The mountains of Pi
an article from the New Yorker, March 2, 1992.
[Image]
Gregory V. Chudnovsky and David V. Chudnovsky
GREGORY VOLFOVICH CHUDNOVSKY recently built a
supercomputer in his apart-
ment from mail-order parts. Greg-
ory Chudnovsky is a number theo-
rist. His apartment is situated near
the top floor of a run-down build-
ing on the West Side of Manhat-
tan, in a neighborhood near Co-
lumbia University. Not long ago,
a human corpse was found dumped
at the end of the block. The world's
most powerful supercomputers in-
clude the Cray Y-MP C90, the
Thinking Machines CM-5, the
Hitachi S-820/80, the nCube, the
Fujitsu parallel machine, the
Kendall Square Research parallel
machine, the NEC SX-3, the
Touchstone Delta, and Gregory
Chudnovskv's apartment. The
apartment seems to be a kind of con-
tainer for the supercomputer at least
as much as it is a container for
people.
Gregory Chudnovsky's partner in
the design and construction of the
supercomputer was his older brother,
David Volfovich Chudnovsky, who is
also a mathematician, and who lives
five blocks away from Gregory. The
Chudnovsky brothers call their ma-
chine m zero. It occupies the former
living room of Gregory's-apartment,
and its tentacles reach into other rooms.
The brothers claim that m zero is a
"true, general-purpose supercomputer,"
and that it is as fast and powerful
as a somewhat older Cray Y-MP, but
it is not as fast as the latest of the
Y-MP machines, the C90, an ad-
vanced supercomputer made by Cray
Research. A Cray Y-MP C90 costs
more than thirty million dollars. It is
a black monolith, seven feet tall and
eight feet across, in the shape of a squat
cylinder, and is cooled by liquid freon.
So far, the brothers have spent around
seventy thousand` dollars on parts for
their supercomputer, and much of the
money has come out of their wives'
pockets.
Gregory Chudnovsky is thirty-nine
years old, and he has a spare frame
and a bony, handsome face. He has a
long beard, streaked with gray, and
dark, unruly hair, a wide forehead,
and wide-spaced brown eyes. He walks
in a slow, dragging shuffle, leaning on
a bentwood cane, while his brother,
David, typically holds him under one
arm, to prevent him from toppling
over. He has severe myasthenia gravis,
an auto-immune disorder of the muscles.
The symptoms, in his case, are mus-
cular weakness and difficulty in breath-
ing. "I have to lie in bed most of the
time," Gregory once told me. His
condition doesn't seem to be getting
better, and doesn't seem to be getting
worse. He developed the disease when
he was twelve years old, in the city of
Kiev, Ukraine, where he and David
grew up. He spends his days sitting or
lying on a bed heaped with pillows, in
a bedroom down the hall from the
room that houses the supercomputer.
Gregory's bedroom is filled with pa-
per; it contains at least a ton of
paper. He calls the place his junk yard.
The room faces east, and would be
full of sunlight in the morning if he
ever raised the shades, but he keeps
them lowered, because light hurts
his eyes.
You almost never meet one of the
Chudnovsky brothers without the other.
You often find the brothers conjoined,
like Siamese twins, David holding
Gregory by the arm or under the
armpits. They complete each other's
sentences and interrupt each other,
but they don't look alike. While
Gregory is thin and bearded, David
has a stout body and a plump,
clean-shaven face. He is in his
early forties. Black-and-gray curly
hair grows thickly on top of David's
head, and he has heavy-lidded
deep-blue eyes. He always wears
a starched white shirt and, usu-
ally, a gray silk necktie in a fou-
lard print. His tie rests on a bulg-
ing stomach.
The Chudnovskian supercom-
puter, m zero, burns two thousand
watts of power, and it runs day
and night. The brothers don't dare
shut it down; if they did, it might
die. At least twenty-five fans blow
air through the machine to keep it cool;
otherwise something might melt. Waste
heat permeates Gregory's apartment,
and the room that contains m zero
climbs to a hundred degrees Fahren-
heit in summer. The brothers keep the
apartment's lights turned off as much
as possible. If they switched on too
many lights while m zero was run-
ning, they might blow the apartment's
wiring. Gregory can't breathe city air
without developing lung trouble, so he
keeps the apartment's windows closed
all the time, with air-conditioners
running in them during the summer,
but that doesn't seem to reduce the
heat, and as the temperature rises in-
side the apartment the place can smell
of cooking circuit boards, a sign that
m zero is not well. A steady stream
of boxes arrives by Federal Express,
and an opposing stream of boxes flows
back to mail-order houses, contain-
ing parts that have bombed, along
with letters from the brothers demand-
ing an exchange or their money back.
The building superintendent doesn't
know that the Chudnovsky brothers
have been using a supercomputer in
Gregory's apartment, and the broth-
ers haven't expressed an eagerness to
tell him.
The Chudnovskys, between them,
have published a hundred and fifty-
four papers and twelve books, mostly
in collaboration with each other, and
mostly on the subject of number theory
or mathematical physics. They work
together so closely that it is possible to
argue that they are a single mathema-
tician&emdash;anyway, it's what they claim.
The brothers lived in Kiev until 1977,
when they left the Soviet Union and,
accompanied by their parents, went to
France. The family lived there for six
months, then emigrated to the United
States and settled in New York; they
have become American citizens.
The brothers enjoy an official rela-
tionship with Columbia University:
Columbia calls them senior research
scientists in the Department of Math-
ematics, but they don't have tenure and
they don't teach students. They are
really lone inventors, operating out of
Gregory's apartment in what you might
call the old-fashioned Russo-Yankee
style. Their wives are doing well.
Gregory's wife, Christine Pardo Chud-
novsky, is an attorney with a midtown
law firm. David's wife, Nicole Lanne-
grace, is a political-affairs officer at
the United Nations. It is their salaries
that help cover the funding needs of
the brothers' supercomputing complex
in Gregory and Christine's apart-
ment. Malka Benjaminovna Chud-
novsky, a retired engineer, who is
Gregory and David's mother, lives in
Gregory's apartment. David spends his
days in Gregory's apartment, taking
care of his brother, their mother, and
m zero.
When the Chudnovskys applied to
leave the
Soviet Union, the fact that
they are Jewish and mathematical
attracted at least a dozen K.G.B. agents
to their case. The brothers' father, Volf
Grigorevich Chudnovsky, who was
severely beaten by the K.G.B. in 1977,
died of heart failure in 1985. Volf
Chudnovsky was a professor of civil
engineering at the Kiev Architectural
Institute, and he specialized in the
structural stability of buildings, towers,
and bridges. He died in America, and
not long before he died he constructed
in Gregory's apartment a maze of book-
shelves, his last work of civil engineer-
ing. The bookshelves extend into ev-
ery corner of the apartment, and today
they are packed with literature and
computer books and books and papers
on the subject of numbers. Since almost
all numbers run to infinity (in digits)
and are totally unexplored, an apart-
mentful of thoughts about numbers
holds hardly any thoughts at all, even
with a supercomputer on the premises
to advance the work.
The brothers say that the "m" in
"m zero" stands for "machine," and
that they use a small letter to imply that
the machine is a work in progress.
They represent the name typographi-
cally as "mO." The "zero" stands for
success. It implies a dark history of
failure&emdash;three duds (in Gregory's apart-
ment) that the brothers now refer to
as negative three, negative two, and
negative one. The brothers broke up
the negative machines for scrap, got on
the telephone, and waited for Federal
Express to bring more parts.
M zero is a parallel supercomputer,
a type of machine that has lately come
to dominate the avant-garde in super-
computer architecture, because the
design offers succulent possibilities for
speed in solving problems. In a par-
allel machine, anywhere from half a
dozen to thousands of processors work
simultaneously on a problem, whereas
in a so-called serial machine&emdash;a nor-
mal computer&emdash;the problem is solved
one step at a time. "A serial machine
is bound to be very slow, because the
speed of the machine will be limited
by the slowest part of it," Gregory said.
"In a parallel machine, many circuits
take on many parts of the problem at
the same time." As of last week, m zero
contained sixteen parallel processors,
which ruminate around the clock on
the Chudnovskys' problems.
The brothers' mail-order super-
computer makes their lives more con-
venient: m zero performs inhumanely
difficult algebra, finding roots of gi-
gantic systems of equations, and it has
constructed colored images of the in-
terior of Gregory Chudnovsky's body.
According to the Chudnovskys, it could
model the weather or make pictures of
air flowing over a wing, if the brothers
cared about weather or wings. What
they care about is numbers. To them,
numbers are more beautiful, more nearly
perfect, possibly more complicated, and
arguably more real than anything in
the world of physical matter.
The brothers have lately been using
m zero to explore the number pi. Pi,
which is denoted by the Greek letter
Pi, is the most famous ratio in math-
ematics, and is one of the most ancient
numbers known to humanity. Pi is
approximately 3.14&emdash;the number of
times that a circle's diameter will fit
around the circle. Here is a circle, with
its diameter:
Pi goes on forever, and can't
be calculated to perfect
precision:
3.1415926535897932384626433832
795028841971693993751.... This is
known as the decimal expansion of pi.
It is a bloody mess. No apparent pat-
tern emerges in the succession of dig-
its. The digits of pi march to infinity
in a predestined yet unfathomable code:
they do not repeat periodically, seem-
ing to pop up by blind chance, lacking
any perceivable order, rule, reason, or
design&emdash;"random" integers, ad infini-
tum. If a deep and beautiful design
hides in the digits of pi, no one knows
what it is, and no one has ever been
able to see it by staring at the digits.
Among mathematicians, there is a nearly
universal feeling that it will never be
possible, in principle, for an inhabitant
of our finite universe to discover the
system in the digits of pi. But for the
present, if you want to attempt it, you
need a supercomputer to probe the
endless scrap of leftover pi.
Before the Chudnovsky brothers built
m zero, Gregory had to derive pi over
the telephone network while lying in
bed. It was inconvenient. Tapping at
a small keyboard, which he sets on the
blankets of his bed, he stares at a
computer display screen on one of the
bookshelves beside his bed. The key-
board and the screen are connected to
Internet, a network that leads Gregory
through cyberspace into the heart of a
Cray somewhere else in the United
States. He calls up a Cray through
Internet and programs the machine to
make an approximation of pi. The job
begins to run, the Cray trying to es-
timate the number of times that the
diameter of a circle goes around the
periphery, and Gregory sits back on
his pillows and waits, watching mes-
sages from the Cray flow across his
display screen. He eats dinner with his
wife and his mother and then, back in
bed, he takes up a legal pad and a red
felt-tip pen and plays
with number theory,
trying to discover hid-
den properties of num-
bers. Meanwhile, the
Cray is reaching toward
pi at a rate of a hundred
million operations per
second. Gregory dozes
beside his computer
screen. Once in a while,
he asks the Cray how
things are going, and
the Cray replies that the
job is still active. Night
passes, the Cray run-
ning deep toward pi.
Unfortunately, since the
exact ratio of the circle's
circumference to its di-
ameter dwells at infin-
ity, the Cray has not
even begun to pinpoint
pi. Abruptly, a message
appears on Gregory's
screen:
LINE IS
DISCONNECTED.
"What the hell is
going on?" Gregory ex-
claims. It seems that
the Cray has hung up
the phone, and may
have crashed. Once
again, pi has demon-
strated its ability to give
a supercomputer a heart
attack.
MYASTHENIA GRA-
VIS iS a funny
thing Gregrory Chudnovsky said one day from his bed in
the junk yard. "In a sense, I'm very
lucky, because I'm alive, and I'm alive
after so many years." He has a reso-
nant voice and a Russian accent. "There
is no standard prognosis. It sometimes
strikes young women and older women.
I wonder if it is some kind of sluggish
virus."
