The Mountains of Pi

WHEN HE WAS THIRTY-SIX, Gregory Volfovich Chudnovsky began to build a supercomputer in his apartment from mail-order parts. Gregory Chudnovsky was a number theorist, a mathematician who studies numbers, and he felt that he needed a supercomputer to do it. His apartment was situated near the top floor of a run-down building at the corner of 120th Street and Amsterdam Avenue, on the West Side of Manhattan. Around the time he decided to build the supercomputer, a corpse was found stuffed into a garbage can at the end of his block. The project officially took two years, though in reality it never ended. At the time he began the project, the world’s most powerful supercomputers included the Cray series, the Thinking Machines arrays, the Hitachi line of supercomputers, the nCube, the Fujitsu machine, the Kendall Square Research machine, the NEC supercomputer, the Touchstone Delta, and Gregory Chudnovsky’s apartment. The apartment was a kind of container for the supercomputer at least as much as it was a container for people.

Gregory Chudnovsky’s partner in the design and construction of the supercomputer was his older brother, David Volfovich Chudnovsky. (“Volfovich” means “Son of Wolf.”) David was also a number theorist, and he lived five blocks away from Gregory. The Chudnovsky brothers were reluctant to give a name to their machine. To them, it was a household appliance that could help with their investigation of numbers. You didn’t give a name to your toaster oven, so why would you give a name to your supercomputer?

When I pressed the Chudnovsky brothers to give me some sort of a name for it, they shrugged and said it was nothing.

“I don’t want to call it nothing,” I said to the brothers.

“Why not?” David answered. However, he said, as a convenience I could refer to it as “m zero.”

At any rate, the “zero” in the machine’s name hinted at a history of failures—three previous duds in Gregory’s apartment, three homemade supercomputers that hadn’t worked. The brothers referred to these machines as negative three, negative two, and negative one. The brothers broke them up for scrap, and they got on the telephone and ordered more parts.

Whatever the supercomputer was, it filled the former living room of Gregory’s apartment, and its tentacles reached into other rooms. The brothers claimed that m zero was a “true, general-purpose supercomputer” and that it would turn out to be as fast and powerful as a Cray Y-MP. A Cray Y-MP had a sticker price of more than thirty million dollars. A Cray was a black cylinder seven feet tall, and it was cooled by liquid freon. The brothers spent around seventy thousand dollars on parts for their supercomputer, and much of the money came out of their wives’ pockets. Seventy thousand dollars was a little more than two-tenths of one percent of the cost of a Cray.

It was safe to say that Gregory Chudnovsky was one of the world’s leading architects of supercomputers. He had an ability to see the design of a supercomputer in his mind’s eye. He liked to imagine supercomputers that might never be built, like an architect who dreams of towers and cities in a splendid future. M zero was incredibly fast. Gregory called it a relativistic machine, because he had woven the design of the machine around Einstein’s theory of special relativity. M zero’s network of processors shuttled numbers around it so fast that the different parts of the machine operated in slightly different space-times.

Gregory Chudnovsky had a spare frame and a bony, handsome face. He had a long beard, streaked with gray, and dark, unruly hair, a wide forehead, and wide-spaced brown eyes. He walked in a slow, dragging shuffle, leaning on a bentwood cane, while his brother, David, typically held him under one arm, to prevent him from toppling over. He had myasthenia gravis, an autoimmune disorder of the muscles. The symptoms, in his case, were muscular weakness and difficulty in breathing. “I have to lie in bed most of the time,” Gregory told me. His condition seemed to be getting gradually worse. He developed the disease when he was twelve years old, in the city of Kiev, Ukraine, where he and David grew up. In those days, Ukraine was part of the old Soviet Union. Now Gregory spent his days sitting or lying in a bed heaped with pillows, in his bedroom down the hall from the room that housed the supercomputer. Gregory’s bedroom was filled with paper. It contained, by my estimate and the calculation of a New Yorker fact-checker, at least one ton of paper. He called his bedroom his junkyard. The room faced east. It would have been full of sunlight in the morning if he’d ever raised the shades, but he kept them lowered, because light hurt his eyes.

You almost never met one of the Chudnovsky brothers without the other. You usually found the brothers conjoined, like Siamese twins, David holding Gregory by the arm or under the armpits, speaking to him tenderly, cautioning him to be careful not to fall or hurt himself. They worked together so closely that they claimed to be a single mathematician who by chance happened to occupy two human bodies. They completed each other’s sentences and interrupted each other, but they didn’t look completely alike. While Gregory was thin and bearded, David was portly, with a plump, clean-shaven face. David’s manner was refined and aristocratic. Black-and-gray curly hair grew thickly on top of his head, and he had heavy-lidded pale blue eyes, which had a melancholy look. He always wore a starched white shirt and, usually, a muted silk necktie. His tie rested on a bulging stomach.

The Chudnovskian supercomputer, m zero, burned two thousand watts of power. It ran day and night. The brothers didn’t dare shut it down; they were afraid it would die if they did. At least twenty-five fans blew air through the machine to keep it cool; otherwise something might melt. Waste heat permeated Gregory’s apartment, and the room that contained the supercomputer climbed to more than a hundred degrees Fahrenheit in the summer. The brothers kept the apartment’s lights turned off as much as possible. If they switched on too many lights while m zero was running, they feared they might start an electrical fire. Gregory couldn’t breathe city air without developing lung trouble, so he kept the apartment’s windows closed all the time. He had air conditioners running in them during the summer, but that didn’t seem to reduce the heat. As the temperature climbed on hot days, the inside of the apartment smelled of cooking circuit boards, a sign that m zero was not well. A steady stream of boxes arrived by Federal Express, and an opposing stream of boxes flowed back to mail-order houses, containing parts that had overheated, failed, bombed, or acted strange, along with letters from the brothers demanding an exchange or their money back. The building superintendent didn’t know that the Chudnovsky brothers were using a supercomputer in Gregory’s apartment. The brothers were afraid he would find out.

The Chudnovskys, between them, had published more than a hundred and fifty papers and twelve books, mostly on the subject of number theory or mathematical physics. They 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, where David fell in love with a French diplomat named Nicole Lannegrace, and they were married. The Chudnovsky brothers, along with their parents and Nicole Lannegrace, immigrated to the United States and settled in New York, where Nicole became a diplomat with the United Nations. The brothers eventually became American citizens.

