It is hard to ignore 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 spectra of all kinds, and therefore pi occurs in colors and music, in earthquakes, in surf. Pi is everywhere in superstrings, the hypothetical loops of energy that may vibrate in many dimensions, forming the essence of matter. 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 should be so. Eugene Wigner, the physicist, once said that the miracle in the way the language of mathematics fits the laws of physics “is a wonderful gift which we neither understand nor deserve.” We may not understand or deserve pi, but nature is aware of it, as Captain O. C. Fox learned while he was recovering in a hospital from a wound that he got in the American Civil War. Having nothing 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 on the floor eleven hundred times, Captain Fox was able to derive pi to two places after the decimal point—he got it to the same accuracy that Archimedes did. But Captain Fox’s method was not efficient. Each digit took far more time to get than the previous one. If he had had a thousand years to recover from his wound, he might have gotten pi to perhaps another decimal place. To go deeper into pi, it is necessary to use a machine.
The race toward pi happened in cyberspace, inside supercomputers. In the beginning, computer scientists used pi as an ultimate test of a machine. Pi is to a computer what the East Africa rally is to a car. In 1949, George Reitwiesner, at the Ballistic Research Laboratory, in Maryland, derived pi to 2,037 decimal places with the 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 IBM 7090 mainframe computer, and saw nothing. This was the Shanks-Wrench pi, a milestone. The race continued in a desultory fashion. Eventually, in 1981, Yasumasa Kanada, the head of a team of computer scientists at Tokyo University, used an 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. They noticed 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 places with a smallish workstation, beating Kanada’s team by a million-and-a-half digits. Gosper saw nothing of interest.
The next year, David H. Bailey, at NASA, used a Cray supercomputer and a 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, Kanada and his Tokyo team got 134 million digits of pi. They saw no patterns anywhere. Kanada stayed in to the game. He went past two hundred million digits, and saw further amounts of nothing. Then the Chudnovsky brothers (who had not previously been known to have any interest in calculating pi) suddenly announced that they had obtained 480 million digits of pi—a world record—using supercomputers at two sites in the United States. Kanada’s Tokyo team seemed to be taken by surprise. The emergence of the Chudnovskys as competitors sharpened the Tokyo team’s appetite for more pi. They got on a Hitachi supercomputer and ripped through 536 million digits of pi, beating the Chudnovsky brothers and setting a new world record. They saw nothing new in pi. The brothers responded by smashing through one billion digits. Kanada’s restless boys and their Hitachi were determined not to be beaten, and they soon pushed into slightly more than a billion digits. The Chudnovskys took up the challenge and squeaked past the Japanese team again, having computed pi to 1,130,160,664 decimal places, without finding anything special. It was another world record. At this 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 a question: What is the point of computing pi from New York to Kansas? That question was indeed 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 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 of today. The mere thought of a billion decimals of pi gave some mathematicians a feeling of indefinable horror, and they declared the Chudnoskys’ effort trivial.
I asked Gregory if an impression I had of mathematicians was true, that they spend a certain amount of time declaring one another’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.”
For that final, record-breaking, Hitachi-beating, transbillion-digit push into pi, Gregory did the calculation from his bed in New York, working on the Internet with the Cray supercomputer in Minneapolis, occasionally answering the phone when the system operator called to ask why the Cray had crashed. Gregory also did some of the pi work on a massive IBM dreadnought mainframe at the Thomas J. Watson Research Center, in Yorktown Heights, New York, where he also triggered some dramatic crashes. The calculation of more than a billion digits of pi took half a year. This was because the Chudnovsky brothers could get time on the supercomputers only in bits and pieces, usually during holidays and in the dead of night.
Meanwhile, supercomputer system operators had become leery of Gregory. They worried that he might really toast a $30 million supercomputer. The work of calculating pi was also very expensive for the Chudnovskys. They had to rent time on the Cray. This cost the Chudnovskys $750 an hour. At that rate, a single night of driving the Cray into pi could easily cost the Chudnovskys close to ten thousand dollars. The money came from the National Science Foundation. Eventually the brothers concluded that it would be cheaper to build their own supercomputer in Gregory’s apartment. They could crash their machine all they wanted in privacy at home, while they opened doors in the house of numbers.
When I first met them, the brothers had got an idea that they would compute pi to two billion digits with their new machine. They would try to almost double their old world record and leave the Japanese team and their sleek Hitachi burning in a gulch, as it were. They thought that testing their new supercomputer with a massive amount of pi would put a terrible strain on their machine. If the machine survived, it would prove its worth and power. Provided the machine didn’t strangle on digits, they planned to search the huge resulting string of pi for signs of hidden order. In the end, if what the Chudnovsky brothers ended up seeing in pi was a message from God, the brothers weren’t sure what God was trying to say.
* * *
GREGORY SAID, “Our knowledge of pi was barely in the millions of digits—”
“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—He didn’t want to t
hink about it; it would set back the project by months. He said, “Galileo had to build his telescope—”
“Because he couldn’t afford the Dutch model,” Gregory said.