It was a cold afternoon, and rain
pelted the windows; the shades were
drawn, as always. He lay against a
heap of pillows, with his legs folded
under him. He wore a tattered gray
lamb's-wool sweater that had multiple
patches on the elbows, and a starched
white shirt, and baggy blue sweat pants,
and a pair of handmade socks. I had
never seen socks like Gregory's. They
were two-tone socks, wrinkled and
floppy, hand-sewn from pieces of dark-
blue and pale-blue cloth, and they
looked comfortable. They were the
work of Malka Benjaminovna, his
mother. Lines of computer code flickered
on the screen beside his bed.
This was an apartment built for
long voyages. The paper in the room
was jammed into the bookshelves, from
floor to ceiling. The brothers had
wedged the paper, sheet by sheet, into
manila folders, until the folders had
grown as fat as melons. The paper also
flooded two freestanding bookshelves
(placed strategically around Gregory's
bed), five chairs (three of them in a
row beside his bed), two steamer trunks,
and a folding cocktail table. I moved
carefully around the room, fearful of
triggering a paper slide, and sat on the
room's one empty chair, facing the foot
of Gregory's bed, my knees touching
the blanket. The paper was piled in
three-foot stacks on the chairs. It
guarded his bed like the flanking tow-
ers of a fortress, and his bed sat at the
center of the keep. I sensed a profound
happiness in Gregory Chudnovsky's
bedroom. His happiness, it occurred to
me later, sprang from the delicious
melancholy of a life chained to a bed
in a disordered world that breaks open
through the portals of mathematics
into vistas beyond time or decay.
"The system of this paper is
archeological," he said. "By
looking at a slice, I know the
year. This slice is 1986. Over
here is some 1985. What you
see in this room is our work-
ing papers, as well as the papers
we used as references for them.
Some of the references we pull
out once in a while to look at,
and then we leave them some-
where else, in another pile.
Once, we had to make a Xerox
copy of a book three times, and
we put it in three different
places in the piles, so we would
be sure to find it when we
needed it. Unfortunately, once
we put a book into one of these
piles we almost never go back
to look for it. There are books
in there by Kipling and Macau-
lay. Actually, when we want
to find a book it's easier to go
back to the library. Eh. this
place is a mess. Eventually, these papers
or my wife will turn me out of the
house."
Much of the paper consists of legal
pads covered with Gregory's hand-
writing. His holograph is dense and
careful, a flawless minuscule written
with a red felt-tip pen&emdash;a mixture of
theorems, calculations, proofs, and con-
lectures concerning numbers. He uses
a felt-tip pen because he doesn't have
enough strength in his hand to press
a pencil on paper. Mathematicians who
have visited Gregory Chudnovsky's
bedroom have come away dizzy, won-
dering what secrets the scriptorium
may hold. Some say he has published
most of his work, while others wonder
if his bedroom holds unpublished dis-
coveries. He cautiously refers to his
steamer trunks as valises. They are
filled to the lids with compressed pa-
per. When Gregory and David used
to fly to Europe to speak at conferences,
they took both "valises" with them, in
case they needed to refer to a theorem,
and the baggage particularly annoyed
the Belgians. "The Belgians were
always fining us for being overweight,"
Gregory said.
Pi is by no means the only unex-
plored number in the Chudnovskys'
inventory, but it is one that interests
them very much. They wonder whether
the digits contain a hidden rule, an as
vet unseen architecture. close to the
mind of God. A subtle and fantastic
order may appear in the digits of pi
way out there somewhere; no one
knows. No one has ever proved, for
example, that pi does not turn into
nothing but nines and zeros, spattered
to infinity in some peculiar arrange-
ment. If we were to explore the digits
of pi far enough, they might resolve
into a breathtaking numerical pattern,
as knotty as "The Book of Kells," and
it might mean something. It might be
a small but interesting message from
God, hidden in the crypt of the circle,
awaiting notice by a mathematician.
On the other hand, the digits of pi may
ramble forever in a hideous cacophony,
which is a kind of absolute perfection
to a mathematician like Gregory Chud-
novsky. Pi looks "monstrous" to him.
"We know absolutely nothing about
pi," he declared from his bed. "What
the hell does it mean? The definition
of pi is really very simple&emdash;it's just the
ratio of the circumference to the diam-
eter&emdash;but the complexity of the se-
quence it spits out in digits is really
unbelievable. We have a sequence of
digits that looks like gibberish."
"Maybe in the eyes of God pi looks
perfect," David said, standing in a cor-
ner of the room, his head and shoulders
visible above towers of paper.
Pi, or 1t, has had various names
through the ages, and all of them are
either words or abstract symbols, since
pi is a number that can't be shown
completely and exactly in any finite
form of representation. Pi
is a transcendental number. A transcendental
number is a number that exists but
can't be expressed in any finite series
of either arithmetical or algebraic op-
erations. For example, if you try to
express pi as the solution to an equa-
tion you will find that the equation
goes on forever. Expressed in digits, pi
extends into the distance as far as the
eye can see, and the digits never repeat
periodically, as do the digits of a ra-
tional number. Pi slips away from all
rational methods used to locate it. Pi
is a transcendental number because it
transcends the power of algebra to dis-
play it in its totality. Ferdinand Lin-
demann, a German mathematician,
proved the transcendence of pi in 1882;
he proved, in effect, that pi can't be
written on a piece of paper, not even
on a piece of paper as big as the
universe. In a manner of speaking, pi
is indescribable and can't be found.
Pi possibly first entered human con-
sciousness in Egypt. The earliest known
reference to pi occurs in a Middle
Kingdom papyrus scroll, written around
1650 B.C. by a scribe named Ahmes.
Showing a restrained appreciation for
his own work that is not uncommon
in a mathematician, Ahmes began his
scroll with the words "The Entrance
Into the Knowledge of All Existing
Things." He remarked in passing that
he composed the scroll "in likeness to
writings made of old," and then he led
his readers through various mathemati-
cal problems and their solutions, along
several feet of papyrus, and toward the
end of the scroll he found the area of
a circle, using a rough sort of pi.
Around 200 B.C., Archimedes of
Syracuse found that pi is somewhere
between 3 1O/7l and 3 1/7_that's about
3.14. (The Greeks didn't use deci-
mals.) Archimedes had no special term
for pi, calling it "the perimeter to the
diameter." By in effect approximating
pi to two places after the decimal point,
Archimedes narrowed the known value
of pi to one part in a hundred. There
knowledge of pi bogged down until the
seventeenth century, when new for-
mulas for approximating pi were dis-
covered. Pi then came to be called the
Ludolphian number, after Ludolph van
Ceulen, a German mathematician who
approximated it to thirty-five decimal
places, or one part in a hundred mil-
lion billion billion billion&emdash;a calcula-
tion that took Ludolph most of his life
to accomplish, and gave him such
satisfaction that he had the digits en-
graved on his tombstone, at the Ladies'
Church in Leiden, in the Netherlands.
Ludolph and his tombstone were later
moved to Peter's Church in Leiden, to
be installed in a special graveyard for
professors, and from there the stone
vanished, possibly to be turned into a
sidewalk slab. Somewhere in Leiden,
people may be walking over Ludolph's
digits. The Germans still call pi the
Ludolphian number. In the eighteenth
century, Leonhard Euler, mathemati-
cian to Catherine the Great, called it
p or c. The first person to use the
Greek letter Pi for the number was
William Jones, an English mathema-
tician, who coined it in 1706 for his
book "A New Introduction to the Math-
ematics." Euler read the book and
switched to using the symbol Pi, and
the number has remained Pi ever since.
Jones probably meant Pi to stand for the
English word "periphery."
Physicists have noted the ubiquity of
pi in nature. Pi is obvious in the disks
of the moon and the sun. The double
helix of DNA revolves around pi. Pi
hides in the rainbow, and sits in the
pupil of the eye, and when a raindrop
falls into water pi emerges in the
spreading rings. Pi can be found in
waves and ripples and spectra of all
kinds, and therefore pi occurs in colors
and music. Pi has lately turned up in
superstrings, the hypothetical loops of
energy vibrating inside subatomic par-
ticles. Pi occurs naturally in tables of
death in what is known as a Gaussian
distribution of deaths in a
population; that is, when a
person dies, the event
"feels" the Ludolphian
number.
It is one of the great
mysteries why nature seems
to know mathematics. No
one can suggest why this
necessarily has to be so.
Eugene Wigner, the physi-
cist, once said, "The mir-
acle of the appropriateness
of the language of math-
ematics for the formula-
tion of the laws of physics
is a wonderful gift which
we neither understand nor
deserve." We may not un-
derstand pi or deserve it,
but nature at least seems to
be aware of it, as Captain
0. C. Fox learned while
he was recovering in a
hospital from a wound sus-
tained in the American
Civil War. Having noth-
in better to do with his
time than lie in bed and derive pi,
Captain Fox spent a few weeks tossing
pieces of fine steel wire onto a wooden
board ruled with parallel lines. The
wires fell randomly across the lines in
such a way that pi emerged in the
statistics. After throwing his wires elev-
en hundred times, Captain Fox was
able to derive pi to two places after the
decimal point, to 3.14. If he had had
a thousand years to recover from his
wound, he might have derived pi to
perhaps another decimal place. To go
deeper into pi, you need a powerful
machine.
The race toward pi happens in
cyberspace, inside supercomputers. In
1949, George Reitwiesner, at the Bal-
listic Research Laboratory, in Mary-
land, derived pi to two thousand and
thirty-seven decimal places with the
ENIAC, the first general-purpose elec-
tronic digital computer. Working at
the same laboratory, John von Neu-
mann (one of the inventors of the
ENIAC) searched those digits for signs of
order, but found nothing he could put
his finger on. A decade later, Daniel
Shanks and John W. Wrench, Jr.,
approximated pi to a hundred thousand
decimal places with an I.B.M. 7090
mainframe computer, and saw noth-
ing. The race continued desultorily,
through hundreds of thousands of digits,
until 1981, when Yasumasa Kanada,
the head of a team of computer scien-
tists at Tokyo University, used a NEC
supercomputer, a Japanese machine, to
compute two million digits of pi. People
were astonished that anyone would
bother to do it, but that was only the
beginning of the affair. In 1984, Kanada
and his team got sixteen million digits
of pi, noticing nothing remarkable. A
year later, William Gosper, a math-
ematician and distinguished hacker em-
ployed at Symbolics, Inc., in Sunny-
vale, California, computed pi
to seventeen and a half mil-
lion decimal places with a
Symbolics workstation, beat-
ing Kanada's team by a mil-
lion digits. Gosper saw noth-
ing of interest.
The next year, David H.
Bailey, at the National Aeronautics
and Space Administration, used a Cray 2
supercomputer and a formula discov-
ered by two brothers, Jonathan and
Peter Borwein, to scoop twenty-nine
million digits of pi. Bailey found noth-
ing unusual. A year after that, in 1987,
Yasumasa Kanada and his team got a
hundred and thirty-four million digits
of pi, using a NEC SX-2 supercom-
puter. They saw nothing of interest.
In 1988, Kanada kept going, past two
hundred million digits, and saw fur-
ther amounts of nothing. Then, in the
spring of 1989, the Chudnovsky broth-
ers (who had not previously been known
to have any interest in calculating pi)
suddenly announced that they had
obtained four hundred and eighty mil-
lion digits of pi&emdash;a world record&emdash;
using supercomputers at two sites in
the United States, and had seen noth-
ing. Kanada and his team were a little
surprised to learn of unknown compe-
tition operating in American cyberspace,
and they got on a Hitachi supercom-
puter and ripped through five hundred
and thirty-six million digits, beating
the Chudnovksys, setting a new world
record, and seeing nothing. The broth-
ers kept calculating and soon cracked
a billion digits, but Kanada's restless
boys and their Hitachi then nosed into
a little more than a billion digits. The
Chudnovskys pressed onward, too, and
by the fall of 1989 they had squeaked
past Kanada again, having computed
pi to one billion one hundred and
thirty million one hundred and sixty
thousand six hundred and sixty-four
decimal places, without finding any-
thing special. It was another world
record. At that point, the brothers gave
up, out of boredom.