The brothers enjoyed an official relationship with Columbia University: Columbia called them senior research scientists in the Department of Mathematics, but they didn’t have tenure, they didn’t teach students, and they didn’t attend faculty meetings. They were lone inventors, operating out of Gregory’s apartment. Gregory’s wife, Christine Pardo Chudnovsky, was an attorney with a midtown law firm. She had been an undergraduate at Columbia University when Gregory arrived there, and she’d fallen in love with him at first sight. Nicole Lannegrace’s salary as a U.N. diplomat and Christine’s as a lawyer helped cover much of the funding needs of the brothers’ supercomputing complex in Gregory and Christine’s apartment. Gregory and David’s mother, Malka Benjaminovna Chudnovsky, a retired engineer, was living with Gregory and Christine and was in poor health. David spent 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, it attracted the attention of the KGB. The brothers happened to be friends with the physicist Andrei Sakharov, a key inventor of the Soviet hydrogen bomb, who had later become a human-rights activist and a proponent of nuclear disarmament, getting himself into serious trouble with the Kremlin. The Chudnovskys’ association with Sakharov, as well as the fact that they were Jewish and mathematical, attracted at least a dozen KGB agents to their case. The brothers’ father, Volf Grigorevich Chudnovsky (“Wolf, Son of Gregory”) was severely beaten by KGB agents in 1977. Volf died in 1985, in New York City, of what the brothers believed were lingering effects of his torture. 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. Not long before he died, he constructed in Gregory’s apartment a labyrinth of bookshelves, his last work of civil engineering. Volf’s bookshelves extended into every corner of the apartment, and they had become packed with literature and computer books and books on history and art and, above all, books and papers on the subject of numbers. Since almost all numbers run to infinity (in digits) and are totally unexplored, an apartment full of writings on numbers holds hardly any knowledge about numbers at all. Numbers, and the patterns of relationships among them, are powerful, deep, and mysterious. It is not at all clear that the human mind evolved in such a way that it is very much able to understand numbers. But it helps to have a supercomputer on the premises to advance the work.

* * *

ONE DAY, I called the Columbia University math department trying to find out how to make contact with the Chudnovskys. I had read a short news item about them but could learn very little that was definite. They were reportedly living somewhere in New York City. However, they did not seem to be listed in the Manhattan telephone book, and they didn’t have an unlisted telephone number, either. (I learned later that they actually were listed in the Manhattan telephone book but under a nonexistent name.) “The Chudnovskys?” the person who answered the phone at Columbia said. “I have no idea where they are. We haven’t seen them around here in a long time. I have an old phone number for them. Somebody said it doesn’t work anymore.”

I dialed the number and got a fax tone. I handwrote a message on a piece of paper and faxed it, asking if this number belonged to the Chudnovskys and, if so, would they be able to meet with me? There was no reply. Weeks passed. I gave up. But then one day my phone rang; it was David Chudnovsky. “Look, you are welcome,” he said. He had a genteel-sounding voice with a Russian accent.

On a cold winter day soon afterward, I rang the bell of Gregory’s apartment on 120th Street. I was carrying a little notebook and a mechanical pencil in my shirt pocket. David answered the door. He pulled the door open a few inches, and then it stopped. It was jammed against an empty cardboard box and a mass of hanging coats. He nudged the box out of the way with his foot. “Don’t worry,” he said. “Nothing unpleasant will happen to you here. We will not turn you into digits.” A Mini Maglite flashlight protruded from his shirt pocket.

We were standing in a long, dark hallway. The place was a swamp of heat. My face and armpits began to drip with sweat. The lights were off, and it was hard to see anything. This was the reason for David’s flashlight. The hall was lined on both sides with bookshelves supporting huge stacks of paper and books. The shelves took up most of the space, leaving a passage about two feet wide running down the length of the hallway. At the end of the hallway was a French door. Its mullioned glass panes were covered with translucent paper. The panes glowed.

We went along the hallway. We passed a bathroom and a bedroom door, which was closed. The bedroom belonged to Malka Benjaminovna Chudnovsky. We passed a sort of cave containing vast amounts of paper. This was Gregory’s bedroom, his junkyard. We passed a small kitchen, our feet rolling on computer cables. David opened the French door, and we entered the living room. This was the chamber of the supercomputer. A bare lightbulb burned in a ceiling fixture. The room contained seven display screens, two of which were filled with numbers; the other screens were turned off. The windows were closed and the shades were drawn. Gregory Chudnovsky sat on a chair facing the lit-up screens. He wore a tattered and patched lamb’s wool sweater, a starched white shirt, blue sweatpants, 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. “Right now, our goal is to compute pi,” he said. “For that we have to build our own computer.” He had a resonant voice and a Russian accent.

The Chudnovsky Mathematician: Gregory and David Chudnovsky in Gregory’s New York City apartment, 1992.
Irena Roman

“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-open shells of personal computers—cheap PC clones, knocked wide open like cracked walnuts, their meat spilling all over the place. The brothers had crammed superfast logic boards inside the PCs. 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 lettering on their spines), and screwdrivers, and data-storage tapes, and software manuals by the cubic yard, and stalagmites of obscure trade magazines, and a twenty-thousand-dollar engineering computer that they no longer used. “We use it as a place to stack paper,” Gregory explained. From an oval photograph on the wall, the face of Volf Chudnovsky, their late father, 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. Various disk drives were scattered around the room. The drives were humming, and there was a continuous whir of fans. A strong warmth emanated from the equipment, as if a steam radiator were going in the room. The brothers were heating the apartment with silicon chips.

* * *

“MYASTHENIA GRAVIS is a funny thing,” Gregory Chudnovsky said one day from his bed in his bedroom, the junkyard. “In a sense, I’m very lucky, because I’m alive, and I’m alive after so many years. 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, and the room was stiflingly warm. He lay against a heap of pillows with his legs folded under him. His bed was surrounded by freestanding bookshelves packed and piled with ramparts of stacked paper. That day, he wore the same tattered wool sweater, a starched white shirt, blue sweatpants, and another 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 bookshelves along the wall, too, 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 was also stacked chest-high to chin-high on five chairs (three of them in a row beside his bed). It was heaped on top of and filled two steamer trunks that sat beneath the window, and the paper had accumulated in a sort of unstable-looking lava flow on a small folding cocktail table. I moved carefully around the room, fearful of triggering a paperslide, and I sat down on the room’s one empty chair, facing the foot of Gregory’s bed, my knees touching the blanket. The paper surrounded his bed like the walls of a fortress, and his bed sat at the center of the keep. I sensed a profound happiness in Gregory Chudnovsky. His happiness, it occurred to me later, sprang from the delicious melancholy of a life spent largely in bed while he explored a more perfect world that opened through the portals of mathematics into vistas beyond time or decay.

