Within this universe it does suggest, however, that our pic­ of "pure thought," Greek mathematics tures of ancient mathematics are in the could still retain its honored place. En­ process of change, and this applies to the shrined in the language of modem indigenous traditions of Greece as well axiomatics, it took on new form in as to interaction with other cultures. By accenting the plural in traditions, countless English-language texts that presented Greek geometry as a wa­ I mean to emphasize that there were tered-down version of Heath's Euclid. several different currents of Greek The history of mathematics abounds mathematical thought. They continued with examples of this kind: a good the­ to flourish in the Hellenistic world and orem, so the adage goes, is always beyond: we should not imagine Greek worth proving twice (or thrice), just as mathematics monolithically, as if a sin­ a good theory is one worthy of being gle mathematical style dominated all renovated. In the case of an old others. warhorse like Euclidean geometry, we Nor should we overestimate the take this for granted. But if mathemati- unity of Greek mathematics even

within the highbrow tradition of Eu­

introduced infmitesimals in geometry,

likely to keel over as soon as it caught

clid, Archimedes, and Apollonius. In

and by so doing had found the volume

its first strong gust of wind.)

his

Conica and the other minor works,

Apollonius systematically exploits an

of a cone, presumably arguing along

Archimedes's work presumably was

led

related to his other duties as an advisor to the Syracusan court, which later

lines similar

to

the

ideas

that

impressive repertoire of geometrical

Bonaventura Cavalieri to his general

operations and techniques in order to

principle for finding the volumes of

called upon him when the city was be­

derive a series of complex metrical the­

solids of known cross-sectional area.

sieged by the Roman armies of Marcel­

orems whose significance is often ob­

As is well known, Eudoxus is cred­

lus. Plutarch immortalized the story of

scure. In this respect, his style con­

ited with having introduced the "method

how Archimedes single-handedly held

Elements.

of exhaustion" in order to demonstrate

back the Roman legions with all manner

When we compare the works of

theorems involving areas and volumes

of

Apollonius and Euclid with those of

of curvilinear figures, including the re­

These legendary exploits inspired Italian

Archimedes, whose inventiveness is

sults obtained earlier by Democritus.

far more striking than any single styl­

Archimedes used the Eudoxian method

Archimedes's feats of prowess as a mil­

istic element, the contrasts only widen.

with impressive virtuosity, but because

itary engineer. No longer content with

trasts sharply with Euclid's

strange, terrifying

war machines.

Renaissance writers to elaborate on

Unlike Apollonius, Archimedes appar­

this technique could only be applied af­

ently had little interest in showcasing

ter one knew the correct result, he had

Archimedes devises a system of mirrors

visional results. His inspiration came

sails of Roman ships, setting them all

He was first and foremost a problem­

from mechanics. By performing sophis­

ablaze. These mythic elements reflect

solver, not a systematizer, and many of

ticated thought experiments with a fic­

the imaginative reception of Archimedes

the problems he tackled were inspired

titious

during the Renaissance as a symbol of

all possible variant results merely to demonstrate his arsenal of techniques.

by ancient mechanics. Ivo Schneider

to rely first on ingenuity to obtain pro­

balance,

Archimedes

could

"weigh" various kinds of geometrical ob­

has suggested that Archimedes's early

jects as if they were composed of "geo­

career in Syracuse was probably closer

metrical atoms"-indivisible slivers of

to what we would today call "mechan­

lower dimension.As he clearly realized,

mechanical contraptions, the new-age that could focus the sun's rays on the

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

ical engineering" than to mathematics.

this mechanical method was a definite

derwent a long, convoluted journey dur­

Not that this was unusual; practical

no-no for a Eudoxian geometer, but he

ing the Middle Ages, so that by Galileo's

and applied mathematics flourished in

also knew that there was "method" to

time they had begun to exert a deep

ancient Greece, and again in early

this

influence on a new style of mathemat­

madness, since it enabled him to

modern Europe when Galileo taught

"guess" the areas and volumes of curvi­

ics. By the seventeenth century, the

these subjects as professor of mathe­

linear figures such as the segment of a

Archimedean tradition had

become

matics at the University of Padua,

parabola, cylinders, and spheres. As

strongly interwoven with the Euclidean

which belonged to the Venetian Re­

Heath once put it, here we gain a glimpse

tradition, but these two currents were by

public. Like Venice, Syracuse had an

of Archimedes in his workshop, forging

no means identical from their inception.