If a billion decimals of pi were
printed in ordinary type, they would
stretch from New York City to the
middle of Kansas. This notion raises
the question: What is the point of
computing pi from New York to Kan-
sas? The question has indeed been
asked among mathematicians, since an
expansion of pi to only forty-seven
decimal places would be sufficiently
precise to inscribe a circle around the
visible universe that doesn't
deviate from perfect circu-
larity by more than the dis-
tance across a single proton.
A billion decimals of pi go so
far beyond that kind of pre-
cision, into such a lunacy of
exactitude, that physicists will
never need to use the quantity in any
experiment&emdash;at least, not for any phys-
ics we know of today&emdash;and the thought
of a billion decimals of pi oppresses
even some mathematicians, who de-
clare the Chudnovskys' effort trivial. I
once asked Gregory if a certain im-
pression I had of mathematicians was
true, that they spent immoderate
amounts of time declaring each other's
work trivial. "It is true," he admitted.
"There is actually a reason for this.
Because once you know the solution to
a problem it usually is trivial."
Gregory did the calculation from his
bed in New York, working through
cyberspace on a Cray 2 at the Minne-
sota Supercomputer Center, in Minne-
apolis, and on an I.B.M. 3090-VF
supercomputer at the I.B.M. Thomas J.
Watson Research Center, in York-
town Heights, New York. The calcu-
lation triggered some dramatic crashes,
and took half a year, because the broth-
ers could get time on the supercomputers
only in bits and pieces, usually during
holidays and in the dead-of night. It
was also quite expensive&emdash;the use of
the Cray cost them seven hundred and
fifty dollars an hour, and the money
came from the National Science Foun-
dation. By the time of this agony, the
brothers had concluded that it would
be cheaper and more convenient to
build a supercomputer in Gregory's
apartment. Then they could crash their
own machine all they wanted, while
they opened doors in the house of
numbers. The brothers planned to
compute two billion digits of pi on their
new machine&emdash;to try to double their
old world record. They thought it
would be a good way to test their
supercomputer: a maiden voyage into
pi would put a terrible strain on their
machine, might blow it up. Presuming
that their machine wouldn't overheat
or strangle on digits, they planned to
search the huge resulting string of pi
for signs of hidden order. If what the
Chudnovsky brothers have seen in the
Ludolphian number is a message from
God, the brothers aren't sure what
God is trying to say.
ON a cold winter day, when the
Chudnovskys were about to be-
gin their two-billion-digit expedition
into pi, I rang the bell of Gregory
Chudnovsky's apartment, and David
answered the door. He pulled the door
open a few inches, and then it stopped,
jammed against an empty cardboard
box and a wad of hanging coats. He
nudged the box out of the way with his
foot. "Look, don't worry," he said.
"Nothing unpleasant will happen to
you. We will not turn you into digits."
A Mini Mag-Lite flashlight protruded
from his shirt pocket.
We were standing in a long, dark
hallway. The lights were off, and it
was hard to see anything. To try to
find something in Gregory's apartment
is like spelunking; that was the reason
for David's flashlight. The hall is
lined on both sides with bookshelves,
and they hold a mixture of paper and
books. The shelves leave a passage
about two feet wide down the length
of the hallway. At the end of the
hallway is a French door, its mul-
lioned glass covered with translucent
paper, and it glowed.
The apartment's rooms are strung
out along the hallway. We passed a
bathroom and a bedroom. The bed-
room belonged to Malka Benjaminovna
Chudnovsky. We passed a cave of
paper, Gregory's junk yard. We passed
a small kitchen, our feet rolling on
computer cables. David opened the
French door, and we entered the room
of the supercomputer. A bare light bulb
burned in a ceiling fixture. The room
contained seven display screens: two of
them were filled with numbers; the
others were turned off. The windows
were closed and the shades were drawn.
Gregory Chudnovsky sat on a chair
facing the screens. He wore the usual
outfit&emdash;a tattered and patched lamb's-
wool sweater, a starched white shirt,
blue sweat pants, and the hand-stitched
two-tone socks. From his toes trailed
a pair of heelless leather slippers. His
cane was hooked over his shoulder,
hung there for convenience. I shook
his hand. "Our first goal is to compute
pi," he said. "For that we have to build
our own computer."
"We are a full-service company,"
David said. "Of course, you know
what 'full-service' means in New York.
It means 'You want it? You do it
yourself."'
A steel frame stood in the center of
the room, screwed together with bolts.
It held split shells of mail-
order personal computers&emdash;
cheap P.C. clones, knocked
wide open, like cracked wal-
nuts, their meat spilling all
over the place. The brothers
had crammed special logic
boards inside the personal
computers. Red lights on the
boards blinked. The floor
was a quagmire of cables.
The brothers had also managed to
fit into the room masses of empty
cardboard boxes, and lots of books
(Russian classics, with Cyrillic letter-
ing on their spines), and screwdrivers,
and data-storage tapes, and software
manuals by the cubic yard, and stalag-
mites of obscure trade magazines, and
a twenty-thousand-dollar computer
workstation that the brothers no longer
used. ("We use it as a place to stack
paper," Gregory said.) From an oval
photograph on the wall, the face of
their late father&emdash;a robust man, squint-
ing thoughtfully&emdash;looked down on the
scene. The walls and the French door
were covered with sheets of drafting
paper showing circuit diagrams. They
resembled cities seen from the air: the
brothers had big plans for m zero.
Computer disk drives stood around the
room. The drives hummed, and there
was a continuous whirr of fans, and
a strong warmth emanated from the
equipment, as if a steam radiator were
going in the room. The brothers heat
their apartment largely with chips.
Gregory said, "Our knowledge of pi
was barely in the millions of digits&emdash;"
"We need many billions of digits,"
David said. "Even a billion digits is a
drop in the bucket. Would you like a
Coca-Cola?" He went into the kitchen,
and there was a horrible crash. "Never
mind, I broke a glass," he called.
"Look, it's not a problem." He came
out of the kitchen carrying a glass of
Coca-Cola on a tray, with a paper
napkin under the glass, and as he
handed it to me he urged me to hold
it tightly, because a Coca-Cola spilled
into&emdash;He didn't want to think about
it; it would set back the project by
months. He said, "Galileo had to build
his telescope&emdash;"
"Because he couldn't afford the Dutch
model," Gregory said.
"And we have to build our machine,
because we have&emdash;"
"No money," Gregory said. "When
people let us use their computer, it's
alwavs done as a kindness." He grinned
and pinched his finger and
thumb together. "They say,
'You can use it as long as
nobody complains."'
I asked the brothers
when they planned to build
their supercomputer.
They burst out laughing.
"You are sitting inside it!"
David roared.
"Tell us how a super-
computer should look," Gregory said.
I started to describe a Cray to the
brothers.
David turned to his brother and
said, "The interviewer answers our
questions. It's Pirandello! The inter-
viewer becomes a person in the story."
David turned to me and said, "The
problem is, you should change your
thinking. If I were to put inside this
Cray a chopped-meat machine, you
wouldn't know it was a meat chopper."
"Unless you saw chopped meat
coming out of it. Then you'd suspect
it wasn't a Cray," Gregory said, and
the brothers cackled.
"In ten years, a Cray will fit in your
pocket," David said.
Supercomputers are evolving incred-
ibly fast. The notion of what a super-
computer is and what it can do changes
from year to year, if not from month
to month, as new machines arise. The
definition of a supercomputer is simply
this: one of the fastest and most pow-
erful scientific computers in the world,
for its time. The power of a super-
computer is revealed, generally speak-
ing, in its ability to solve tough prob-
lems. A Cray Y-MP8, running at its
peak working speed, can perform more
than two billion floating-point opera-
tions per second. Floating-point opera-
tions&emdash;or flops, as they are called&emdash;are
a standard measure of speed. Since a
Cray Y-MP8 can hit two and a half
billion flops, it is considered to be a
gigaflop supercomputer. Giga (from
the Greek for "giant") means a bil-
lion. Like all supercomputers, a Cray
often cruises along significantly below
its peak working speed. (There is a
heated controversy in the supercom-
puter industry over how to measure the
typical working performance of any
given supercomputer, and there are
many claims and counterclaims.) A
Cray is a so-called vector-processing
machine, but that design is going out
of fashion. Cray Research has an-
nounced that next year it will begin
selling an even more powerful parallel
machine.
"Our machine is a gigaflop super-
computer," David Chudnovsky told
me. "The working speed of our ma-
chine is from two hundred million flops
to two gigaflops&emdash;roughly in the range
of a Cray Y-MP8. We can probably go
faster than a Y-MP8, but we don't
want to get too specific about it."
M zero is not the only ultrapowerful
silicon engine to gleam in the Chud-
novskian |uvre. The brothers recently
fielded a supercomputer named Little
Fermat, which they designed with
Monty Denneau, an I.B.M. super-
computer architect, and Saed Younis,
a graduate student at the Massachu-
setts Institute of Technology. Younis
did the grunt work: he mapped out
circuits containing more than fifteen
thousand connections and personally
plugged in some five thousand chips.
Little Fermat is seven feet tall, and sits
inside a steel frame in a laboratory at
M.I.T., where it considers numbers.
What m zero consists of is a group
of high-speed processors linked by cables
(which cover the floor of the room).
The cables form a network of connec-
tions among the processors&emdash;a web.
Gregory sketched on a piece of paper
the layout of the machine. He drew a
box and put an "x" through it, to show
the web, or network, and he attached
some processors to the web:
"Each processor is connected to a
high-speed switching network that
connects it to all the others," he said.
"It's like a telephone network&emdash;every-
body is talking to everybody else. As
far as I know, no one except us has
built a machine that has this type of
web. In other parallel machines, the
processors are connected only to near
neighbors, while they have to talk to
more distant processors through inter-
vening processors. Think of a phone
system: it wouldn't be very pleasant if
you had to talk to distant people by
sending them messages through your
neighbors. But the truth is that nobody
really knows how the hell parallel
machines should perform, or the best
design for them. Right now we have
eight processors. We plan to have
two hundred and fifty-six processors.
We will be able to fit them into the
apartment."
He said that each processor had its
own memory attached to it, so that
each processor was in fact a separate
computer. After a processor was fed
some data and had got a result, it could
send the result through the web to
another processor. The brothers wrote
the machine's application software in
FORTRAN, a programming language
that is "a dinosaur from the late fifties,"
Gregory said, adding, "There is al-
ways new life in this dinosaur." The
software can break a problem into
pieces, sending the pieces to the ma-
chine's different processors. "It's the
principle of divide and conquer," Greg-
ory said. He said that it was very hard
to know what exactly was happening
in the web when the machine was
running&emdash;that the web seemed to have
a life of its own.
"Our machine is mostly made of
connections," David said. "About ninety
per cent of its volume is cables. Your
brain is the same way. It is mostly
made of connections. If I may say so,
your brain is a liquid-cooled parallel
supercomputer." He pointed to his nose
"This is the fan."
The design of the web is the key
element in the Chudnovskian architec-
ture. Behind the web hide several new
r findings in number theory, which the
Chudnovskys have not yet published
The brothers would not disclose to me
the exact shape of the web, or the
discoveries behind it, claiming that
they needed to protect their competitive
edge in a worldwide race to develop
faster supercomputers. "Anyone with a
hundred million dollars and brains
could be our competitor," David said
dryly.
The Chudnovskys have formidable
competitors. Thinking Machines Cor-
poration, in Cambridge, Massachu-
setts, sells massively parallel super-
computers. The price of the latest model,
the CM-S, starts at one million four
hundred thousand dollars and goes up
from there. If you had a hundred
mil1ion dollars, you could order a CM-S
that would be an array of black mono-
liths the size of a Burger King, and
it would burn enough electricity to
light up a neighborhood. Seymour Cray
is another competitor of the brothers,
as it were. He invented the original
Cray series of supercomputers, and is
now the head of the Cray Computer
Corporation, a spinoff from Cray Re-
search. Seymour Cray has been work-
ing to develop his Cray 3 for several
years. His company's effort has re-
cently been troubled by engineering
delays and defections of potential cus-
tomers, but if the machine ever is
released to customers it may be an
octagon about four feet tall and four
feet across, and it will burn more than
two hundred thousand watts. It would
melt instantly if its cooling system
were to fail.