“The system of this paper is archaeological,” he said. “By looking at a slice, I know the date. This slice is from five years ago. Over here is some paper from four years ago. What you see in this room is our working papers, as well as the papers we used as references. Some of the references we pull out once in a while to look at, and then we leave them in another pile. Once, we had to make a Xerox copy of the same book three times, and we put it in three different piles, so we could be sure to find it when we needed it. There are books in there by Kipling and Macaulay. Eh, this place is a mess. Actually, when we want to find a book it’s easier to go to the library.”

Much of the paper consisted of legal pads covered with Gregory’s handwriting. His handwriting was dense and careful, a flawless minuscule written with a felt-tipped pen. The writing contained a mixture of theorems, calculations, proofs, and conjectures concerning numbers. He used a felt-tip pen because he didn’t have enough strength in his hand to press a pencil on paper. Mathematicians who had visited Gregory Chudnovsky’s bedroom had come away dizzy, wondering what secrets the scriptorium might hold. He cautiously referred to the steamer trunks beneath the window as valises. They were filled to the lids with compressed paper. When Gregory and David flew to Europe to speak at conferences on the subject of numbers, they took both “valises” with them, in case they needed to refer to a proof or a theorem. Their baggage particularly annoyed Belgian officials. “The Belgians were always fining us for being overweight,” Gregory said.

The brothers’ mail-order supercomputer made their lives more convenient. It performed inhumanly difficult algebra, finding roots of gigantic systems of equations, and it constructed colored images of the interior of Gregory Chudnovsky’s body. They used the supercomputer to analyze and predict fluctuations in the stock market. They had been working with a well-known Wall Street investor named John Mulheren, helping him get a profitable edge in computerized trades on the stock market. One day I called John Mulheren to find out what the brothers had been doing for him. “Gregory and David have certainly made us money,” Mulheren said, but he wouldn’t give any details on what the brothers had done. Mulheren had been paying the Chudnovskys out of his trading profits; they used the money to help fund their research into numbers. To them, numbers were more beautiful, more nearly perfect, possibly more complicated, and arguably more real than anything in the world of physical matter.

* * *

THE NUMBER PI, or π, is the most famous ratio in mathematics. It is also one of the most ancient numbers known to humanity. Nobody knows when pi first came to the awareness of the human species. Pi may very well have been known to the builders of Stonehenge, around 2,600 B.C.E. Certainly it was known to the ancient Egyptians. Pi is approximately 3.14—it is the number of times that a circle’s diameter will fit around a circle. On the following page is a circle with its diameter.

Landscape with a circle and its diameter. This drawing shows a rough visual approximation of pi.
Drawing by Richard Preston

Pi is an exact number; there is only one pi. Even so, pi cannot be expressed exactly using any finite string of digits. If you try to calculate pi exactly, you get a chain of random-looking digits that never ends. Pi goes on forever, and can’t be calculated to perfect precision: 3.1415926535897932384626433832795028841971693993751…. This is known as the decimal expansion of pi. It is a bloody mess. If you try to express pi in another way, using an algebraic equation rather than digits, the equation goes on forever. There is no way to show pi using digits or an equation that doesn’t get lost in the sands of infinity. Pi can’t be shown completely or exactly in any finite form of mathematical representation. There is only one way to show pi exactly, and that is with a symbol. See the illustration on the following page for a symbol for pi.

The pizza pi I baked and drew, here, is as good a symbol for pi as any other. (It tasted good, too.) The digits of pi march to infinity in a predestined yet unfathomable code. When you calculate pi, its digits appear, one by one, endlessly, while no apparent pattern emerges in the succession of digits. They never repeat periodically. They seem 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 caught a glimpse of the pattern by staring at the digits. There is certainly a design in pi, no doubt about it. It is also almost certain that the human mind is not equipped to see that design. Among mathematicians, there is a feeling that it may never be possible for an inhabitant of our 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 sea of pi.

Pi.
Drawing by Richard Preston

Before the Chudnovsky brothers built m zero, Gregory had to derive pi over the Internet while lying in bed. It was inconvenient. The work typically went like this:

Tapping at a small wireless keyboard, which he places on the blankets of his bed, he stares at a computer display screen on one of the bookshelves beside his bed.

The keyboard and screen are connected through cyberspace into the heart of a Cray supercomputer at the Minnesota Supercomputer Center, in Minneapolis. He calls up the Cray through the Internet. When the Cray answers, he sends into the Cray a little software program that he has written. This program—just a few lines of code—tells the supercomputer to start making an approximation of pi. The job begins to run. The Cray starts trying to estimate the number of times the diameter of a circle goes around the periphery.

While this is happening, Gregory sits back on his pillows and waits. He watches messages from the Cray flow across his display screen. The supercomputer is estimating pi. He gets hungry and wanders into the dining room to eat dinner with his wife and his mother. An hour or so later, back in bed, he takes up a legal pad and a red felt-tip pen and plays around with number theory, trying to discover hidden properties of numbers. All the while, the Cray in Minneapolis has been trying to get closer to pi at a rate of a hundred million operations per second. Midnight arrives. Gregory dozes beside his computer screen. Once in a while, he taps on the keys, asking the Cray how things are going. The Cray replies that the job is still active. The night passes and dawn comes near, and the Cray is still 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’s going on?” Gregory exclaims.

Moments later, his telephone rings. It’s a guy in Minneapolis who’s working the night shift as the system operator of the Cray. He’s furious. “What the hell did you do? You’ve crashed the Cray! We’re down!”

Once again, pi has demonstrated its ability to give the most powerful computers a heart attack.