impressive navy, and we can be fairly

the tools he would need before he could

sure that Archimedes spent a consid­

proceed to formal demonstration. Going one step further, he carried

erable amount of time around ships

Another major significant tradition within

Greek

mathematics

can

be

traced back to Pythagorean idealism,

and the machines used to build them.

out thought experiments inspired by a

which continued to live on side-by-side

From these, he must have learned the

problem of major importance to the

with the rationalism represented by

Elements.

principles behind the various mechan­

economic and political welfare of Syra­

Euclid's

ical devices that Heron and Pappus of

cuse: the stability of ships.Archimedes's

dogma that "all is number" could no

Alexandria would later describe and

idealized vessels had hulls whose cross­

longer hold sway after the discovery

classify under the five classical types

sections were parabolic in shape, en­

of incommensurable magnitudes, this

of machines for generating power. Archimedes was neither an atomist

abling him to determine the location of

does

their centers of gravity precisely. Had

Pythagorean

not

mean

If the Pythagorean

that

all

mathematics

traces

of

vanished.

nor a follower of Democritus. Never­

he

in

Far from it: we have every reason to

theless, the parallels between these two

seventeenth-century Sweden for King

believe that the Pythagorean and Eu­

bold thinkers are both striking and sug­

gestive. In one of his flights of fancy,

performed

a

similar service

Gustav Adolfus, the latter might have

clidean traditions interpenetrated one

been spared from witnessing one of the

another, influencing both over a long pe­

riod of time. Euclid's approach to number

Archimedes devised a number system

great blunders in maritime history: the

capable of expressing the "atoms" in the

disaster that befell his warship, the

theory in Books VII-IX differs markedly

universe. For this purpose he took a

Vassa,

from that found in the

sand grain as the prototype for these

which flippe<;l over and sank in

the harbor on her maiden voyage. (If

Arithmetica of

Nicomachus of Gerasa, who continued

tiny, indivisible corpuscles.Archimedes

you've ever visited the Vassa Museum

to give expression to the Pythagorean

must have seen Democritus's atomic

in Stockholm, you'll realize that it

tradition during the first century

theory as at least a powerful heuristic

wouldn't have taken an Archimedes to

Still, the distinctive Pythagorean doc­

device in mathematics. Democritus had

guess that this magnificent vessel was

trine of number types (even and odd,

VOLUME 24, NUMBER 1, 2002

A.D.

63

perfect, etc.) can be found in both Eu­

ing that the five Platonic solids are the

deed, for him the third law vindicated

clid and Nicomachus, albeit in very dif­

only regular polyhedra, Euclid deter­

his cosmology of nested Platonic solids by revealing the divine cosmic har­

ferent guises. Thabit ibn Qurra knew

mines the ratio of the side length to the

both works and assimilated these arith­

radius of the circumscribed sphere ac­

monies that God conceived for this sys­

metical traditions into Islamic mathe­

cording to the classification scheme pre­

tem as elaborated by Kepler in Book

matics. Finding Nicomachus's treat­

sented in Book X for incommensurable

of Harmonice

This body of mathematical knowl­

ment of amicable numbers inadequate

lines.

(Euclid ignores it completely), Thabit

edge shows its connection with the doc­

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

Mundi.

Kepler knew Euclid's

V

Elements per­

haps better than any of his contempo­

trine of celestial harmonies, an idea

raries, and his imagination ran wild with

whose origins are obscure, but which un­

it in

doubtedly stems from Pythagoreanism.

early moderns, he saw his work as the

The doctrine that the heavens pro­ duce a sublime astronomical

Harmonice Mundi.

Like so many

continuation of a quest first undertaken

music

by the ancient Greeks. Kepler believed

through the movements of invisible

that the Ancients had already discovered

spheres that carry the stars and planets

deep and immortal truths, none more

continued to ring forth in the works of

important than those found in the thir­

thought into account, we seen an even

Plato and Cicero. Johannes Kepler went

deeper interpenetration of mythic ele­

further,

proclaiming

in

Harmonice

teen books of the

Elements. And since

truth, for Kepler, meant Divine Truth, he

ments into the Euclidean tradition. For

Mundi

(1619) the underlying musical,

saw his quest as inextricably interwoven

Plutarch, a writer whose imagination

astrological, and cosmological signifi­

with theirs. His historical sensibilities

Elements.