Then, there's the Intel Corporation.
Intel, together with a consortium of
federal agencies, has invested more
than twenty-seven million dollars in
the Touchstone Delta, a five-foot-high,
fifteen-foot-long parallel supercomputer
that sits in a computer room at Caltech.
The machine consumes twenty-five
thousand watts of power, and is kept
from overheating by chilled air flow-
ing through its core. One day, I called
Paul Messina, a Caltech research sci-
entist, who is the head of the Touch-
stone Delta project, to get his opinion
of the Chudnovsky brothers. It turned
out that Messina hadn't heard of
them. As for their claim to have built
a pi-computing gigaflop supercomputer
out of mail-order parts for around
seventy thousand dollars, he flatly
believed it. "It can be done, definitely,"
Messina said. "Of course, seventy
thousand dollars is just the cost of the
components. The Chudnovskys are
counting very little of their human
time."
Yasumasa Kanada, the brothers' pi
rival at Tokyo University, uses a Hitachi
S-820/80 supercomputer that is be-
lieved to be considerably faster than a
Cray Y-MP8, and it burns close to
half a million watts&emdash;half a megawatt,
practically enough power to melt steel.
The Chudnovsky brothers particularly
hoped to leave Kanada and his Hita-
chi in the dust with their mail-order
funny car.
"We want to test our hardware,"
Gregory said.
"Pi is the best stress test for a su-
percomputer," David said.
"We also want to find out what
makes pi different from other num-
bers. It's a business."
"Galileo saw the moons of Jupiter
through his telescope, and he tried to
figure out the laws of gravity by look-
ing at the moons, but he couldn't,"
David said. "With pi, we are at the
stage of looking at the moons of Ju-
piter." He pulled his Mini Mag-Lite
flashlight out of his pocket and shone
it into a bookshelf, rooted through
some file folders, and handed me a
color photograph of pi. "This is a pi-
scape," he said. The photograph showed
a mountain range in cyberspace: bony
peaks and ridges cut by valleys. The
mountains and valleys were splashed
with colors&emdash;yellow, green, orange,
violet, and blue. It was the first eight
million digits of pi, mapped as a fractal
landscape by an I.B.M. GF-l 1 super-
computer at Yorktown Heights, which
Gregory had programmed from his
bed. Apart from its vivid colors, pi
looks like the Himalayas.
Gregory thought that the mountains
of pi seemed to contain structure. "I
see something systematic in this land-
scape, but it may be just an attempt by
the brain to translate some random
visual pattern into order," he said. As
he gazed into the nature beyond na-
ture, he wondered if he stood close to
a revelation about the circle and its
diameter. "Any very high hill in this
picture, or any flat plateau, or deep
valley, would be a sign of something in
pi," he said. "There are slight varia-
tions from randomness in this land-
scape. There are fewer peaks and
valleys than you would expect if pi
were truly random, and the peaks and
valleys tend to stay high or low a little
longer than you'd expect." In a man-
ner of speaking, the mountains of pi
looked to him as if they'd been molded
by the hand of the Nameless One,
Deus absconditus (the hidden God), but
he couldn't really express in words
what he thought he saw and, to his
great frustration, he couldn't express it
in the language of mathematics, either.
"Exploring pi is like exploring the
universe," David remarked.
"It's more like exploring underwa-
ter," Gregory said. "You are in the
mud, and everything looks the same.
You need a flashlight. Our computer
is the flashlight."
David said, "Gregory&emdash;I think,
really&emdash;you are getting tired."
A fax machine in a corner beeped
and emitted paper. It was a message
from a hardware dealer in Atlanta.
David tore off the paper and stared at
it. "They didn't ship it! I'm going to
kill them! This a service economy. Of
course, you know what that means&emdash;
the service is terrible."
"We collect price quotes by fax,"
Gregory said.
"It's a horrible thing. Window-
shopping in supercomputerland. We
can't buy everything&emdash;"
"Because everything won't exist,"
Gregory said.
"We only want to build a ma-
chine to compute a few transcendental
numbers&emdash;"
"Because we are not licensed for
transcendental meditation," Gregory
said.
"Look, we are getting nutty," David
said.
"We are not the only ones," Greg-
ory said. "We are getting an average
of one letter a month from someone or
other who is trying to prove Fermat's
Last Theorem."
I asked the brothers if they had
published any of their digits of pi in
a book.
Gregory said that he didn't know
how many trees you would have to
grind up in order to publish a billion
digits of pi in a book. The brothers' pi
had been published on fifteen hundred
microfiche cards stored somewhere in
Gregory's apartment. The cards held
three hundred thousand pages of data,
a slug of information much bigger
than the Encylopaedia Britannica, and
containing but one entry, "Pi." David
offered to find the cards for me; they
had to be around here somewhere. He
switched on the lights in the hallway
and began to shift boxes. Gregory
rifled bookshelves.
"Please sit down, Gregory," David
said. Finally, the brothers confessed
that they had temporarily lost their pi.
"Look, it's not a problem," David said.
"We keep it in different places." He
reached inside m zero and pulled out
a metal box. It was a naked hard-disk
drive, studded with chips. He handed
me the object. "There's pi stored on
this drive." It hummed gently. "You
are holding some pi in your hand. It
weighs six pounds."
MONTHS passed before I visited
the Chudnovskys again. The
brothers had been tinkering with their
machine and getting it ready to go for
two billion digits of pi, when Gregory
developed an abnormality related to
one of his kidneys. He went to the
hospital and had some CAT scans made
of his torso, to see what things looked
like, but the brothers were disappointed
in the pictures, and persuaded the doctors
to give them the CAT data on a mag-
netic tape. They took the tape home,
processed it in m zero, and got spec-
tacular color images of Gregory's torso.
The images showed cross-sectional
slices of his body, viewed through
different angles, and they were far
more detailed than any image from a
CAT scanner. Gregory wrote the im-
aging software. It took him a few
weeks. "There's a lot of interesting
mathematics in the problem of imag-
ing a body," he remarked. For the
moment, it was more interesting than
pi, and it delayed the brothers' probe
into the Ludolphian number.
Spring came, and Federal Express
was active at the Chudnovskys' build-
ing. Then the brothers began to cal-
culate pi, slowly at first, more intensely
as they gained confidence in their
machine, but in May the weather
warmed up and Con Edison betrayed
the brothers. A heat wave caused a
brownout in New York City, and as
it struck, m zero automatically shut
itself down, to protect its circuits, and
died. Afterward, the brothers couldn't
get electricity running properly through
the machine. They spent two weeks
restarting it, piece by piece.
Then, on Memorial Day weekend,
as the calculation was beginning to
progress, Malka Benjaminovna suffered
a heart attack. Gregory was alone with
his mother in the apartment. He gave
her chest compressions and breathed
air into her lungs, although David
later couldn't understand how his
brother didn't kill himself saving her.
An ambulance rushed her to St. Luke's
Hospital. The brothers were terrified
that they would lose her, and
the strain almost killed David. ;Û
One day, he fainted in his
mother's hospital room and
threw up blood. He had devel-
oped a bleeding ulcer. "Look,
it's not a problem," he said
later. After Malka Benjaminovna had
been moved out of intensive care,
Gregory rented a laptop computer,
plugged it into the telephone line in
her hospital room, and talked to m zero
at night through cyberspace, driving
the supercomputer toward pi and
watching his mother's blood pressure
at the same time.
Malka Benjaminovna improved
slowly. When St. Luke's released her,
the brothers settled her in her room in
Gregory's apartment and hired a nurse
to look after her. I visited them shortly
after that, on a hot day in early sum-
mer. David answered the door. There
were blue half circles under his eyes,
and he had lost weight. He smiled
weakly and greeted me by saying, "I
believe it was Oliver Heaviside, the
English physicist, who once said, 'In
order to know soup, it is not necessary
to climb into a pot and be boiled.' But,
look, if you want to be boiled you are
welcome to come inside." He led me
down the dark hallway. Malka Benja-
minovna was asleep in her bedroom,
and the nurse was sitting beside her.
Her room was lined with bookshelves,
packed with paper&emdash;it was an overflow
repository.
"Theoretically, the best way to cool
a supercomputer is to submerge it in
water," Gregory said, from his bed in
the junk yard.
"Then we could add goldfish," David
said.
"That would solve all our problems."
"We are not good plumbers, Greg-
ory. As long as I am alive, we will not
cool a machine with water."
"What is the temperature in there?"
Gregory asked, nodding toward m zero's
room.
"It grows to thirty-four degrees Cel-
sius. Above ninety Fahrenheit. This is
not good. Things begin to fry."
David took Gregory under the arm,
and we passed through the French
door into gloom and pestilential heat.
The shades were drawn, the lights
were off, and an air-conditioner in a
window ran in vain. Sweat immedi-
ately began to pour down my body. "I
don't like to go into this room," Greg-
ory said. The steel frame in the center
of the room&emdash;the heart of
m zero&emdash;had acquired more
logic boards, and more red lights
blinked inside the machine. I
could hear disk drives murmur-
ing. The drives were copying
and recopying segments of tran-
scendental numbers, to check the digits
for perfect accuracy. Gregory knelt on
the floor, facing the steel frame.
David opened a cardboard box and
removed an electronic board. He be-
gan to fit it into m zero. I noticed that
his hands were marked with small
cuts, which he had got from reaching
inside the machine.
"David, could you give me the
flashlight?" Gregory said.
David pulled the Mini Mag-Lite
from his shirt pocket and handed it to
Gregory. The brothers knelt beside
each other, Gregory shining the flash-
light into the supercomputer. David
reached inside with his fingers and
palpated a logic board.
"Don't!" Gregory said. "O.K., look.
No! No!" They muttered to each other
in Russian. "It's too small," Gregory
said.
David adjusted an electric fan. "We
bought it at a hardware store down the
street," he said to me. "We buy our
fans in the winter. It saves money."
He pointed to a gauge that had a
dial on it. "Here we have a meat
thermometer."
The brothers had thrust the ther-
mometer between two circuit boards in
order to look for hot spots inside m zero.
The thermometer's dial was marked
"Beef Rare&emdash;Ham&emdash;Beef Med&emdash;
Pork."
"You want to keep the machine
below 'Pork,' " Gregory remarked. He
lifted a keyboard out of the steel frame
and typed something on it, staring at
a display screen. Numbers filled the
screen. "The machine is checking its
memory," he said. A buzzer sounded.
"It shut down!" he said. "It's a disk-
drive controller. The stupid thing
obviously has problems."
"It's mentally deficient," David
commented. He went over to a book-
shelf and picked up a hunting knife.
I thought he was going to plunge it
into the supercomputer, but he used it
to rip open a cardboard box. "We're
going to ship the part back to the
manufacturer," he said to me. "You
had better send it in the original box
or you may not get your money back.
Now you know the reason this apart-
ment is full of empty boxes. We have
to save them. Gregory, I wonder if you
are tired."
"If I stand up now, I will fall
down," Gregory said, from the floor.
"Therefore, I will sit in my center of
gravity. I will maintain my center of
gravity. Let me see, meanwhile, what
is happening with this machine." He
typed something on his keyboard. "You
won't believe it, Dave, but the control-
ler now seems to work."
"We need to buy a new one," David
said.
"Try Nevada."
David dialled a mail-order house in
Nevada that here will be called Search-
light Computers. He said loudly, in a
thick Russian accent, "Hi, Searchlight.
I need a fifteen-forty controller....
No! No! No! I don't need any-
thing else! Just the controller! Just a
naked unit! Naked! How much you
charge? . . . Two hundred and fifty-
seven dollars?"
Gregory glanced at his brother and
shrugged. "Eh."