* * *

PI WAS BY NO MEANS the only unexplored number in the Chudnovskys’ inventory, but it was one that interested them. They wondered whether the digits contained a hidden rule, an as yet unseen architecture, close to the mind of God. A subtle and fantastic order might appear in the digits of pi way out there somewhere; no one knew. No one had ever proved, for example, that pi did not turn into a string of nines and zeros, spattered in some peculiar arrangement. It could be any sort of arrangement, just so long as it didn’t repeat periodically; for it has been proven that pi never repeats periodically. Pi could, however, conceivably start doing something like this: 122333444455555666666…. That is, the digits might suddenly shift into a strong pattern. Such a pattern is very regular, but it doesn’t repeat periodically. (Mathematicians felt it was very unlikely that pi would ever become obviously regular in some way, but no one had been able to prove that it didn’t.)

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 might ramble forever in a hideous cacophony, which was a kind of absolute perfection to a mathematician like Gregory Chudnovsky. Pi looked “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—it’s just the circumference to the diameter—but the complexity of the sequence 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 corner of the bedroom, his head and shoulders visible above towers of paper.

Mathematicians call pi a transcendental number. In simple terms, a transcendental number is a number that exists but can’t be expressed in any finite series of finite operations.[2] For example, if you try to express pi as the solution to an algebraic equation made up of terms that have integer coefficients in them, 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 don’t repeat periodically, as do the digits of a rational 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 display it in its totality.

It turns out that almost all numbers are transcendental, yet only a tiny handful of them have ever actually been discovered by humans. In other words, humans don’t know anything about almost all numbers. There are certainly vast classes and categories of transcendental numbers that have never even been conjectured by humans—we can’t even imagine them. In fact, it’s very difficult even to prove that a number is transcendental. For a while, mathematicians strongly suspected that pi was a transcendental number, but they couldn’t prove it. Eventually, in 1882, a German mathematician named Ferdinand von Lindemann proved the transcendence of pi. He proved, in effect, that pi can’t be written on any piece of paper, no matter how big: a piece of paper as big as the universe would not even begin to be large enough to hold the tiniest droplet of pi. In a manner of speaking, pi is undescribable and cannot be found.

The earliest known reference to pi in human history occurs in a Middle Kingdom papyrus scroll, written around 1650 B.C.E. by a scribe named Ahmes. He titled his scroll “The Entrance into the Knowledge of All Existing Things.” He led his readers through various mathematical problems and solutions, and toward the end of the scroll he found the area of a circle, using a rough sort of pi.

Around 200 B.C.E., Archimedes of Syracuse found that pi is somewhere between 310/71 and 31/7. That’s about 3.14. (The Greeks didn’t use decimals.) 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 down the suspected location of pi to one part in a hundred. After that, knowledge of pi bogged down. Finally, in the seventeenth century, a German mathematician named Ludolph van Ceulen approximated pi to thirty-five decimal places, or one part in a hundred million billion billion billion—a calculation that took Ludolph most of his life to accomplish. It gave him such satisfaction that he had the thirty-five digits of pi engraved on his tombstone, which ended up being installed in a special graveyard for professors in St. Peter’s Church in Leiden, in the Netherlands. Ludolph was so admired for his digits that pi came to be called the Ludolphian number. But then his tombstone vanished from the graveyard, and some people think it was turned into a sidewalk slab. If so, somewhere in Leiden people are probably walking over Ludolph’s digits. The Germans still call pi the Ludolphian number.

In the eighteenth century, Leonhard Euler, mathematician to Catherine the Great, empress of Russia, began calling it p or c. The first person to use the Greek letter π was William Jones, an English mathematician, who coined it in 1706. Jones probably meant π to stand for “periphery.”

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 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 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. You 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.

Malka Benjaminovna improved slowly. When she got home from the hospital, the brothers settled her back 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 summer. 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, my dear fellow, if you want to be boiled you are welcome to come in.” He led me down the dark hallway. Malka Benjaminovna was asleep in her bedroom, and the nurse was sitting beside her. Her room was lined with her late husband Volf’s bookshelves, and they were packed with paper. 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 junkyard.

“Then we could add goldfish,” David said.

“That would solve all our problems.”

“We are not good plumbers, Gregory. 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 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 immediately began to pour down my body. “I don’t like to go into this room,” Gregory said. The steel frame in the center of the room—the heart of m zero—seemed to have acquired more guts, and red lights blinked inside it. I could hear disk drives murmuring. The drives were copying and recopying huge segments of transcendental numbers, to check the digits for perfect accuracy. “If the machine makes an error in a single digit of pi, then every digit after that is nonsense. What comes out is not pi at all, it’s just some random number.” Thus they had to keep checking and rechecking the digits to make sure they were exactly pi to the last digit.

Gregory knelt on the floor, facing the steel frame.

David opened a cardboard box and removed an electronic board. He began to fit it into m zero. I noticed that his hands were marked with small cuts, which he had got from reaching into the machine.

“David, could you give me the flashlight?” Gregory asked.

David pulled the Mini Maglite from his shirt pocket and handed it to Gregory. The brothers knelt beside each other, Gregory shining the flashlight into the supercomputer. David reached inside with his fingers and palpated a logic board.

“Don’t!” Gregory said. “Okay, 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 meat thermometer between two circuit boards inside m zero, in order to look for hot spots. The thermometer’s dial was marked “Beef Rare—Ham—Beef Med—Pork.”

“You want to keep the machine below ‘Pork,’” Gregory remarked.

He lifted a keyboard out of a steel frame and typed something on it. Numbers filled a display 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 bookshelf 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 apartment is full of empty boxes. Gregory, I wonder if you are tired.”

“If I stand up now, I will fall down,” Gregory said. “Therefore, I will sit in 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 controller now seems to work.”

“We need to buy a new one anyway,” David said.

“Try Nevada.”

David dialed a computer-parts wholesaler in Nevada called Searchlight Compugear. He said loudly, in a Russian accent, “Hi, Searchlight. I need a fifteen-forty controller…. No! No! No! I don’t need anything else! Just a naked unit! How much you charge? What? 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 FedEx? Come on! What about afternoon delivery?…Twenty-nine dollars before three P.M.? Relax. What is your name?…Bob. Fine. Okay. So, Bob, it is two hundred and fifty-seven dollars plus twenty-nine dollars for FedEx?”

“Twenty-nine dollars for Fed Ex!” Gregory burst out. “It should be fifteen.” He pulled a second keyboard out of the frame and tapped the keys. Another display screen came alive and filled with numbers.

“Tell me this,” David said to Bob in Nevada. “Do you have a 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 usually it doesn’t.”