For him,

were shaped by a profound religious

often outran his critical judgment, Eu­

cance of Euclid's

clid's Elements was itself imbued with

Book N, on the theory of constructible

faith that led him to identify his Christ­ ian God with the Deity that pagan

beautiful Proposition VI

Pythagorean lore. He linked Euclid's

polygons, contained the keys to the plan­

25 with the

etary aspects, the cornerstone of his

Greeks described in the mythic language

creation myth in Plato's

a

"scientific" astrology. Historians of sci­

of Pythagorean symbols. We gasp at the

work rife with Pythagorean symbol­

ence have long overlooked the inspira­

gulf that separates our post-historicist

ism. Plato's Demiurge, the Craftsman of

tion behind Kepler's self-acknowledged

Timaeus,

world from Kepler's naive belief in a

the universe, fashions his cosmos out of

magnum opus from 1619, preferring in­

chaos following a metaphysical princi­

stead to emphasize his "positive contri­

can only marvel in the realization that it

ple, one that Plutarch identified with the­

butions" to the history of astronomy,

was Kepler's sense of a shared past that

transcendent realm of bare truth. We

enabled him to compose his Harmonice

orem VI 25: given two rectilinear figures,

namely Kepler's three laws. Few seem

to construct a third equal in area to the

to have been puzzled about the connec­

Mundi

tion between these laws and Kepler's

he thought he saw in the works of an­

cosmological views as first set forth in

cient Greek writers.

first figure and similar to the second. In other words, Euclid's geometrical crafts­

while contemplating the truths

man must transform a given quantity of

Mysterium Cosmographicum

(1596),

These brief reflections suggest some

matter into a desired form.

where he tries to account for the dis­

broader conclusions for the history of

But we need no Plutarchian wings

tances between the planets by a famous

mathematics: that mathematical knowl­

El­

system of nested Platonic solids. Kepler

edge, as a general rule, is related to var­

ements contain numerous and striking

published his first two laws (that the

ious other types of knowledge, that its

allusions to Pythagorean/Platonic cos­

planets move around the sun in ellipti­

sources are varied, and that the form and

mological thought, as noted by Proclus

cal orbits, and that from the sun's posi­

content of its results are affected by the

of imagination to see that Euclid's

and other commentators. The theories

tion they sweep out equal areas in equal

cultures within which it is produced.

of constructible regular polygons and

times) thirteen years later in

Astrono­

Those who have produced mathematics

polyhedra appear in Books N and XIII,

mia Nova (1609), which presents the as­

have done so in quite different societies,

respectively, thereby culminating the

tronomical results of his long struggle to

within which these producers have had

first and last major structural divisions

grasp the motion of Mars. The third law

in the

Elements

(Books I-N on the

(that for all planets the ratio of the

quite varied functions. Western mathe­ matics owes much of course to ancient

congruence properties of plane figures;

square of their mean distance to the sun

Greek mathematicians, but even within

Books XI-XIII on solid geometry). In

to the cube of their period is the same

the scope of the Greeks' traditions we

both cases, the figures are constructed

constant) only appeared another ten

encounter considerable variance in the

as

years later in Harmonice Mundi. Unlike

styles and even the content and pur­ poses of their mathematics. For this rea­

inscribed

figures

in

circles

or

spheres, the perfect celestial objects

the first two astronomical laws, the third

that pervade all of Greek astronomy

had a deeper cosmic significance for Ke­

son, we should avoid the temptation to

and cosmology. Perhaps most striking

pler, who never abandoned the cosmo­

reduce Greek mathematics to one dom­

logical views he advanced in

inant paradigm or style.

of all, in Book XIII, which ends by prov-

64

THE MATHEMATICAL INTELLIGENCER

1596.