"Look, Searchlight, can you ship it
to me Federal Express? For tomorrow
morning. How much?. . . Thirty-nine
dollars for Fed Ex? Come on! What
about afternoon delivery? . . . Twenty-
nine dollars before 3 P.M.? Relax. What
is your name? . . . Bob. Fine. O.K. So
it's two hundred and fifty-seven dol-
lars plus twenty-nine dollars for Fed-
eral Express?"
"Twenty-nine dollars for Fed Ex!"
Gregory burst out. "It should be fifteen."
He pulled a second keyboard out of the
steel frame and tapped the keys. An-
other display screen came alive and
filled with numbers.
"Tell me this," David said to Bob
in Nevada. "Do you have thirty-day
money-back guarantee? . . . No? Come
on! Look, any device might not work."
"Of course, a part might work,"
Gregory muttered to his brother. "But
it usually doesn't."
"Question Number Two: The Fed
Ex should not cost twenty-nine bucks,"
David said to Bob. "No, nothing! I'm
just asking." David hung up the phone.
"I'm going to call A.K.," he said. "Hi,
A.K., this is David Chudnovsky, call-
ing from New York. A.K., I need
another controller, like the one you
sent. Can you send it today Fed
Ex? . . . How much you charge? . . .
Naked! I want a naked unit! Not in
a shoebox, nothing!"
A rhythmic clicking sound came
from one of the disk drives. Gregory
remarked to me, "We are calculating
pi right now."
"Do you want my MasterCard?
Look, it's really imperative that I
get my unit tomorrow. A.K., please,
I really need my unit bad." David
hung up the telephone and sighed.
"This is what has happened to a pure
mathematician."
GREGORY and David are both ex-
tremely childlike, but I don't
mean childish at all," Gregory's wife,
Christine Pardo Chudnovsky, said one
muggy summer day, at the dining-
room table. "There is a certain amount
of play in everything they do, a certain
amount of fooling around between two
52
brothers." She is six years younger
than Gregory; she was an undergradu-
ate at Barnard College when she first
met him. "I fell in love with Gregory
immediately. His illness came with the
package." She is still in love with him,
even if at times they fight over his
heaps of paper. ("I don't have room
to put my things down," she says to
him.) As we talked, though, pyramids
of boxes and stacks of paper leaned
against the dining-room windows,
pressing against the glass and blocking
daylight, and a smell of hot electrical
gear crept through the room. "This
house is an example of mathematics in
family life," she said. At night, she
dreams that she is dancing from room
to room through an empty apartment
that has parquet floors.
David brought his mother out of her
bedroom, settled her at the table, and
kissed her on the cheek. Malka Ben-
jaminovna seemed frail but alert. She
is a small, white-haired woman with
a fresh face and clear blue eyes, who
speaks limited English. A mathemati-
cian once described Malka Ben-
jaminovna as the glue that holds the
Chudnovsky family together. She was
an engineer during the Second World
War, when she designed buildings,
laboratories, and proving grounds in
the Urals for testing the Katyusha
rocket; later, she taught engineering at
schools around Kiev. She handed me
plates of roast chicken, kasha, pickles,
cream cheese, brown bread, and little
wedges of The Laughing Cow cheese
in foil. "Mother thinks you aren't
getting enough to eat," Christine said.
Malka Benjaminovna slid a jug of
Gatorade across the table at me.
After lunch, and fortified with
Gatorade, the brothers and I went into
the chamber of m zero, into a pool of
thick heat. The room enveloped us like
noon on the Amazon, and it teemed
with hidden activity. The disk drives
clicked, the red lights flashed, the air-
conditioner hummed, and you could
hear dozens of whispering fans. Greg-
ory leaned on his cane and contem-
plated the machine. "It's doing many
jobs at the moment," he said. "Frankly,
I don't know what it's doing. It's doing
some algebra, and I think it's also
backing up some pieces of pi."
"Sit down, Gregory, or you will
fall," David said.
"What is it doing now, Dave?"
"It's blinking."
"It will die soon."
"Gregory, I heard a funny noise."
"You really heard it? Oh, God, it's
going to be like the last time&emdash;"
"That's it!"
"We are dead! It crashed!"
"Sit down, Gregory, for God's sake!"
Gregory sat on a stool and tugged
at his beard. "What was I doing before
the system crashed? With God's help,
I will remember." He jotted a few
notes in a laboratory notebook. David
slashed open a cardboard box with his
hunting knife and lifted out a board
studded with chips, for making color
images on a display screen, and plugged
it into m zero. Gregory crawled under
a table. "Oh, shit," he said, from
beneath the table.
"Gregory, you killed the system
again!"
"Dave, Dave, can you get me a
flashlight?"
David handed his Mini Mag-Lite
under the table. Gregory joined some
cables together and stood up. "Whoo!
Very uncomfortable. David, boot it up."
"Sit down for a moment."
Gregory slumped into a chair.
"This monster is going on the blink,"
David said, tapping a keyboard.
"It will be all right."
On a screen, m zero declared, "The
system is ready."
"Ah," David said.
The drives began to click, and the
parallel processors silently multiplied
and conjoined huge numbers. Gregory
headed for bed, David holding him by
the arm.
In the junk yard, his nest, his paper-
lined oubliette, Gregory kicked off his
gentleman's slippers, lay down on the
bed, and predicted the future. He said,
"The gigaflop supercomputers of today
are almost useless. What is needed is
a teraflop machine. That's a machine
that can run at a trillion flops, a trillion
floating-point operations per second, or
roughly a thousand times as fast as a
Cray Y-MP8. One such design for a
teraflop machine, by Monty Denneau,
at I.B.M., will be a parallel super-
computer in the form of a twelve-foot-
wide box. You want to have at least
MARCH 2, 1992
sixty-four thousand processors in the
machine, each of which has the power
of a Cray. And the processors will be
joined by a network that has the total
switching capacity of the entire tele-
phone network in the United States.
I think a teraflop machine will exist
by 1993. Now, a better machine is a
petaflop machine. A petaflop is a qua-
drillion flops, a quadrillion floating-
point operations per second, so a petaflop
machine is a thousand times as fast as
a teraflop machine, or a million times
as fast as a Cray Y-MP8. The petaflop
machine will exist by the year 2000,
or soon afterward. It will fit into a
sphere less than a hundred feet in
diameter. It will use light and mir-
rors&emdash;the machine's network will con-
sist of optical cables rather than copper
wires. By that time, a gigaflop 'super-
computer' will be a single chip. I think
that the petaflop machine will be used
mainly to simulate machines like itself,
so that we can begin to design some
real machines."
I N the nineteenth century, math-
ematicians aKacked pi with the help
of human computers. The most pow-
erful of these was Johann Martin
Zacharias Dase, a prodigy from Ham-
burg. Dase could multiply large num-
bers in his head, and he made a living
exhibiting himself to crowds in Ger-
many, Denmark, and England, and
hiring himself out to mathematicians.
A mathematician once asked Dase to
multiply 79,532,853 by 93,758,479, and
Dase gave the right answer in fifty-
four seconds. Dase extracted the square
root of a hundred-digit number in
fifty-two minutes, and he was able to
multiply a couple of hundred-digit num-
bers in his head during a period of
eight and three-quarters hours. Dase
could do this kind of thing for weeks
on end, running as an unattended
supercomputer. He would break off a
calculation at bedtime, store every-
thing in his memory for the night, and
resume calculation in the morning.
Occasionally, Dase had a system crash.
In 1845, he bombed while trying to
demonstrate his powers to a mathema-
tician and astronomer named Hein-
rich Christian Schumacher, reckoning
wrongly every multiplication that he
attempted. He explained to Schumacher
that he had a headache. Schumacher
also noted that Dase did not in the least
understand theoretical mathematics. A
mathematician named Julius Petersen
once tried in vain for six weeks to
teach Dase the rudiments of Euclidean
geometry, but they absolutely baffled
Dase. Large numbers Dase could
handle, and in 1844 L. K. Schulz
von Strassnitsky hired him to compute
pi. Dase ran the job for almost two
months iri his brain, and at the end of
the time he wrote down pi correctly to
the first two hundred decimal places&emdash;
then a world record.
To many mathematicians, math-
ematical objects such as the number pi
seem to exist in an external,
objective reality. Numbers seem
to exist apart from time or the
world; numbers seem to tran-
scend the universe; numbers
might exist even if the uni-
verse did not. I suspect that in
their hearts most working
mathematicians are Platonists,
in that they take it as a matter of
unassailable if unprovable fact that
mathematical reality stands apart from
the world, and is at least as real as the
world, and possibly gives shape to the
world, as Plato suggested. Most math-
ematicians would probably agree that
the ratio of the circle to its diameter
exists brilliantly in the nature beyond
nature, and would exist even if the
human mind was not aware of it, and
might exist even if God had not both-
ered to create it. One could imagine
that pi existed before the universe came
into being and will exist after the
universe is gone. Pi may even exist
apart from God, in the opinion of some
mathematicians, for while there is
reason to doubt the existence of Gd,
by their way of thinking there is no
good reason to doubt the existence of
the circle.
"To an extent, pi is more real than
the machine that is computing it,"
Gregory remarked to me one day.
"Plato was right. I am a Platonist. Of
course pi is a natural object. Since pi
is there, it exists. What we are doing
is really close to experimental phys-
ics&emdash;we are 'observing pi.' Since we
can observe pi, I prefer to think of pi
as a natural object. Observing pi is
easier than studying physical phenom-
ena, because you can prove things in
mathematics, whereas you can't prove
anything in physics. And, unfortu-
nately, the laws of physics change once
every generation."
"Is mathematics a form of art?" I
asked.
"Mathematics is partially an art,
even though it is a natural science," he
said. "Everything in mathematics does
exist now. It's a matter of naming it.
The thing doesn't arrive from God in
a fixed form; it's a matter of represent-
ing it with symbols. You put it through
your mind in order to make sense
of it."
Mathematicians have sorted num-
bers into classes in order to make sense
of them. One class of numbers is that
of the rational numbers. A rational
number is a fraction composed of
inteÛers (whole numbers):
l/l, T/3, 3/5, 1¡/7l, and so on.
Every rational number, when
it is expressed in decimal form,
repeats periodically: I/3, for
example, becomes .333....
Next, we come to the irratio-
nal numbers. An irrational
number can't be expressed as
a fraction composed of whole numbers,
and, furthermore, its digits go to in-
finity without repeating periodically.
The square root of two (Û12) is an
irrational number. There is simply no
way to represent any irrational number
as the ratio of two whole numbers; it
can't be done. Hippasus of Metapontum
supposedly made this discovery in the
fifth century B.C., while travelling in
a boat with some mathematicians who
were followers of Pythagoras. The
Pythagoreans believed that everything
in nature could be reduced to a ratio
of two whole numbers, and they threw
Hippasus overboard for his discovery,
since he had wrecked their universe.
Expanded as a decimal, the square root
of two begins 1.41421 . . . and runs in
"random" digits forever. It looks ex-
actly like pi in its decimal expan-
sion; it is a hopeless jumble, show-
ing no obvious system or design. The
square root of two is not a transcen-
dental number, because it can be found
with an equation. It is the solution
(root) of an equation. The equation is
x2 = 2, and a solution is the square
root of two. Such numbers are called
algebraic.
While pi is indeed an irrational
number&emdash;it can't be expressed as a
fraction made of whole numbers&emdash;
more important, it can't be expressed
with finite algebra. Pi is therefore said
to be a transcendental number, because
it transcends algebra. Simply and gen-
erally speaking, a transcendental num-
ber can't be pinpointed through an
equation built from a finite number of
integers. There is no finite algebraic
54
equation built from whole numbers
that will give an exact value for pi.
The statement can be turned around
this way: pi is not the solution to any
equation built from a less than infinite
series of whole numbers. If equations
are trains threading the landscape of
numbers, then no train stops at pi.
Pi is elusive, and can be approached
only through rational approximations.