“And we have to build our machine, because we have—”
“No money,” Gregory said. “When people let us use their supercomputer, it’s always 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 supercomputer 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 interviewer 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 a few years, a Cray will fit in your pocket,” David said.
Supercomputers are evolving incredibly fast. The definition of a supercomputer is simply this: one of the fastest and most powerful computers in the world, for its time. M zero was not the only ultra-powerful silicon engine to gleam in the Chudnovskys’ designs. They had fielded a supercomputer named Little Fermat, which they had designed with Monty Denneau, a supercomputer architect at IBM, and Saed Younis, a graduate student at the Massachusetts Institute of Technology. Little Fermat was seven feet tall. It sat in a lab at MIT, where it considered numbers.
What m zero consisted of was a group of high-speed processors linked by cables (which covered the floor of the room). The cables formed a network among the processors that the Chudnovskys called a web. On a piece of paper, Gregory sketched the layout of the machine. He drew a box and put an X through it, to show the web, or network. Then he attached some processors to the web.
The design of the supercomputer m zero.
Drawing by Richard Preston
The exact design of this web was a secret. “Each processor is connected to all the others,” Gregory said. “It’s like a telephone network—everybody is talking to everybody else.” This made the machine very fast. They planned to have 256 processors. “We will be able to fit them into the apartment,” Gregory said. The brothers wrote the machine’s software in FORTRAN, a programming language that is “a dinosaur from the late fifties,” Gregory said, adding, “There is always new life in this dinosaur.” He said that it was very hard to know what exactly was happening inside the machine when it was running. It seemed to have a life of its own.
The brothers would not disclose the exact shape of the network inside their machine. The design contained several new discoveries in number theory, which the Chudnovskys hadn’t published. They claimed that they needed to protect their competitive edge in the worldwide race to develop ultrafast computers. “Anyone with a hundred million dollars and brains could be our competitor,” David said dryly.
One day, I called Paul Messina, a Caltech scientist and leading supercomputer designer, 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 true supercomputer out of mail-order parts for around seventy thousand dollars, he flatly believed it. “It can be done, definitely,” Messina said. “Of course, that’s 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, was using a Hitachi supercomputer that burned close to half a million watts when it was running—half a megawatt, practically enough power to drive an electric furnace in a steel mill. The Chudnovskys were particularly hoping to show that their machine was as powerful as the Hitachi.
“Pi is the best stress test for a computer,” David said.
“We also want to find out what makes pi different from other numbers. Eh, it’s a business,” Gregory said.
David pulled his Mini Maglite 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—yellow, green, orange, violet, and blue. It was the first eight million digits of pi, mapped as a fractal landscape by an IBM supercomputer 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, possibly, a hidden structure. “I see something systematic in this landscape,” he said. “It may be just an attempt by the brain to translate some random visual pattern into order.” But as he gazed into the nature beyond nature, 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 seem to be, perhaps, slight variations from randomness in this landscape. There are, perhaps, 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 manner 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). Yet he couldn’t really express in words what he thought he saw. 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 underwater,” Gregory said. “You are in the mud, and everything looks the same. You need a flashlight to see anything. Our computer is a flashlight.”
David said, “Gregory—I think, really—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 is a service economy. Of course, you know what that means—the service is terrible.”
“We collect price quotes by fax,” Gregory said.
“It’s a horrible thing. Window-shopping in computerland. We can’t buy everything—”
“Because everything won’t exist,” Gregory broke in, and cackled.
“We only want to build a machine to compute a few transcendental numbers—”
“Because we are not licensed for transcendental meditation,” Gregory said.
“Look, we are getting nutty,” David said.
“We are not the only ones,” Gregory 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 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 Encyclopaedia 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 rifling through boxes. Gregory got up and began fishing through bookshelves.
“Please sit down, Gregory,” David said. Finally the brothers confessed that they had temporarily lost their billion digits of 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 drive, studded with chips. He handed me the object. It hummed gently. “There’s pi stored on it. Y
ou are holding some pi in your hand.”
* * *
MONTHS PASSED before I visited the Chudnovskys again. They had been tinkering with their machine and getting it ready to go after 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 in there. The brothers were disappointed in the quality of the pictures, and they persuaded the doctors to give them the CAT scan data. They processed it in m zero and got detailed color images of Gregory’s insides, far more detailed than any image from a CAT scanner. Gregory wrote the imaging software; it took him a few weeks. “There’s a lot of interesting mathematics in the problem of making an image of a body,” he remarked. It delayed the brothers’ probe into the Ludolphian number.
Spring arrived, and Federal Express was active at the Chudnovskys’ building, while the superintendent remained in the dark about what was going on. The brothers began to calculate pi. Slowly at first, then faster and faster. 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 down 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 later David couldn’t understand how his brother hadn’t killed himself saving her. An ambulance rushed her to St. Luke’s-Roosevelt 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 developed a bleeding ulcer. “Look, it’s not a problem,” he said to me. After Malka Benjaminovna had been moved out of intensive care, Gregory rented a laptop computer, plugged it into a telephone line in her hospital room, and talked to m zero over the Internet, driving his supercomputer toward pi and watching his mother’s blood pressure at the same time.
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