In-

ATHANASE PAPADOPOU LOS

Mathematics and M usic Theory: From Pythagoras to Rameau usic theory is a wide and beautiful subject, and some basic mathematical ideas are inherent in it. Some of these ideas were intraduced in music theory by mathematicians, and others by musicians 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 Harmonica­ rum 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 Ten­

tamen novae theoriae musicae ex certissimis harmoniae principiis dilucide expositae. Jean d'Alembert wrote in 1 752 his Elements de musique theorique et pr-atique suiv­ ant 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 musical performers. Research and teaching in p R E1 F A C E. music theory were much more prestigious qru i'effirit en confoi�c lu proprim" , a ujfi e •fin , lui efl natNrcU occupations than musical composition or focilcment que /'oreillc les fent. performance. Some famous mathemati­ Vn flu/ homme n'efl pas capable d'epuiflr une matiere auffi cians were also composers or performers, qu'il n'y ou /ic profonde que ctlle-cy ; il e_(l p1·;jque but this is another subject. 3 fes tous malgre , chofl mats dte mom.r , que/que to�jours J. Ph. Rameau, who is certainly the great­ /es nou"Ptlles decou�ertes qu'il pcut j oindrc a cc qui a deja paru est French musicologist of the eighteenth fo ime fojet, font autant de routes ftayees piJur ccux qrei / m e r century, wrote in his Traite de l'harmonie loin. plus a/ler peu�ent reduite a ses principes naturels ( 1 722): lA Mujique eft une ftimc� qui doit a�oir des rcgl�s certaines ; La musique est une science qui doit avoir ces regles doirvmt ltre tirdes d'un principe evident ' � ce principe

impojji�lc foms;

des regles certaines; ces regles doivent etre tirees d'un principe evident, et ce principe ne peut guere nous etre connu sans le sec­ ours 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 mathe­ matiques que mes idees se sont debrouil­ lees, 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.

�

nc pcut gutres nous ltre connufons leflcours 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 ideesfl font debrouil. Lees , @- que Ia lumie1ey 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 con­ fiquei'Jces qu'il m'a fournies enfoite , m'ont fait cvnnottre en el/es autant de rtgles J qui de"Poient fl rapporterpar confl'luent a ce principe ; le 'Veritablefins 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 , @j­ ain.fipar 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 je _.aip/J m'tmplcher de con"Penir qu'il flroit 4 foubaitel' ( commeon mt le difoit, un jour que j'applaN­ dijfois J I• pe'.foiJion de ndm Mujique modtme ) qut les connoi.f­ fonces des Mujicitns de ct jitc/t repotlliijfent II.NX heii.Ntt'{de leurs Compojitiot�s. IL ntfojfit done pas defl#ir les :elfets tfunt Science ou d'un Nt , ilfat 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.

3For instance. Pythagoras. according to his biographers, be· sides being a geometer, a number-theorist, and a musicologist , was a composer. and he also played several instru-

ments; see e.g., [7]. Chapter XV, p. 32.

66

THE MATHEMATICAL INTELLIGENCER

Figure

1.

naturels.

Glowing words from Rameau's Traite de /'harmonia reduite

a ses principes

musical intervals, rhythm, duration, tempi, and several I Z. other musical notions are naturally expressed by num­ cS" bers. The mathematical use of the word "harmonic" It (for instance in "harmonic series" or "harmonic analy­ � : sis") has its origin in music theory. The composer Mil­ s . 9 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 ax­ ioms, definitions and theorems, the proofs of which <SO !) ToNvs o 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 article. 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 mathematical notions in an intuitive manner, before these notions would be possible to devise instrumental assistance to the had been shaped and refined by mathematicians. One such hearing, which could be firm and unerring, such as the sight example is the use of logarithms, also discussed below. obtains through the compass and the rule." Walking Now let us start from the beginning, that is, with through a brazier's shop, Pythagoras heard the different Pythagoras. sounds produced by hammers beating an anvil. He realized that the pitch, that is, the musical note, that was produced Pythagoras and the Theory of Musical Intervals by a particular hammer, depended only on the weight of Historians of science usually agree that Pythagoras (sixth the hammer and not on the particular place where the ham­ c. BC) is at the origin of mathematics as a purely theoreti­ mer hit the anvil, or on the magnitude of the stroke. cal science. 5 At the same time, Pythagoras is regarded as Pythagoras realized also that the compass of a musical in­ the first music theorist (from the point of view of European terval between two notes produced by two different ham­ music). The major musical discovery of Pythagoras is the mers depended only on the relative weights of the ham­ relation of musical intervals with ratios of integers. This is mers, and in particular that the consonant musical described by Jamblichus ([7], Chap. XXVI, p. 62) in these intervals, which in classical Greek music were the intervals terms: Pythagoras was "reasoning with himself, whether it of octave, of fifth and of fourth, correspond, in terms of

�e

.