The approximations hover around the
number, closing in on it, but do not
touch it. Any formula that heads to-
ward pi will consist of a chain of
operations that never ends. It is an
infinite series. In 1674, Gottfried
Wilhelm Leibniz (the co-inventor of
the calculus, along with Isaac New-
ton) noticed an extraordinary pattern
of numbers buried in the circle. The
Leibniz series for pi has been called
one of the most beautiful mathematical
discoveries of the seventeenth century:
In English: pi over four equals one
minus a third plus a fifth minus a
seventh plus a ninth&emdash;and so on. You
follow the odd numbers out to infinity,
and when you arrive there and sum
the terms, you get pi. But since you
never arrive at infinity you never get
pi. Mathematicians find it deeply mys-
terious that a chain of discrete rational
numbers can connect so easily to ge-
ometry, to the smooth and continuous
circle.
As an experiment in "observing pi,"
as Gregory Chudnovsky puts it, I
computed the Leibniz series on a pocket
calculator. It was easy, and I got
results that did seem to wander slowly
toward pi. As the series progresses, the
answers touch on 2.66, 3.46, 2.89, and
3.34, in that order. The answers land
higher than pi and lower than pi,
skipping back and forth across pi, and
gradually closing in on pi. A math-
ematician would say that the series
"converges on pi." It converges on pi
forever, playing hopscotch over pi but
never landing on pi.
You can take the Leibniz series out
a long distance&emdash;you can even dra-
matically speed up its movement to-
ward pi by adding a few corrections to
it&emdash;but no matter how far you take
the Leibniz series, and no matter
how many corrections you hammer into
it, when you stop the operation and
sum the terms, you will get a rational
number that is somewhere around pi
but is not pi, and you will be damned
if you can put your hands on pi.
Transcendental numbers continue
forever, as an endless non-repeating
string, in whatever rational form you
choose to display them, whether as
digits or as an equation. The Leibniz
series is a beautiful way to represent
pi, and it is finally mysterious, because
it doesn't tell us much about pi. Look-
ing at the Leibniz series, you feel the
independence of mathematics from
human culture. Surely, on any world
that knows pi the Leibniz series will
also be known. Leibniz wasn't the first
mathematician to discover the Leibniz
series. Nilakantha, an astronomer,
grammarian, and mathematician who
lived on the Kerala coast of India,
described the formula in Sanskrit po-
etry around the year 1500.
It is worth thinking about what a
decimal place means. Each decimal
place of pi is a range that shows the
approximaÛe location of pi to an accu-
racy ten times as great as the previous
range. But as you compute the next
decimal place you have no idea where
pi will appear in the range. It could
pop up in 3, or just as easily in 9, or
in 2. The apparent movement of pi as
you narrow the range is known as the
random walk of pi.
Pi does not move; pi is a fixed point.
The algebra wobbles around pi. There
is no such thing as a formula that is
steady enough or sharp enough to stick
a pin into pi. Mathematicians have
discovered formulas that converge on
pi very fast (that is, they skip around
pi with rapidly increasing accuracy),
but they do not and cannot hit pi. The
Chudnovsky brothers discovered their
own formula in 1984, and it attacks pi
with great ferocity and elegance. The
Chudnovsky formula is the fastest series
for pi ever found which uses rational
numbers. Various other series for pi,
which use irrational numbers, have
also been found, and they converge on
pi faster than the Chudnovsky for-
mula, but in practice they run more
slowly on a computer, because irratio-
nal numbers are harder to compute.
The Chudnovsky formula for pi is
thought to be "extremely beautiful" by
persons who have a good feel for
numbers, and it is based on a torus (a
doughnut), rather than on a circle. It
uses large assemblages of whole num-
bers to hunt for pi, and it owes much
to an earlier formula for pi worked out
in 1914 by Srinivasa Ramanujan, a
mathematician from Madras, who was
a number theorist of unsurpassed ge-
nius. Gregory says that the Chudnovsky
formula "is in the style of Ramanujan,"
and that it "is really very simple, and
can be programmed into a computer
with a few lines of code."
In 1873, Georg Cantor, a Russian-
born mathematician who was one of
the towering intellectual figures of the
nineteenth century, proved that the set
of transcendental numbers is infinitely
more extensive than the set of alge-
braic numbers. That is, finite algebra
can't find or describe most numbers. To
put it another way, most numbers are
infinitely long and non-repeating in
any rational form of representation. In
this respect, most numbers are like pi.
Cantor's proof was a disturbing piece
of news, for at that time very few
transcendental numbers were actually
known. ( Not until nearly a decade
later did Ferdinand Lindemann finally
prove the transcendence of pi; before
that, mathematicians had only conjec-
tured that pi was transcendental.) Per-
haps even more disturbing, Cantor
offered no clue, in his proof, to what
a transcendental number might look
like, or how to construct such a beast.
Cantor's celebrated proof of the exis-
tence of uncountable multitudes of tran-
scendental numbers resembled a proof
that the world is packed with micro-
scopic angels&emdash;a proof, however, that
does not tell us what the angels look
like or where they can be found; it
merely proves that they exist in un-
countable multitudes. While Cantor's
proof lacked any specific description of
a transcendental number, it showed
that algebraic numbers (such as the
square root of two) are few and far
between: they poke up like marker
buoys through the sea of transcenden-
tal numbers.
Cantor's proof disturbed some math-
ematicians because, in the first place,
it suggested that they had not yet
discovered most numbers, which were
transcendentals, and in the second
place that they lacked any tools or
methods that would determine whether
a given number was transcendental
or not. Leopold Kronecker, an influ-
ential older mathematician, rejected
Cantor's proof, and resisted the whole
notion of "discovering" a number. (He
once said, in a famous remark, "God
made the integers, all else is the work
of man.") Cantor's proof has with-
stood such attacks, and today the de-
bate over whether transcendental num-
bers are a work of God or man has
subsided, mathematicians having de-
cided to work with transcendental
numbers no matter who made them.
The Chudnovsky brothers claim that
the digits of pi form the most nearly
perfect random sequence of digits that
has ever been discovered. They say
that nothing known to humanity ap-
pears to be more deeply unpredictable
than the succession of digits in pi,
except, perhaps, the haphazard clicks
of a Geiger counter as it detects the
decay of radioactive nuclei. But pi is
not random. The fact that pi can be
produced by a relatively simple for-
mula means that pi is orderly. Pi looks
random only because the pattern in the
digits is fantastically complex. The
Ludolphian number is fixed in eter-
nity&emdash;not a digit out of place, all char-
acters in their proper order, an endless
sentence written to the end of the
world by the division of the circle's
diameter into its circumference. Vari-
ous simple methods of approximation
will always yield the same succession
of digits in the same order. If a single
digit in pi were to be changed any-
where between here and infinity, the
resulting number would no longer be
pi; it would be "garbage," in David's
word, because to change a single digit
in pi is to throw all the following digits
out of whack and miles from pi.
"Pi is a damned good fake of a
random number," Gregory said. "I
just wish it were not as good a fake.
It would make our lives a lot easier."
Around the three-hundred-millionth
decimal place of pi, the digits go
88888888&emdash;eight eights pop up in a
row. Does this mean anything? It
appears to be random noise. Later,
ten sixes erupt: 6666666666. What
does this mean? Apparently nothing,
only more noise. Somewhere past the
half-billion mark appears the string
123456789. It's an accident, as it were.
"We do not have a good, clear, crys-
tallized idea of randomness," Gregory
said. "It cannot be that pi is truly
random. Actually, a truly random se-
quence of numbers has not yet been
discovered."
No one knows what happens to the
digits of pi in the deeper regions, as the
number is resolved toward infinity. Do
the digits turn into nothing but eights
and fives, say? Do they show a pre-
dominance of sevens? Similarly, no
one knows if a digit stops appearing in
pi. This conjecture says that after a
certain point in the sequence a digit
drops out completely. For example, no
more fives appear in pi&emdash;something
like that. Almost certainly, pi does not
do such things, Gregory Chudnovsky
thinks, because it would be stupid, and
nature isn't stupid. Nevertheless, no
one has ever been able to prove or
disprove a certain basic conjecture about
pi: that every digit has an equal chance
of appearing in pi. This is known as
the normality conjecture for pi. The
normality conjecture says that, on
average, there is no more or less of any
digit in pi: for example, there is no
excess of sevens in pi. If all digits do
appear with the same average fre-
quency in pi, then pi is a "normal"
number&emdash;"normal" by the narrow
mathematical definition of the word.
"This is the simplest possible conjec-
ture about pi," Gregory said. "There
is absolutely no doubt that pi is a
'normal' number. Yet we can't prove
it. We don't even know how to try to
prove it. We know very little about
transcendental numbers, and, what is
worse, the number of conjectures about
them isn't growing." No one knows
even how to tell the difference between
the square root of two and pi merely
by looking at long strings of their
digits, though the two numbers have
completely distinct mathematical prop-
erties, one being algebraic and the
other transcendental.
Even if the brothers couldn't prove
anything about the digits of pi, they felt
that by looking at them through the
window of their machine they mighÛ
at least see something that could lead
to an important conjecture about pi or
about transcendental numbers as a class.
You can learn a lot about all cats by
looking closely at one of them. So if
you wanted to look closely at pi how
much of it could you see with a very
large supercomputer? What if you
turned the universe into a supercom-
puter? What then? How much pi could
you see? Naturally, the brothers had
considered this project. They had
imagined a computer built from the
universe. Here's how they estimated
the machine's size. It has been calcu-
lated that there are about 1079 electrons
and protons in the observable universe;
this is the so-called Eddington number
of the universe. (Sir Arthur Stanley
EddingtonÛ the astrophysicist, first came
up with the number.) The Edding-
ton number is the digit 1 followed by
seventy-nine zeros: 10,000,000,000,000,
000,000,000,000,000,000,000,000,000,
ooo,ooo,ooo,ooojooo,ooo,ooo,ooo,ooo,
000,000,000,000. Ten vigintsextillion.
The Eddington number. It declares
the power of the Eddington machine.
The Eddington machine would be
the universal supercomputer. It would
be made of all the atoms in the uni-
verse. The Eddington machine would
contain ten vigintsextillion parts, and
if the Chudnovsky brothers could figure
out how to program it with FORTRAN
they might make it churn toward pi.
"In order to study the sequence of pi,
you have to store it in the Eddington
machine's memory," Gregory said. To
be realistic, the brothers thought that
a practical Eddington machine wouldn't
be able to store pi much beyond 1077
digits&emdash;a number that is only a hun-
dredth of the Eddington number. Now,
what if the digits of pi only begin to
show regularity beyond 1077 digits?
Suppose, for example, that pi manifests
a regularity starting at 101¡¡ decimal
places? That number is known as a
googol. If the design in pi appears only
after a googol of digits, then not even
the Eddington machine will see any
system in pi; pi will look totally dis-
ordered to the universe, even if pi
contains a slow, vast, delicate struc-
ture. A mere googol of pi might be only
the first knot at the corner of a kind
of limitless Persian rug, which is woven
into increasingly elaborate diamonds,
cross-stars, gardens, and cosmogonies.
It may never be possible, in principle,
to see the order in the digits of pi. Not
even nature itself may know the nature
of pi.
"If pi doesn't show systematic be-
havior until more than ten to the
seventy-seven decimal places, it would
MARCH 2, 1992
really be a disaster," Gregory said. "It
would be actually horrifying."
"I wouldn't give up," David said.
"There might be some other way of
leaping over the barrier&emdash;"
"And of attacking the son of a bitch,"
Gregory said.
