an article from the New Yorker, March 2, 1992.

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 Ïuvre. 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. ;¥

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

inte¥ers (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 (¥12) 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

approxima¥e 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 migh¥

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

Eddington¥ 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 honor¥s causa from

Bard College, the Moscow Mathemati-

cal Society Prize&emdash;but there is one

defect in their résumé, 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