8

�· DJA:trs: rs

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

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

I I I

I

·�fourth� I

I

I E-<-I

' <:: Figure

3.

fifth

octave

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

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 dif ference 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

erect them unnatural and a threat to their philosophical sys­

of Theon of Smyrna

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

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

duction, he says, "Harmony is spread in the world, and of­

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

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

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 su­

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

ages, and it was at the ba­ sis of a strong feeling of cosmic structure and or­ der. There are important

For the Pythag o reans , deal­ i ng with

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 beyond the scope of this paper. The following is a list of "useful" musical intervals, which was known to Gioseffo Zarlino and Descartes:

i rrational

philosophical and esoteric traditions behind this idea, which led eventually to explanations of physical phenomena, like the mo­ 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 semitone 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-

perparticular ratios de­ rivedfrom 2, 3, and 5, that

n u m bers

wou ld have been incom pat i ­ ble with their p h i losophy .

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 ma­ jor 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 musi­ cal 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 descrip­ tion 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:

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 num­ ber 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 val­ ues being 3/2 and 4/3. For instance, the value of the inter­ val 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. =

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 ra­ tio 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 be­ tween 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 ex­ pected, since the Pythagorean scale needs to contain these three conso­ nant intervals. The intervals between consecutive notes, except those be­ tween 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. There is a logic behind the defini-

70

THE MATHEMATICAL INTELUGENCER

Figure 4. Jean-Philippe Rameau. Portrait by Jean Bernard Restout, titled "The inspired poet."

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) 1 2 is larger than the one with value (2/1F The difference is a small (but never­ theless audible) interval, (3/2) 1 2/27. 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]). Ram eau and the Harmonic S equence

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' D E L,H A R. M D

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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 Nou­

4o. donnc: a le Son grav e de: cc: dcrnicr ! Ac,ord amfi 5 <40: 6o, 75• 9 0• IoB.l , La� Ud E 1 Mi� Sol J

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de la'Quintc-fupc:dlui!,

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. 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

En 2500 ans d'histoire ecrite, la musique n'a peut-etre connu que deux veritables theoriciens, dont les autres n'ont guere fait qu'amenager ou rapetasser les proposi­ tions. L 'un, au VIe siecle avant notre ere, jut le jabuleux Pythagore. L 'autre mourut a Paris en 1 764: ce jut Jean­ Philippe Rameau.

A diagram in Rameau's Traite, discussing ratios of frequencies of a dissonant

chord (containing a minor seventh).

derived from the existence and the properties of the harmonic 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.

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

T H E MATHEMATICAL INTELLIGENCER

[2] Milton Babbitt, Past and present concepts of the nature and lim�

A U T H O R

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�

ATHANASE PAPADOPOULOS

Institute de Mathematiques, CNRS

tion by Thomas Taylor, London, John M. Watkins, 1 965.

7 rue Rene Descartes

[8] Johannes Kepler, Harmonicas Mundi. (I have used the French

67084 Strasbourg Cedex

translation with comments by J. Peyroux, Librairie A. Blanchard, 9

France

rue de Medicis, Paris, 1 97 7 . ) [9] A.

L.

e-mail: [email protected]

Le1gh Silver, Musimatics o r t h e nun 's fiddle, Amer. Math.

Monthly 78 ( 1 9 7 1 ) , 351 -35 7 . [1 0] J . Ph. Rameau, Complete Theoretical Writings, edited b y

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

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

1 27 (1 898), 752-754. [1 2] Theon of Smyrna (beginning of the second c. AD), Exposition of

tion, he teaches a course on mathematics and music at the

the mathematical knowledge useful for the reading of Plato. A bilin�

Universite Louis Pasteur, Strasbourg. He was choir director of

gual (Greek�French) edition due to J. Dupuis (Paris 1 892) is

the Russian Orthodox Church of Strasbourg from 1 989 to

reprinted by Culture et Civilisation, 1 1 5 Av. Gabriel Lebon, Brus�

1 999.

sels, 1 966.