THE brothers first came in contact
with the membrane that divides
the dreamlike earth from mathemati-
cal reality when they were boys, grow-
ing up in Kiev, and their father gave
David a book entitled "What Is Math-
ematics?," by two mathematicians named
Richard Courant and Herbert Rob-
bins. The book is a classic&emdash;millions
of copies of it have been printed in
unauthorized Russian and Chinese edi-
tions alone&emdash;and after the brothers
finished reading "Robbins," as the book
is called in Russia, David decided to
become a mathematician, and Gregory
soon followed his brother's footsteps
into the nature beyond nature. Gregory's
first publication, in the journal Soviet
Mathematics&emdash;Doklady, came when he
was sixteen years old: "Some Results
in the Theory of Infinitely Long Ex-
pressions." Already you can see where
he was headed. David, sensing his
younger brother's power, encouraged
him to grapple with central problems
in mathematics. Gregory made his first
major discovery at the age of seven-
teen, when he solved Hilbert's Tenth
Problem. (It was one of twenty-three
great problems posed by David Hilbert
in 1900.) To solve a Hilbert problem
would be an achievement for a life-
time; Gregory was a high-school stu-
dent who had read a few books on
mathematics. Strangely, a young Rus-
sian mathematician named Yuri Matya-
sevich had just solved Hilbert's Tenth
Problem, and the brothers hadn't heard
the news. Matyasevich has recently
said that the Chudnovsky method is
the preferred way to solve Hilbert's
Tenth Problem.
The brothers enrolled at Kiev State
University, and both graduated summa
cum laude. They took their Ph.D.s at
the Institute of Mathematics at the
Ukrainian Academy of Sciences. At
first, they published their papers sepa-
rately, but by the mid-nineteen-seventies
they were collaborating on much of
their work. They lived with their parents
in Kiev until the family decided to try
to take Gregory abroad for treatment,
and in 1976 Volf and Malka Chud-
novsky applied to the government to
emigrate. Volf was immediately fired
from his job.
The K.G.B. began tailing the broth-
ers. "Gregory would not believe me
until it became totally obvious," David
said. "I had twelve K.G.B. agents on
my tail. No, look, I'm not kidding!
They shadowed me around the clock
in two cars, six agents in each car.
Three in the front seat and three in
the back seat. That was how the K.G. B.
operated." One day, in 1976, David
was walking down the street when
K.G.B. officers attacked him, breaking
his skull. He went home and nearly
died, but didn't go to the hospital. "If
I had gone to the hospital, I would
have died for sure," he told me. "The
hospital is run by the state. I would
forget to breathe."
On July 22, 1977, plainclothesmen
from the K.G.B. accosted Volf and
Malka on a street in Kiev and beat
them up. They broke Malka's arm and
fractured her skull. David took his
mother to the hospital. "The doctor in
the emergency room said there was no
fracture," David said.
Gregory, at home in bed, was not
so vulnerable. Also, he was conspicu-
ous in the West. Edwin Hewitt, a
mathematician at the University of
Washington, in Seattle, had visited
Kiev in 1976 and collaborated with
Gregory on a paper, and later, when
Hewitt learned that the Chudnovsky
family was in trouble, he persuaded
Senator Henry M. Jackson, the pow-
erful member of the Senate Armed
Services Committee, to take up the
Chudnovskys' case. Jackson put pres-
sure on the Soviets to let the family
leave the country. Just before the K.G.B.
attacked the parents, two members of
a French parliamentary delegation that
was in Kiev made an unofficial visit to
the Chudnovskys to see what was going
on. One of the visitors, a staff member
of the delegation, was Nicole Lanne-
grace, who married David in 1983.
Andrei Sakharov also helped to draw
attention to the Chudnovskys' increas-
ingly desperate situation. Two months
after the parents were attacked, the
Soviet government unexpectedly let the
family go. "That summer when I was
geKing killed by the K.G.B., I could
never have imagined that the next year
I would be in Paris or that I would
wind up in New York, married to a
beautiful Frenchwoman," David said.
The Chudnovsky family settled in New
York, near Columbia University.
I F pi is truly random, then at
times pi will appear to be ordered.
Therefore, if pi is random it contains
accidental order. For example, some-
where in pi a sequence may run
07070707070707 for as many decima]
places as there are, say, hydrogen at-
oms in the sun. It's just an accident.
Somewhere else the same sequence of
zeros and sevens may appear, only this
time interrupted by a single occurrence
of the digit 3. Another accident. Those
and all other "accidental" arrange-
ments of digits almost certainly erupl
in pi, but their presence has never been
proved. "Even if pi is not truly ran-
dom, you can still assume that you
get every string of digits in pi," Greg-
ory said.
If you were to assign letters of the
alphabet to combinations of digits, and
were to do this for all human alpha-
bets, syllabaries, and ideograms, then
you could fit any written character in
any language to a combination of digits
in pi. According to this system, pi could
be turned into literature. Then, if you
could look far enough into pi, you
would probably find the expression
"See the U.S.A. in a Chevrolet!" a
billion times in a row. Elsewhere, you
would find Christ's Sermon on the
Mount in His native Aramaic tongue,
and you would find versions of the
Sermon on the Mount that are pure
blasphemy. Also, you would find a
dictionary of Yanomamo curses. A
guide to the pawnshops of Lubbock.
The book about the sea which James
Joyce supposedly declared he would
write after he finished "Finnegans
Wake." The collected transcripts of
"The Tonight Show" rendered into
Etruscan. "Knowledge of All Existing
Things," by Ahmes the Egyptian scribe.
Each occurrence of an apparently- or-
dered string in pi, such as the words
t "Ruin hath taught me thus to rumi-
nate/That Time will come and take
my love away," is followed by unimag-
inable deserts of babble. No book and
none but the shortest poems will ever
be seen in pi, since it is infinitesimally
unlikely that even as brief a text as
an English sonnet will appear in the
first 1077 digits of pi, which is the
longest piece of pi that can be calcu-
lated in this universe.
Anything that can be produced by a
simple method is by definition orderly.
Pi can be produced by various simple
methods of rational approximation, and
those methods yield the same digits in
a fixed order forever. Therefore, pi is
- orderly in the extreme. Pi may also be
a powerful random-number generator,
spinning out any and all possible com-
binations of digits. We see that the
distinction between chance and fixity
dissolves in pi. The deep connection
between disorder and order, between
cacophony and harmony, in the most
famous ratio in mathematics fascinated
Gregory and David Chudnovsky. They
wondered if the digits of pi had a
personality.
"We are looking for the appearance
of some rules that will distinguish the
digits of pi from other numbers,"
Gregory explained. "It's like studying
writers by studying their use of words,
their grammar. If you see a Russian
sentence that extends for a whole page,
with hardly a comma, it is definitely
Tolstoy. If someone were to give you
a million digits from somewhere in pi,
could you tell it was from pi? We don't
really look for patterns; we look for
rules. Think of games for children. If
I give you the sequence one, two,
three, four, five, can you tell me what
the next digit is? Even a child can do
it; the next digit is six. How about this
game? Three, one, four, one, five,
nine. Just by looking at that sequence,
can you tell me the next digit? What
if I gave you a sequence of a million
digits from pi? Could you tell me the
next digit just by looking at the se-
quence? Why does pi look like a totally
unpredictable sequence with the high-
est complexity? We need to find out the
rules that govern this game. For all we
know, we may never find a rule in pi."
HERBERT ROBBINS, the co-author of
"What Is Mathematics?," is an
emeritus professor of mathematical
statistics at Columbia University. For
the past six years, he has been teaching
at Rutgers. The Chudnovskys call him
once in a while to get his advice on
how to use statistical tools to search for
signs of order in pi. Robbins lives in
a rectilinear house that has a lot of
glass in it, in the woods on the out-
skirts of Princeton. Some of the twen-
tieth century's most creative and pow-
erful discoveries in statistics and
probability theory happened inside his
head. Robbins is a tall, restless man
in his seventies, with a loud voice,
furrowed cheeks, and penetrating eyes.
One recent day, he stretched himself
out on a daybed in a garden room
in his house and played with a rub-
ber band, making a harp across his
fingertips.
"It is a very difficult philosophical
question, the question of what 'ran-
dom' is," he said. He plucked the
rubber band with his thumb, boink,
boink. "Everyone knows the famous
remark of Albert Einstein, that God
does not throw dice. Einstein just would
not believe that there is an element of
randomness in the construction of the
world. The question of whether the
universe is a random process or is
determined in some way is a basic
philosophical question that has nothing
to do with mathematics. The question
is important. People consider it when
they decide what to do with their lives.
It concerns religion. It is the question
of whether our fate will be revealed or
whether we live by blind chance. My
God, how many people have been
murdered over an answer to that ques-
tion! Mathematics is a lesser activity
than religion in the sense that we've
agreed not to kill each other but to
discuss things."
Robbins got up from the daybed and
sat in an armchair. Then he stood up
and paced the room, and sat at a table
in the room, and sat on a couch, and
went back to the table, and finally
returned to the daybed. The man was
in constant motion. It looked random
to me, but it may have been systematic.
It was the random walk of Herbert
Robbins.
"Mathematics is broken into tiny
specialties today, but Gregory Chud-
novsky is a generalist who knows the
whole of mathematics as well as any-
one," he said as he moved around.
"You have to go back a hundred years,
to David Hilbert, to find a mathema-
tician as broadly knowledgeable as
Gregory Chudnovsky. He's like Mozart:
he's the last of his breed. I happen to
think the brothers' pi project is a will-
o'-the-wisp, and is one of the least
interesting things they've ever done.
But what do I know? Gregory seems
to be asking questions that can't be
answered. To ask for the system in pi
is like asking 'Is there life after death?'
When you die, you'll find out. Most
mathematicians are not interested in
the digits of pi, because the question is
of no practical importance. In order for
a mathematician to become interested
in a problem, there has to be a possi-
bility of solving it. If you are an
athlete, you ask yourself if you can
jump thirty feet. Gregory likes to ask
if he can jump around the world. He
likes to do things that are impossible."
At some point after the brothers
settled in New York, it became obvi-
ous that Columbia University was not
going to be able to invite them to
become full-fledged members of the
faculty. Since then, the brothers have
always enjoyed cordial personal rela-
tionships with various members of the
faculty, but as an institution the Math-
ematics Department has been unable
to create permanent faculty positions
for them. Robbins and a couple of
fellow-mathematicians&emdash;Lipman Bers
and the late Mark Kac&emdash;once tried to
raise money from private sources for
an endowed chair at Columbia to be
shared by the brothers, but the effort
failed. Then the John D. and Cath-
erine T. MacArthur Foundation award-
ed Gregory Chudnovsky a "genius"
fellowship; that happened in 1981, the
first year the awards were given, as if
to suggest that Gregory is a person
for whom the MacArthur prize was
invented. The brothers can exhibit
other fashionable paper&emdash;a Prix Peccot-
Vimont, a couple of Guggenheims, a
Doctor of Science honorÛs causa from
Bard College, the Moscow Mathemati-
cal Society Prize&emdash;but there is one
defect in their rŽsumŽ, which is the fact
that Gregory has to lie in bed most of
the day. The ugly truth is that Gregory
Chudnovsky can't get a permanent job
at any American institution of higher
learning because he is physically dis-
abled. But there are other, more per-
plexing reasons that have led the Chud-
novsky brothers to pursue their work
in solitude, outside the normal aca-
demic hierarchy, since the day they
arrived in the United States.
Columbia University has awarded
each brother the title of senior research
scientist in the Department of Math-
ematics. Their position at Columbia is
ambiguous. The university officially
considers them to be members of the
faculty, but they don't have tenure, and
Columbia doesn't spend its own funds
to pay their salaries or to support their
research. However, Columbia does give
them heakh-insurance benefits and a
housing subsidy.
The brothers have been living on
modest grants from the National Sci-
ence Foundation and various other
research agencies, which are funnelled
through Columbia and have to be
applied for regularly. Nicole Lanne-
grace and Christine Chudnovsky fi-
nanced m zero out of their paychecks.
Christine's father, Gonzalo Pardo, who
is a professor of dentistry at the State
University of New York at Stony Brook,
built the steel frame for m zero in his
basement during a few weekends, using
a wrench and a hacksaw.
The brothers' mode of existence has
come to be known among mathema-
ticians as the Chudnovsky Problem.
Herbert Robbins eventually decided
that it was time to ask the entire
American mathematics profession why
it could not solve the Chudnovsky
Problem. Robbins is a member of the
National Academy of Sciences, and in
1986 he sent a letter to all of the
mathematicians in the academy:
I fear that unless a decent and honorable
position in the American educational and re-
search system is found for the brothers soon,
a personal and scientific tragedy will take
place for which all American mathematicians
will share responsibility....