“Question Number Two: The FedEx 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 A.K.,” he said. “Hi, A.K., this is David Chudnovsky calling from New York. A.K., I need another controller, like the one you sent. Can you send it today FedEx?…How much you charge?…Naked! I want a naked unit! Not in a shoe box, 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. 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 extremely 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 brothers.” She was six years younger than Gregory; she had been an undergraduate at Barnard College when she first met him. “I fell in love with Gregory immediately. His illness came with the package.” She remained in love with him, even if at times they fought over the heaps of paper. (“I don’t have room to put my things down anywhere,” she told 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 dreamed that she was dancing from room to room through an empty apartment that had parquet floors.

David brought his mother out of her bedroom, settled her at the table, and kissed her on the cheek. Malka Benjaminovna seemed frail but alert. She was a small, white-haired woman with a fresh face and clear blue eyes who spoke limited English. A mathematician once described Malka Benjaminovna as the glue that held the Chudnovsky family together. She’d been an engineer during the Second World War, when she designed buildings, laboratories, and proving grounds for testing the Katyusha rocket. Later, she taught engineering at schools around Kiev. Smiling, she handed me plates of roast chicken, kasha, pickles, cream cheese, brown bread, and little wedges of 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 we finished lunch, and were 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 machine clicked, the red lights flashed, the air conditioner hummed, and you could hear dozens of whispering fans. Gregory leaned on his cane and stared into the machine. “Frankly, I don’t know what it’s doing right now. 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.”

“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 notebook. David slashed open a cardboard box with his hunting knife and lifted something out of the box and plugged it into m zero. Gregory crawled under a table. “Oh, crap,” he said from beneath the table.

“Gregory! You killed the system again!”

“Dave, Dave, can you get me a flashlight?”

David handed his Mini Maglite 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 in 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 machine began to click, while its processors silently multiplied and joined huge numbers, heading ever deeper into pi. Gregory went off to bed, David holding him by the arm.

In his junkyard, his nest, his chamber of memory and imagination, Gregory kicked off his gentleman’s slippers, lay down on his bed, and brought into his mind’s eye the shapes of computing machines yet un-built.

* * *

IN THE NINETEENTH CENTURY, mathematicians attacked pi with the help of human computers. The most powerful of these was a man named Johann Martin Zacharias Dase. A prodigy from Hamburg, Dase could multiply large numbers in his head. He made a living exhibiting himself to crowds and hiring himself out as a computer for use by 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 numbers in his head in slightly less than nine hours. Dase could do this kind of thing for weeks on end, even as he went about his daily business. He would break off a calculation at bedtime, store everything in his memory for the night, and resume the calculation in the morning. Occasionally, Dase had a system crash. In 1845, he crashed while he was trying to demonstrate his powers to an astronomer named Heinrich Christian Schumacher—he got wrong every multiplication he tried. He explained to Schumacher that he had a headache. Schumacher also noticed that Dase did not in the least understand mathematics. A mathematician named Julius Petersen once tried in vain for six weeks to teach Dase the rudiments of geometry—such things as an equilateral triangle and a circle—but they absolutely baffled Dase. He had no problem with large numbers. In 1844, L. K. Schulz von Strassnitsky hired him to compute pi. Dase ran the job in his brain for almost two months. At the end of that time he wrote down pi correctly to the first two hundred decimal places—then a world record.

To many mathematicians, mathematical objects such as a circle seem to exist in an external, objective reality. Numbers, as well, seem to have a reality apart from time or the world. Numbers seem to transcend the universe. Numbers might even exist if the universe did not. I suspect that in their hearts most working mathematicians are Platonists, in that they accept the notion 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 mathematicians would probably agree that the ratio of the circle to its diameter exists luminously and eternally in the nature beyond nature, and would exist even if the human mind was not aware of it. Pi might exist even if God had not bothered 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. This is in the opinion of some mathematicians, anyway, who would argue that while there is at least some reason to doubt the existence of God, 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. Since pi is there, it exists. What we are doing is really close to experimental physics—we are ‘observing pi.’ Observing pi is easier than studying physical phenomena, because you can prove things in mathematics, whereas you can’t prove anything in physics. And, unfortunately, 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 representing it with symbols. You put it through your mind to make sense of it.”

Pi is elusive and can be approached only through approximations. There is no equation built from whole numbers that will give an exact value for pi. If equations are trains threading the landscape of numbers, no train stops at pi. A formula that heads toward pi will never get there, though it can get ever closer to pi. It will consist of a chain of operations that never ends. It is an infinite series. In 1674, Gottfried Wilhelm Leibniz (the coinventor of calculus, along with Isaac Newton) discovered an extraordinary pattern of numbers buried in the circle. This string of numbers—the Leibniz series for pi—has been called one of the most beautiful mathematical discoveries of the seventeenth century:

π/4 = 1/11/3 + 1/51/7 + 1/9 – …

In English: pi divided by four equals one minus a third plus a fifth minus a seventh plus a ninth—and so on. It seems almost musical in its harmony. You follow this chain of 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 mysterious that a chain of discrete rational numbers can connect so easily to the smooth and continuous circle.

As an experiment in “observing pi,” as Gregory put it, I got a pocket calculator and started computing the Leibniz series, to see what would happen. It was easy to do. I got answers that seemed to wander slowly toward pi. As I pushed the buttons on the calculator, the answers touched on 2.66, then 3.46, then 2.89, and 3.34, in that order. The answers landed higher than pi and lower than pi, skipping back and forth across pi, and were gradually closing in on pi. A mathematician would say that the series “converges on pi.” It converges on pi forever, playing hopscotch over pi, narrowing it down, but never landing on pi. No matter how far you take it, it never exactly touches pi. Transcendental numbers continue forever, as an endless nonrepeating string, in whatever rational form you choose to display them, whether as digits or 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. Looking 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.

It is worth thinking about what a decimal place means. Each decimal place of pi is a range that shows the approximate location of pi to an accuracy 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 wanders around pi. This is no such thing as a formula that is steady enough and 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, a powerful one, and it attacked pi with ferocity and elegance. The Chudnovsky formula for pi was the fastest series for pi ever found that uses rational numbers. It was very fast on a computer. The Chudnovsky formula for pi was thought to be “extremely beautiful” by persons who had a good feel for numbers, and it was based on a torus (a doughnut), rather than on a circle.

