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
+ 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
-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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VOLUME 24, NUMBER 1, 2002
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
This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.
Please send all submissions to the
Marjorie Senechal,
A
Editor
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 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 =
