Volfovich Chudnovsky recently build a super computer in his appartment
from mail-order parts. Gregory Chudnovsky is a number theorist. His appartment
is situated near the top floor of a run-down building on the West Side
of Manhattan, in a neighborhood near Columbia University. Not long ago,
a human corpse was found dumped at the end of the block. The world's most
powerful supercomputers include 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 Touchtone delta, and
Gregory Chudnovsky's apartment. The apartment seems to be a kind of container
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 machine m zero. It occupies the former living room of Grogory'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 not as fast as the latest of the Y-MP machines, the C90, an advanced 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 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 toppling over. He has severe myasthenia gravis, an auto-immune disorder of muscles. The symptoms, in his case, are muscular weakness and difficulty in breathing. "I have to lie in bed most of the time", Gregory once told me. His condition does not seem to be getting better, and does not seem to be getting worse. He developed the desease 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 room is filled with paper; 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 the top of David's head, and he has heavy-lidded deep-blue eyes. He always wears starched white shirt and, usually, a gray silk necktie in a foulard print. His tie rests on a bulging stomach.
The Chudnovskian supercomputer, m zero, burns two thousand watts of power, and it runs day and night. The brothers don't dare shut it down; if it 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 Fahrenheit in summer. The brothers keep apartment's lights turned off as much as possible. If they switched on too many lights while m zero was running, 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 inside 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, containing parts that have bombed, along with letters from the brothers demanding 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 brothers 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 mathematician-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 relationship with Columbia University: Columbia calls them senior research scientists in the Department of Mathematics, 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 Chudnovsky, 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 brother's supercomputing complex in Gregory and Christine's appartment. Malka Benjaminovna Chudnovsky, 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 brother's 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 Kiev Architectual Institute, and he specialized in the structual 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 engineering. The bookshelves extend into every 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 apartmentful of thoughts about numbers holds hardly any thoughts at all, even with a supercomputer on the premises to advance their 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 typographically as "m0". The "zero" stands for success. It implies a dark history of failure-three duds (in Gregory's apartment) 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 supercomputer architecture, because the design offers succulent possibilities for speed in solving problems. In a parallel machine, anywhere from half a dozen to thousands of processors work simultaneously on a problem, whereas in a so-called serial machine- a normal computer-the problwem 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 Chednovskys' problems.
The brothers' mail-order supercomputer makes their lives more convenient: m zero performs inhumanly difficult algebra, finding roots of gigantic systems of equations, and it has constructed colored images of the interior 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, is the most famous rati in mathematics, and is one of the most ancient numbers known to humanity. Pi is approximately 3.14 - 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:
This is known as the decimal expansion of pi. It is a bloody mess. No apparent pattern emerges in the succession of digits. The digits of pi marches to infinity in a predestined yet unfathomable code: they do not repeat periodically, seeming to pop up by blind chance, lacking any perceivable order, rule, reason, or design - "random" integers, ad infinitum. 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 out finite universe to discover the system in the digits of pi. But for the present, if you want 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 keyboard 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 estimate the number of times that the diameter of a circle goes around the periphery, and Gregory sits back on his pillows and waits, watching messages from the Cray flow across his display screen. He eats his 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 hidden properties of numbers. Meanwhile, the Cray is reaching toward pui 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 running deep toward pi. Unfortunately, since the exact ratio of the circle's circumference to its diameter dwells at infinity, 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 exclaims. It seems that the Cray has hung up the phone, and may have crashed. Once again, pi has demonstrated its ability to give a supercomputer a heart attack.
"MYASTHENIA GRAVIS is a funny thing", Gregory Chudnovsky said one day from his bed in his junk yard. "In a sense, I'm very lucky, because I'm alive, and I'm alive after so many years". He has a resonant voice and a Russian accent. "There are 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 are 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 sweatpants, and a pair of handmade socks. I had never seen socks like Gregory's. They were two-tone socks, wrincled and floppy, hand-sewn from pieces of dark-blue and pale-blue cloth, and they look 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 has wedged the paper, sheet by sheet, into manila folders, until the folders had grown as fast 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 paperslide, 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 towers 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 breakes open through the portals of mathematics into vistas beyomd time or decay.
The race toward pi
happens in cyberspace, inside supercomputers. In 1949, Geoerge Reitwiesner,
at the Ballistic Research Laboratory, in Maryland, derived pi to two thousand
and thirty-seven decimal places with ENIAC, the first general-purpose electronic
digital computer. Working at the same laboratory, John von Neumann (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 nothing. The race
continued desultorily, through hundreds of thousands of digits, until 1981,
when Yasumasa Kanada, the head of a team of computer scientists 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 mathematician and distinguished hacker
employed at Symbolics, Inc., in Sunnyvale, California, computed pi to seventeen
and a half million decimal places with a Symbolics workstation, beating
Kanada's team by a million digits. Gosper saw nothing of interest.
The next year, David H. Bailey, at the National Aeronautics and Space Administration, used a Cray 2 supercomputer and formula discovered by two brothers, Jonathan and Peter Borwein, to scoop twenty-nine million digits of pi. Bailey found nothing 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 supercomputer. They saw nothing of interest. In 1988, Kanada kept going, past two hundred million digits, and saw further amounts of nothing. Then, in the spring of 1989, the Chudnovsky brothers (who had not previously been known to have any interest in calculating pi) suddenly announced that they had obtained four hundred and eighty million digits of pi - a world record - using supercomputers at two sites in the United States, and had seen nothing. Kanada and his team were a little surprised to learn of unknown competition operating in American cyberspace, and they got on a Hitachi supercomputer and ripped through five hundred and thirty-six million digits, beating the Chudnovskys, setting a new world record, and seeing nothing. The brothers 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 sity-four decimal places, without finding anything special. It was another world record. At that pouint, 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 Kansa ? The question has indeed been asked among mathematicians, since an expansion of pi tp only forty-seven decimal places would be sufficiently precise to inscribe a circle around the visible universe that doesn't deviate from perfect circularity by more than the distance across a single proton. A billion decimals of pi go so far beyond that kind of precision, into such a lunacy of exactitude, that physicists will never need to use the quantity in any experiment - at least, not for any physics we know today - and the thought of a billion decimals of pi oppresses even some mathematicians, who dexclare the Chudnovskys' effort trivial. I once asked Gregory if a certain impression 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 Minnesota Supercomputer Center, in Minneapolis, and on an I.B.M. 3090-VF supercomputer at the I.B.M. Thomas J. Watson Research Center, in New Yorktown heights, New York. The calculation triggered some dramatic crashes, and took half a year, because the brothers could get time on the supercomputer only in bits and pieces, usually during holidays and in the dead of night. It was also quite expensive - the use of the Cray cost them seven hundred and fifty dollars an hour, and the money came from the National Science Foundation. 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 - 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 begin their two-billion-digits expedition
into pi, I rang the bell of Gregory Chudnovsky's apartment, and David answers
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 aprtment 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 mullioned 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 bedroom 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 Frwench 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 - a tattered and patched lamb's-wool sweater, a starched white shirt, blue sweatpants, and 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 - cheap P.C. clones, knocked wide open, like cracked walnuts, their meat spilling all over the place. The brothers crammed special logic boards inside the personal computers. Red lights on the board blinked. The floor was a quagmire of cables. (...)