The Chudnovsky brothers claimed that the digits of pi form the most nearly perfect random sequence of digits that has ever been discovered. They said that nothing known to humanity appeared to be more deeply unpredictable than the sequence of digits in pi, except, perhaps, the haphazard clicks of a Geiger counter as it detects the decay of radioactive nuclei. But pi isn’t random. Not at all. The fact that pi can be produced by a relatively simple formula means that pi is orderly. Pi only looks random. In fact, there has to be a pattern in the digits. No doubt about it, because pi comes from the most perfectly symmetrical of all mathematical objects, the circle. But the pattern in pi is very, very complex. The Ludolphian number is something fixed in eternity—not a digit out of place, all characters 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.

“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—eight eights come up in a row. Does this mean anything? It seems to be random noise. Later, ten sixes erupt: 6666666666. 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, crystallized idea of randomness,” Gregory said. “It cannot be that pi is truly random. Actually, a truly random sequence of numbers has not yet been discovered.”

He explained that the “random” combinations of a slot machine in a casino are not random at all. They’re generated by simple computer programs, and, according to Gregory, the pattern is easy to figure out. “You might need only five consecutive tries on a slot machine to figure out the pattern,” he said.

“Why don’t you go to Las Vegas and make some money this way?” I asked.

“Eh.” Gregory shrugged, leaning on his cane.

“But look, this is not interesting,” David said. Besides, he pointed out, Gregory’s health would be threatened by a trip to Las Vegas.

No one knew what happened to the digits of pi in the deeper regions, as the number resolved toward infinity. Did the digits turn into nothing but eights and fives, say? Did they show a predominance of sevens? In fact, no one knew if a digit simply stopped appearing in pi. For example, there might be no more fives in pi after a certain point. Almost certainly, pi doesn’t do this, Gregory Chudnovsky thinks, because it would be stupid, and nature isn’t stupid. Nevertheless, no one has ever been able to prove or disprove it. “We know very little about transcendental numbers,” Gregory said.

If you take a string of digits from the square root of two and you compare it to a string of digits from pi, they look the same. There’s no way to tell them apart just by looking at the digits. Even so, the two numbers have completely distinct properties. Pi and the square root of two are as different from each other as a Rembrandt is from a Picasso, but human beings don’t have the ability to tell the two numbers apart by looking at their digits. (A sufficiently intelligent race of beings could probably do it easily.) Distressingly, the number pi makes the smartest humans into blockheads.

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 might have a chance of at least seeing 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 entire universe into a computer? What if you took every particle of matter in the universe and used all of it to build a computer? What then? How much pi could you see? Naturally, the brothers had considered this project. They had imagined a supercomputer built from the whole universe.

Here’s how they estimated the machine’s size. It has been calculated that there may be around 1079 electrons and protons in the observable universe. This is the so-called Eddington number of the universe. (Sir Arthur Stanley Eddington, an astrophysicist, first came up with it.) The Eddington number is the digit 1 followed by seventy-nine zeros: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. Ten vigintsextillion. The Eddington number. It gives you an idea of the power of the device that the Chudnovskys referred to as the Eddington machine.

The Eddington machine was the entire universe turned into a computer. It was made of all the atoms in the universe. 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 felt that a practical Eddington machine wouldn’t be able to store more than about 1077 digits of pi. That’s only a hundredth of the Eddington number. Now, what if the digits of pi were to begin to show regularity only beyond 1077 digits? Suppose, for example, that pi were only to begin manifesting a regularity starting at 10100 decimal places? That number is known as a googol. If the design in pi appeared only after a googol of digits, then not even the largest possible computer would ever be able to penetrate pi far enough to reveal any order in it. Pi would look totally disordered to the universe, even if it contained a slow, vast, delicate structure. A mere googol of pi might be only the first warp and weft, the first knot in a colored thread, in a limitless tapestry woven into gardens of delight and cities and towers and unicorns and unimaginable beasts and impenetrable mazes and unworldly cosmogonies, all invisible forever to us. It may never be possible, in principle, to see the design in the digits of pi. Not even nature itself may know the nature of pi.

“If pi doesn’t show systematic behavior until more than ten to the seventy-seven decimal places, it would really be a disaster,” Gregory said. “It would actually be horrifying.”

“I wouldn’t give up,” David said. “There might be some way of leaping over the barrier—”

“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 the perfect and beautiful world of mathematical reality when they were boys, growing up in Kiev. Their father, Volf, gave David a book entitled What is Mathematics?, written by Richard Courant and Herbert Robbins, two American mathematicians. The book is a classic. Millions of copies of it had been printed in unauthorized Russian and Chinese editions alone. (Robbins wrote most of the book, while Courant got ownership of the copyright and collected most of the royalties but paid almost none of the money to Robbins.) After reading it, David decided to become a mathematician. Gregory soon followed his brother’s footsteps into the nature beyond nature.

Gregory’s first publication, in a Soviet math journal, came when he was sixteen years old: “Some Results in the Theory of Infinitely Long Expressions.” Already you can see where he was headed. David, sensing his younger brother’s power, encouraged him to grapple with central problems in mathematics. In 1900, at the dawn of the twentieth century, the German mathematician David Hilbert had proposed a series of twenty-three great problems in mathematics that remained to be solved, and he’d challenged his colleagues, and future generations, to solve them. They became known as the Hilbert problems. At the age of seventeen, Gregory Chudnovsky made his first major discovery when he solved Hilbert’s Tenth Problem. To solve a Hilbert problem would be an achievement for a lifetime; Gregory was a high school student who had read a few books on mathematics. Strangely, a young Russian mathematician named Yuri Matyasevich had also just solved Hilbert’s Tenth Problem, but Gregory hadn’t heard the news. Eventually, Matyasevich said that the Chudnovsky method was the preferred way to solve Hilbert’s Tenth Problem.

The brothers enrolled at Kiev State University, and took their PhDs at the Ukrainian Academy of Sciences. At first, they published their papers separately, but as Gregory’s health declined, they began collaborating. They lived with their parents in Kiev until the family decided to try to take Gregory abroad for medical treatment. In 1976, Volf and Malka applied to the government of the USSR for permission to emigrate. Volf was immediately fired from his job.