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VOLUME 24. NUMBER 1 . 2002

73

I ;J§Ih§'.lfj

.J et Wim p , Editor

I

The Math Gene: How Mathematical Thinkin g Evolved and Why Numbers Are Like Gossip b y Keith

J. Devlin

NEW YORK: BASIC BOOKS, 2000 328 pp. US $25.00; ISBN 0-4650-1 61 8-9 REVIEWED BY REUBEN HERSH

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.

74

D

evlin 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.

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 Proceed­ ings of the 28th annual Child Lan­ guage 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 ac­ tual 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 Brin g s Mathematics Into Bein g

•

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

T

his book is an attempt by cognitive scientists to launch a new disci­ pline: cognitive science of mathemat­ ics. 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 can­ not, 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 HilbertIf 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 es­ tablished as any result in cognitive sci­ ence." (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-do­ main mapping-a neural mechanism that allows us to use the inferential structure of one conceptual domain (say, geometry) to reason about an­ other (say, arithmetic)." For some cog­ nitive metaphors, cognitive scientists have detected actual neural connec­ tions 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 ob­ jects of the same size (based on our commonest experiences with group­ ing objects)"; a target domain: natural numbers with addition and subtraction (which the authors call "arithmetic"); a cross-domain mapping as de­ scribed 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 entail­ ments, which for this metaphor the au­ thors describe as follows:

Actual Infinity: The authors "hy­ pothesize" that the idea of infinity in mathematics is metaphorical.

Take the basic truths about collec­ tions of physical objects. Map them onto statements about numbers, us­ ing the metaphorical mapping. The result is a set of "truths" about the natural numbers under the opera­ tions of addition and subtraction. (page 56)

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 infi­ nite process in terms of a process that does have an end. (page 158)

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 consider­ ations."

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 com­ pleted iterative process has four parts, all literal: the beginning state; the process that from an intermediate state produces the next state; an intermedi­ ate state; and the final resultant state that is unique and follows every non-fi­ nal state. These are mapped onto four parts (with the same names and de­ scriptions) 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 (us­ ing BMI): How do we conceptualize (or give meaning to) the notion in Projec­ tive Geometry that two parallel lines meet at infinity? If m and l are two par­ allel lines in the plane, then let the line segment AB be a common perpendicu­ lar 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-

Math ematicians 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 the­ ory, infinity (in varied fonns), continu­ ity, space-filling curves, infinitesimals, and the Euler equation em + 1 0. It is also at this point that I think the au­ thors' arguments and discussions need input from mathematicians and teach­ ers 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

----> ----> ----> ----> ---->

=

Parallel Lines Meet at Infinity

Target Domain Isosceles Triangles with Base AB

Source Domain Completed Iterative Processes The beginning state

---+

The process that from (n - 1 )th state produces nth state Intermediate state

---+

The final resultant state, unique and following every non-final state

---+

---+

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 (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)

Isosceles triangle ABG0, where length of AGo ( = EGo) is Do Form ABGn from ABGn - 1 by making "Dn arbitrarily larger than Dn- 1 " "Dn > Dn - 1 and (90° - an) < (90° - an - d" "ax = 90°, Dx is infinitely long", and the sides meet at a unique Gx, a point at x (because Dx > Dn- 1 , for all finite n) 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 point­ of-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 mathemat­ ics 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

Proj ective Metaphor

Source Domain Motion of the particle before T Motion of the particle after T At time T the particle is at a unique point P

Target Domain ---+ ---+ ---+

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 de­ finitions and axioms are themselves understood in cognitive terms. This quotation contains two related misconceptions about mathematics: •

•

Misconception 1: Mathematics is for­ mal, consisting of formal definitions, axioms, theorems, and proof. Misconception 2: Mathematics does not (and cannot) ask what mathe­ matical ideas mean, how they can be understood, and why they are true.

