In the eighteenth century, Leonhard Euler, mathematician to Catherine the Great, empress of Russia, began calling it
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
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 think 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
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.