It was a totalitarian state. The KGB began tailing the brothers. “I had twelve KGB agents on my tail,” David told me. “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 backseat. That was how the KGB operated.” One day in 1976, David was walking down the street when KGB officers attacked him, fracturing his skull. He nearly died. He didn’t dare go to the hospital; he went home instead. “If I had gone to the hospital, I would have died for sure,” he said. “The hospital was run by the state. I would forget to breathe.”

One July day, plainclothesmen from the KGB accosted Volf and Malka on a street corner and beat them up. They broke Malka’s arm and fractured her skull. David took his mother to the hospital, where he found that the doctors feared the KGB. “The doctor in the emergency room said there was no fracture,” David recalled.

By this time, the Chudnovskys were quite well known to mathematicians in the United States. Edwin Hewitt, a mathematician at the University of Washington, in Seattle, had collaborated with Gregory on a paper. He brought the Chudnovskys’ case to the attention of Senator Henry M. “Scoop” Jackson—a powerful politician from Washington State—and Jackson began putting pressure on the Soviets to let the Chudnovsky family leave the country. Not long before that, two members of a French parliamentary delegation made an unofficial visit to Kiev to see what was going on with the Chudnovskys. One of the visitors was Nicole Lannegrace, who would later become David’s wife. The Soviet government unexpectedly let the Chudnovskys go. “That summer when I was getting killed by the KGB, I could never have imagined that the next year I would be in Paris in love, or that I would wind up in New York, married to a beautiful Frenchwoman,” David said.

* * *

IF PI IS TRULY RANDOM, then at times pi will appear to be orderly. Therefore, if pi is random it contains accidental order. For example, somewhere in pi a sequence may run 070707070707070707 for as many digits as there are atoms in the sun. It’s just an accident. Somewhere else the exact same sequence may appear, only this time interrupted, just once, by the digit 3. Another accident. Every possible arrangement of digits probably erupts in pi, though this has never been proved. “Even if pi is not truly random, you can still assume that you get every string of digits in pi,” Gregory told me. In this respect, pi is like the Library of Babel in the story by Jorge Luis Borges. In that story, Borges imagined a library of vast size that contained all possible books.

You could find all possible books in pi. If you were to assign letters of the alphabet to combinations of digits—for example, the letter a might be 12, the letter b might be 34—you could turn the digits of pi into letters. (It doesn’t matter what digits are assigned to what letters—the combination could be anything.) You could do this with all alphabets and ideograms in all languages. Then pi could be turned into strings of written words. Then, if you could look far enough into pi, you would probably find the expression “See the U.S.A. in a Chevrolet!” 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 blasphemy. Also, you would find a guide to the pawnshops of Lubbock, Texas. It might or might not be accurate. Even so, somewhere else you would find the accurate guide to Lubbock’s pawnshops…if you could look far enough into pi. You would find, somewhere in pi, the unwritten book about the sea that James Joyce supposedly intended to tackle after he finished Finnegans Wake. You would find the collected transcripts of Saturday Night Live rendered into Etruscan. You would find a Google-searchable version of the entire Internet with every page on it exactly as it existed at midnight on July 1, 2007, and another version of the Internet from thirty seconds later. Each occurrence of an apparently ordered string in pi, such as the words “Ruin hath taught me thus to ruminate, / That Time will come and take my love away,” is followed by unimaginable deserts of babble. No book and none but the shortest poems will ever actually be seen in pi, for 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 calculated in this universe.

Anything that can be produced by a simple method is orderly. Pi can be produced by very simple methods; it is orderly, for sure. Yet the distinction between chance and fixity dissolves in pi. The deep connection between order and disorder, between cacophony and harmony, seems to be tantalizingly almost visible in pi, but not quite. “We are looking for some rules that will distinguish the digits of pi from other numbers,” Gregory said. “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? A child can do it: the next digit is six. What if I gave you a sequence of a million digits from pi? Could you tell me the next digit just by looking at it? Why does pi look totally unpredictable, with the highest complexity? For all we know, we may never find out the rule in pi.”

* * *

HERBERT ROBBINS, the coauthor of What Is Mathematics?, the book that had turned the Chudnovsky brothers on to math, was an emeritus professor of mathematical statistics at Columbia University and had become friends with the Chudnovskys. He lived in a rectilinear house with a lot of glass in it, in the woods near Princeton, New Jersey. Robbins was a tall, restless man in his seventies, with a loud voice, furrowed cheeks, and penetrating eyes. One day, he stretched himself out on a daybed in a garden room in his house and played with a rubber band, making a harp across his fingertips.

“It is a very difficult philosophical question, the question of what ‘random’ is,” Robbins 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 question! 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, and moved himself to a couch, and went back to the table, and finally returned to the daybed. The man was in constant motion.

“Mathematics is broken into tiny specialties today, but Gregory Chudnovsky is a generalist who knows the whole of mathematics as well as anyone,” he said as he moved around. “He’s like Mozart. I happen to think that his and David’s pi project is a will-o’-the-wisp, 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. In order for a mathematician to become interested in a problem, there has to be a possibility of solving it. Gregory likes to do things that are impossible.”

The Chudnovsky brothers were operating on their own, and they were looking more and more unemployable. Columbia University was never going to make them full-fledged members of the faculty, never give them tenure. This had become obvious. The John D. and Catherine T. MacArthur Foundation awarded Gregory Chudnovsky a “genius” fellowship. The brothers had won other fashionable and distinguished prizes, but there was a problem in their résumé, which was that Gregory had to lie in bed most of the time. The ugly truth was that Gregory Chudnovsky couldn’t get an academic job because he was physically disabled. But there were other, more perplexing reasons that had led the Chudnovskys to pursue their work in solitude. They had been living on modest grants from the National Science Foundation and various other research agencies and, of course, on their wives’ salaries. Christine’s father, Gonzalo Pardo, who was a professor of dentistry, had also chipped in. He had built the steel frame for m zero in his basement, using a wrench and a hacksaw.

The brothers’ solitary mode of existence had become known to mathematicians around the world as the Chudnovsky Problem. Herbert Robbins eventually decided to try to solve it. He was a member of the National Academy of Sciences, and 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 research system is found for the brothers soon, a personal and scientific tragedy will take place for which all American mathematicians will share responsibility.

There wasn’t much of a response. Robbins got three replies to his letter. One, from a professor of mathematics at an Ivy League university, complained about David Chudnovsky’s personality. He remarked that “when David learns to be less overbearing,” the brothers might have better luck.