These misconceptions of mathemat­ ics are prevalent among non-mathe­ maticians. The blame for this lies mostly with us, the mathematicians. Collec­ tively, we have not done an effective job of communicating to the outside world the nature of our discipline. Neverthe­ less, many of us (including all four of the mathematicians whose endorse­ ments 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 For­ malism," and yet he writes: In mathematics, as in any scientific research, we fmd two tendencies present. On the one hand, the ten­ dency toward abstraction seeks to crystallize the logical relations in­ herent 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 ten­ dency toward intuitive understand­ ing fosters a more immediate grasp of the objects one studies, a live rap­ port with them, so to speak, which stresses the concrete meaning of their relations. [Hilbert's emphasis] On Hilbert's "one hand" is the ten­ dency 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

We Need to Work Together

I find the authors' discussion of their philosophy of embodied mathematics to be profound, and I think any math­ ematician who studies it will find her/his own understandings of mathe­ matics stimulated and challenged in constructive ways. But first the math­ ematicians must overcome their reac­ tions 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:

Cognitive scientists and mathemati­ cians need to work together to develop mathematical idea analysis. I believe that most mathematicians and teach­ ers of mathematics are concerned ex­ actly with the things mentioned by these authors:

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. Mathe­ matics is a product of the neural ca­ pacities of our brains, the nature of our bodies, our evolution, our envi­ ronment, and our long social and cultural history. (page 9)

We, mathematicians and teachers would certainly be thankful to cogni­ tive 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)

In Part V of the book the authors ex­ pand on this summary and proceed to dismiss (or "disconfirm") other philosophies of mathematics. I recom­ mend that the mathematical reader skip over all arguments dismissing var­ ious other philosophies of mathemat­ ics, because for the most part these ar­ guments are based on shallow summaries of what the various philoso­ phies assert. Further, I do not think that the settling of these arguments is important or necessary for under­ standing the authors' main points. Re­ gardless of one's philosophical beliefs, I think all mathematicians (and teach­ ers of mathematics) would welcome conceptual foundations [for mathe­ matics that] would consist of a thor­ ough mathematical ideas analysis that worked out in detail the con­ ceptual structure of each mathemat­ ical domain, showing how those con­ cepts are ultimately grounded in bodily experience and just what the network of ideas across mathemati­ cal disciplines looks like. (page 379)

We believe that revealing the cogni­ tive structure of mathematics makes mathematics more accessi­ ble and comprehensible. . . . [M]ath­ ematical ideas . . . can be under­ stood for the most part in everyday tem1s. (page 7).

REFERENCES

[1 ] Keith Devlin,

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 ,

River,

NJ :

Upper Saddle

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, ally?, New

What Is Mathematics, Re­

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,

1 8 79-193 1 ,

Cam­

bridge: Harvard University Press, 1 967. [6] David Hilbert and S. Cohn-Vossen, etry

and

the

Imagination,

New

Geom­

York:

Chelsea Publishing Co. , 1 983. [7] W. P. Thurston, "On Proof and Progress in Mathematics,"

Bull. Amer. Math. Soc.

1 61 -1 77, 1 994.

Department of Mathematics Cornell University Ithaca, NY 1 4853-7901 USA e-mail: [email protected]

30,

K-jfl i.C+J.JQ.t§i ..

on]

Robin Wils

The Development of Computing

Since the 1950s computers have devel­ oped 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 pro­ grams internally and initially used vac­ uum tubes as their switching technol­ ogy. Because such tubes were bulky, hot and unreliable, they were gradually replaced in the 'second generation' of computers by transistors with thou­ sands of interconnected simple circuits. In the late 1960s, the 'third genera­ tion' of computers saw the develop­ ment 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 cir­ cuit, an assembly of many transistors, resistors, capacitors, and other devices,

Visual display unit

interconnected electronically and packaged as a single functional item. In the 1970s the first personal comput­ ers became available, for use in the home and office. Computer-aided design also devel­ oped rapidly, and in 1970 the Nether­ lands produced the first set of com­ puter-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 com­ puter-animated film Dilemma. The invention of the World Wide Web by Tim Berners-Lee in the early 1990s has led to the information su­ perhighway, whereby all types of in­ formation from around the world has become easily accessible. Communica­ tions 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

M A G YA R P O STA

Computer drawing

12�8 Integrated circuit Tim Berners-Lee

Please send all submissions to the Stamp Corner Editor, Robin Wilson,

Faculty of Mathematics,

The Open University, Milton Keynes, MK7

6AA,

England

e-mail: [email protected]

80

THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK

nederland Isometric projection