I have asked many of my colleagues why
this situation exists, and what can be done to
put an end to what I regard as a national
disgrace. I have never received an answer
that satisfies me.... I am asking you, then,
as one of the leaders of American mathemat-
ics, to tell me what you are prepared to do to
acquaint yourselves with their present cir-
cumstances, and if you are convinced of the
merits of their case, to find a suitable position
somewhere in the country for them as a pair.
There wasn't much of a response.
Robbins says that he received three
written replies to his letter. One, from
a faculty member at a well-known
East Coast university, complained about
David Chudnovsky's personality. He
remarked that "when David learns to
be less overbearing" the brothers might
have better luck. He also did not fully
understand the tone of Rob- .
bins' letter: while he agreed
that some resolution to the
Chudnovsky Problem must
be found, he thought that
Herb Robbins ought to ap-
proach the subject realisti
cally and with more candor.
("More candor? How could I have
been more candid?" Robbins asked.)
Academic administrator. I'm sorry I have
nothing more effective to propose."
An emotional reaction to Robbins'
campaign on behalf of the Chudnovskys
came a bit later from Edwin HewiK,
the mathematician who had helped get
the family out of the Soviet Union, and
one of the few Americans who has
ever worked with Gregory Chudnovsky.
Hewitt wrote to colleagues, "I have
collaborated with many excelIent math-
ematicians . . . but with no one else
have I witnessed an outpouring of
mathematics like that from Gregory.
He simply KNOWS what is true and
what is not." In another letter, Hewitt
wrote:
The Chudnovsky situation is a national
disgrace. Everyone says, "Oh, what a crying
shame" & then suggests that they be placed at
somebody else's institution. No one seems to
want the admittedly burdensome task of car-
ing for the Chudnovsky family. I imagine it
would be a full-time, if not an impossible,
job. We may remember that both Mozart and
Beethoven were disagreeable people, to say
nothing of Gauss.
The brothers would have to be hired
as a pair. Gregory won't take any job
unless David gets one, and vice versa.
Physically and intellectually com-
mingled, like two trees that have grown
together at the root and bole, the broth-
ers claim that they can't be separated
without becoming deadfalls and crash-
ing to the ground. To hire the Chud-
novsky pair, a department would have
to create a joint opening for them.
Gregory can't teach classes in the normal
way, because he is more or less confined to bed.
It would require a small degree of
flexibility in a department to
allow Gregory to concen
trate on research, while David
handled the teaching. The
A problem is that Gregory might
Another letter came from a faculty
member at Princeton University, who
offered to put in a good word with the
National Science Foundation to help
the brothers get their grants, but did
not mention a job at Princeton or
anywhere else. The most thoughtful
response came from a faculty member
at M.I.T., who remarked, "It does
seem odd that they have not been more
sought after." He wondered if in some
part this might be a consequence of
their breadth. "A specialist appears as
a safer investment to a cautious aca-
working with a few brilliant graduate
students&emdash;a privilege that might not go
down well in an American academic
department.
"They are prototypical Russians,"
Robbins said. "They combine a rather
grandiose vision of themselves with an
ability to live on scraps rather than
compromise their principles. These are
people the world is not able to cope
with, and they are not making it any
easier for the world. I don't see that
the world is particularly trying to keep
Gregory Chudnovsky alive. The trag-
edy&emdash;the disgrace, so to speak&emdash;is that
the American scientific and educational
establishment is not benefitting from
the Chudnovskys' assistance. Thirteen
years have gone by since the Chud-
novskys arrived here, and where are
all the graduate students who would
have worked with the brothers? How
many truly great mathematicians have
you ever heard of who couldn't get a
job? I think the Chudnovskys are the
only example in history. This vast
educational system of ours has poured
the Chudnovskys out on the sand, to
waste. Yet Gregory is one of the re-
markable personalities of our time.
When I go up to that apartment and
sit by his bed, I think, My God, when
I was a student at Harvard I was in
contact with people far less interesting
than this. What happens to me in
Gregory's room is like that line in the
Gerard Manley Hopkins poem: 'Mar-
garet, are you grieving/Over Golden-
grove unleaving?' I'm grieving, and I
guess it's me I'm grieving for."
T" WO billion digits of pi? Where
do they keep them?" Samuel
Eilenberg said to me. Eilenberg is a
gifted and distinguished topologist, and
an emeritus professor of mathematics
at Columbia University. He was the
chairman of the department when the
question of hiring the brothers first
became troublesome to Columbia.
"There is an element of fatigue in the
Chudnovsky Problem," he said. "In
the academic world, we have to be
careful who our colleagues are. David
is a pain in the neck. He interrupts
people, and he is not interested in
anything except what concerns him
and his brother. He is a nudnick!
Gregory is certainly unusual, but he is
not great. You can spend all your life
computing digits. What for? You know
in advance that you can't see any
regularity in pi. It's about as interest-
ing as going to the beach and counting
sand. I wouldn't be caught dead doing
that kind of work! Most mathemati-
cians probably feel this way. An im-
portant ingredient in mathematics is
taste. Mathematics is mostly about giving
pleasure. The ultimate criterion of
mathematics is aesthetic, and to calcu-
late the two-billionth digit of pi is to
me abhorrent."
"Abhorrent&emdash;yes, most mathemati-
cians would probably agree with that,"
said Dale Brownawell, a respected
number theorist at Penn State. "Tastes
change, though. If something were to
begin to show up in the digits of pi,
it would boggle everyone's mind."
Brownawell met the Chudnovskys at
the Vienna airport when they escaped
from the Soviet Union. "They didn't
bring much with them, just a pile ol
bags and boxes. David would walk
through a wall to do what is right for
MARCH 2, 1992
his brother. In the situation they are
in, how else can they survive? To see
the Chudnovskys carrying on science
at such a high level with such meagre
support is awe-inspiring."
Richard Askey, a prominent math-
ematician at the University of Wis-
consin at Madison, occasionally flies to
New York to sit at the foot of Gregory
Chudnovsky's bed and learn about
mathematics. "David Chudnovsky is a
very good mathematician," Askey said
to me. "Gregory is one of the few great
mathematicians of our time. Gregory
is so much beKer than I am that it is
impossible for me to say how good he
really is. Is he the best in the world,
or one of the three best? I feel uncom-
fortable evaluating people at that level.
The brothers' pi stuff is just a small
part of their work. They are really
trying to find out what the word 'ran-
dom' means. I've heard some people
say that the brothers are wasting their
time with that machine, but Gregory
Chudnovsky is a very intelligent man,
who has his head screwed on straight,
and I wouldn't begin to question his
priorities. The tragedy is that Gregory
has had hardly any students. If he dies
without having passed on not only his
knowledge but his whole way of think-
ing, then it will be a great tragedy.
Rather than blame Columbia Univer-
sity, I would prefer to say that the
blame lies with all American math-
ematicians. Gregory Chudnovsky is a
national problem."
I" T looks like kvetching," Gregory
said from his bed. "It looks cheap,
and it is cheap. We are here in the
United States by our own choice. I
don't think we were somehow wronged.
I really can't teach. So what does one
want to do about it? Attempts to change
the system are very expensive and
time-consuming and largely a waste of
time. We barely have time to do the
things we want to do."
"To reform the system?" David said,
playing his flashlight across the ceil-
ing. "In this country? Look. Come on.
It's much easier to reform a totalitarian
system."
"Yes, you just make a decree,"
Gregory said. "Anyway, this sort of
talk moves into philosophical ques-
tions. What is life, and where does the
money come from?" He shrugged.
F Toward the end of the summer of
1991, the brothers halted their probe
into pi. They had surveyed pi to two
billion two hundred and sixty million
three hundred and twenty-one thou-
sand three hundred and thirty-six dig-
its. It was a world record, doubling the
record that the Chudnovskys had set in
1989. If the digits were printed in
ordinary type, they would stretch from
New York to Southern California. The
brothers had temporarily ditched their
chief competitor, Yasumasa Kanada&emdash;
a pleasing development when the broth-
ers considered that Kanada had access
to a half-megawatt Hitachi monster
that was supposed to be faster than a
Cray. Kanada reacted gracefully to the
Chudnovskys' achievement, and he told
Science News that he might be able to
get at least a billion and a half digits
of pi if he could obtain enough time
on a Japanese supercomputer.
"You see the advantage to being
truly poor. We had to build our ma-
chine, but now we get to use it, too,"
Gregory said.
The Chudnovskys' machine had
spent its time both calculating pi and
checking the result. The job had taken
about two hundred and fifty hours on
m zero. The machine had spent most
of its time checking the answer, to
make sure each digit was correct, rather
than doing the fundamental computa-
tion of pi.
"We have done our tests for pat-
terns, and there is nothing," Gregory
said. "It would be rather stupid if there
were something in a few billion digits.
There are the usual things. The digit
three is repeated nine times in a row,
and we didn't see that before. Unfor-
tunately, we still don't have enough
computer power to see anything in pi."
Such was their scientific conclusion,
and yet the brothers felt that they may
have noticed something in pi. It hov-
ered out of reach, but it seemed a little
closer now. It was a slight sign of
order&emdash;a possible sign&emdash;and it had to
do with the running average of tht
digits. You can take an average of any
string of digits in pi. It is like getting
a batting average, an average height,
an average weight. The average of the
digits in pi should be 4.5. That's the
average of the decimal digits zero
through nine. The brothers noticed
that the average seems to be slightly
skewed. It stays a little high through
most of the first billion digits, and then
it stays a little low through the next
billion digits. The running average of
pi looks like a tide that rises and
retreats through two billion digits, as
if a distant moon were passing over a
sea of digits, pulling them up and
down. It may or may not be a hint of
a rule in pi. "It's unfortunately not
statistically significant yet," Gregory
said. "It's close to the edge of signifi-
cance." The brothers may have glimpsed
only their desire for order. The tide
that seems to flow through pi may be
nothing but aimless gabble, but what
if it is a wave rippling through pi?
What if the wave begins to show a
weird and complicated pulsation as you
go deeper in pi? You could become
obsessive thinking about things like
this. You might have to build more
machines. "We need a trillion digits,"
David said. A trillion digits printed in
ordinary type would stretch from here
to the moon and back, twice. The
brothers thought that if they didn't get
bored with pi and move on to other
problems they would easily collect a
trillion digits in a few years, with the
L help of increasingly powerful super-
computing equipment. They would orbit
the moon in digits, and head for ALpha
Centauri, and if they lived and their
machines held, perhaps someday they
would begin to see the true nature of pi.
Gregory is lying in bed in the junk
yard, a keyboard on his lap. He offers
to show me a few digits of pi, and taps
at the keys.
On the screen beside his bed, m zero
responds: "Please, give the beginning
of the decimal digit to look."
Gregory types a command, and
suddenly the whole screen fills with
the raw Ludolphian number, moving
like Niagara Falls. We observe pi in
silence for quite a while, until it ends
with:
. . . 18820 54573 01261 27678 17413 87779
66981 15311 24707 34258 41235 99801 92693
52561 92393 53870 24377 10069 16106 22971
02523 30027 49528 06378 64067 12852 77857
42344 28836 88521 72435 85924 57786 36741
32845 66266 96498 68308 59920 06168 63376
85976 35341 52906 04621 44710 52106 99079
33563 54625 71001 37490 77872 43403 57690
01699 82447 20059 93533 82919 46119 87044
02125 12329 11964 10087 41341 42633 88249
48948 31198 27787 03802 08989 05316 75375
43242 20100 43326 74069 33751 86349 40467
52687 79749 68922 29914 46047 47109 31678
05219 48702 00877 32383 87446 91871 49136
90837 88525 51575 35790 83982 20710 59298
41193 81740 92975 31.
"It showed the last digits we've
found," Gregory says. "The last shall
be first."
"Thanks for asking," m zero re-
marks, on the screen.
&emdash;RICHARD PRESTON
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