Then Edwin Hewitt, the mathematician who had helped get the Chudnovsky family out of the Soviet Union, got mad, and erupted in a letter to colleagues:

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 caring for the Chudnovsky family.

The brothers, because they insisted that they were one mathematician divided between two bodies, would have to be hired as a pair. Gregory would refuse to take any job unless David got a job, too, and vice versa. To hire them, a math department would have to create two openings. And Gregory couldn’t teach classes in the normal way, because he was more or less confined to bed. And he might die, leaving the Chudnovsky Mathematician bereft of half its brain.

“The Chudnovskys are people the world is not able to cope with, and they are not making it any easier for the world,” Herbert Robbins said. “Even so, this vast educational system of ours has poured the Chudnovskys out on the sand, to waste. When I go up to that apartment and sit by Gregory’s bed, I think, My God, when I was a mathematics student at Harvard I was in contact with people far less interesting than this. I’m grieving about it.”

* * *

“TWO BILLION DIGITS OF PI? Where do they keep them?” Samuel Eilenberg said scornfully. Eilenberg was a distinguished topologist and emeritus professor of mathematics at Columbia University.

“I think they store the digits on a hard drive,” I answered.

Eilenberg snorted. He didn’t care about some spinning piece of metal covered with pi. He was one of the reasons why the Chudnovskys would never get permanent jobs at Columbia; he made it pretty clear that he would see to it that they were denied tenure. “In the academic world, we have to be careful who our colleagues are,” he told me. “David is a nudnik! You can spend all your life computing digits. What for? It’s about as interesting as going to the beach and counting sand. I wouldn’t be caught dead doing that kind of work.”

In his view, there was something unclean about doing mathematics with a machine. Samuel Eilenberg was a member of the famous Bourbaki group. This group, a sort of secret society of mathematicians that was founded in 1935, consisted mostly of French members (though Eilenberg was originally Polish) who published collectively under the fictitious name Nicolas Bourbaki; they were referred to as “the Bourbaki.” In a quite French way, the Bourbaki were purists who insisted on rigor and logic and formalism. Some members of the Bourbaki group looked down on applied mathematics—that is, they seemed to scorn the use of mathematics to solve real-world problems, even in physics. The Bourbaki especially seemed to dislike the use of machinery in pursuit of truth. Samuel Eilenberg appeared to loathe the Chudnovskys’ supercomputer and what they were doing with it. “To calculate the two billionth digit of pi is to me abhorrent,” he said.

“‘Abhorrent’? Yes, most mathematicians would probably agree with that,” said Dale Brownawell, a respected number theorist at Penn State. “Tastes change, though. To see the Chudnovskys carrying on science at such a high level with such meager support is awe-inspiring.”

Richard Askey, a prominent mathematician at the University of Wisconsin at Madison, would occasionally fly to New York to sit at the foot of Gregory Chudnovsky’s bed and talk about mathematics. “David Chudnovsky is a very good mathematician,” Askey said to me. “Gregory is a great mathematician. The brothers’ pi stuff is just a small part of their work. They are really trying to find out what the word ‘random’ 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. Gregory Chudnovsky’s situation is a national problem.”

* * *

“IT LOOKS LIKE KVETCHING,” Gregory said from his bed. “It looks cheap, and it is cheap. I don’t think we were somehow wronged. I really can’t teach. So what does one do about it? We barely have time to do the things we want to do. What is life, and where does the money come from?” He shrugged.

At the end of the summer, the brothers halted their probe into pi. They had other things they wanted to do with their supercomputer, and it was time to move on. They had surveyed pi to 2,260,321,336 digits. It was a world record, doubling their previous world record. If the digits were printed in type, they would stretch from New York to Los Angeles.

In Japan, their competitor Yasumasa Kanada reacted gracefully. He told Science News that he might be able to get a billion and a half digits if he could rent enough time on the Hitachi—the half-megawatt monster.

“You see the advantage of being truly poor,” Gregory said to me. “We had to build our machine, but now we own it.”

M zero had spent most of its time checking the answer to make sure it was correct. “We have done our tests for patterns, and there is nothing,” Gregory said. He was nonchalant about it. “It would be rather stupid if there were a pattern 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. Unfortunately, we still don’t have enough computer power to see anything in pi.”

And yet…and yet…the brothers felt that they might have noticed something in pi. It hovered out of reach, but seemed a little closer now. It was a slight change in pi that seemed to rise and fall like a tide, as if a distant moon were passing over the sea of digits. It was something random, probably. The brothers felt that they might only have glimpsed the human desire for order. Or was it a wave rippling through pi? Would the wave, if it was there, be the first thread in a tapestry of worlds blossoming in pi? “We need a trillion digits,” David said. Maybe one day they would run the calculation into a trillion digits. Or maybe not. A trillion digits of pi printed in ordinary type would stretch from here to the moon and back, twice. Maybe one day, if they lived and if their machines held together, they would orbit the moon in digits, and would head for Alpha Centauri, seeking pi.

Gregory is lying on his bed in the junkyard, now. He offers to show me the last digits the supercomputer found. He types a command, and suddenly the whole screen fills with pi. It’s the raw Ludolphian number, pouring across the screen like Niagara Falls:


72891 51567 97145 46268 92720 56914 19491 70799 30612 27184

95997 75819 61414 47296 81115 92768 25023 87974 42024 32465

81816 25413 12164 96683 83188 86493 16114 55018 80584 26203

71989 99024 98835 10467 22124 63734 94382 70510 64281 32133

84515 75884 47736 80693 93435 69959 13571 88057 62592 60719

58508 38025 73050 11862 43946 99422 06487 07264 08095 58354

41083 43437 83790 00353 73416 69273 76820 40100 54718 28029

00958 45404 09196 25724 40953 10724 75287 88238 71194 22897

36462 82455 69706 19364 35459 84229 95107 39973 54996 68154

14759 50184 95343 60383 37189 76295 12572 70965 58816 94729

09508 25947 06150 01226 73434 26496 86070 41411 62634 95296

69333 80436 51116 81295 92670 33384 07650 40965 11979 85185

50164 21984 40980 27554 25619 05834 95554 34498 43497 55136

88999 51731 69029 01197 60153 45399 73782 80898 99826 36229

28846 77788 04108 11793 89363 51922 14801 13183 14735 68818

49953 27420 48050 19186 07391 11248 22845 78059 61348 96790

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


We observe pi in silence.

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