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 at least see something that could lead to an important conjecture about pi or about transcendental numbers as a class. You can learn a lot about all cats by looking closely at one of them. So if you wanted to look closely at pi how much of it could you see with a very large supercomputer? What if you turned the universe into a supercomputer? What then? How much pi could you see? Naturally, the brothers had considered this project. They had imagined a computer built from the universe. Here’s how they estimated the machine’s size. It has been calculated that there are about 1079 electrons and protons in the observable universe; this is the so-called Eddington number of the universe. (Sir Arthur Stanley Eddington, the astrophysicist, first came up with the number.) The 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 declares the power of the Eddington machine.

The Eddington machine would be the universal supercomputer. It would be made of all the atoms in the universe. The Eddington machine would contain ten vigintsextillion parts, and if the Chudnovsky brothers could figure out how to program it with fortran they might make it churn toward pi. “In order to study the sequence of pi, you have to store it in the Eddington machine’s memory,” Gregory said. To be realistic, the brothers thought that a practical Eddington machine wouldn’t be able to store pi much beyond 1077 digits—a number that is only a hundredth of the Eddington number. Now, what if the digits of pi only begin to show regularity beyond 1077 digits? Suppose, for example, that pi manifests a regularity starting at 10100 decimal places? That number is known as a googol. If the design in pi appears only after a googol of digits, then not even the Eddington machine will see any system in pi; pi will look totally disordered to the universe, even if pi contains a slow, vast, delicate structure. A mere googol of pi might be only the first knot at the corner of a kind of limitless Persian rug, which is woven into increasingly elaborate diamonds, cross-stars, gardens, and cosmogonies. It may never be possible, in principle, to see the order in the digits of pi. Not even nature itself may know the nature of pi.

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

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

“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 mathematical reality when they were boys, growing up in Kiev, and their father gave David a book entitled “What Is Mathematics?,” by two mathematicians named Richard Courant and Herbert Robbins. The book is a classic—millions of copies of it have been printed in unauthorized Russian and Chinese editions alone—and after the brothers finished reading “Robbins,” as the book is called in Russia, David decided to become a mathematician, and Gregory soon followed his brother’s footsteps into the nature beyond nature. Gregory’s first publication, in the journal Soviet Mathematics—Doklady, 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. Gregory made his first major discovery at the age of seventeen, when he solved Hilbert’s Tenth Problem. (It was one of twenty-three great problems posed by David Hilbert in 1900.) To solve a Hilbert problem would be an achievement for a lifetime; Gregory was a high-school student who had read a few books on mathematics. Strangely, a young Russian mathematician named Yuri Matyasevich had just solved Hilbert’s Tenth Problem, and the brothers hadn’t heard the news. Matyasevich has recently said that the Chudnovsky method is the preferred way to solve Hilbert’s Tenth Problem.

The brothers enrolled at Kiev State University, and both graduated summa cum laude. They took their Ph.D.s at the Institute of Mathematics at the Ukrainian Academy of Sciences. At first, they published their papers separately, but by the mid-nineteen-seventies they were collaborating on much of their work. They lived with their parents in Kiev until the family decided to try to take Gregory abroad for treatment, and in 1976 Volf and Malka Chudnovsky applied to the government to emigrate. Volf was immediately fired from his job.

The K.G.B. began tailing the brothers. “Gregory would not believe me until it became totally obvious,” David said. “I had twelve K.G.B. agents on my tail. No, look, I’m not kidding! They shadowed me around the clock in two cars, six agents in each car. Three in the front seat and three in the back seat. That was how the K.G.B. operated.” One day, in 1976, David was walking down the street when K.G.B. officers attacked him, breaking his skull. He went home and nearly died, but didn’t go to the hospital. “If I had gone to the hospital, I would have died for sure,” he told me. “The hospital is run by the state. I would forget to breathe.”

On July 22, 1977, plainclothesmen from the K.G.B. accosted Volf and Malka on a street in Kiev and beat them up. They broke Malka’s arm and fractured her skull. David took his mother to the hospital. “The doctor in the emergency room said there was no fracture,” David said.

Gregory, at home in bed, was not so vulnerable. Also, he was conspicuous in the West. Edwin Hewitt, a mathematician at the University of Washington, in Seattle, had visited Kiev in 1976 and collaborated with Gregory on a paper, and later, when Hewitt learned that the Chudnovsky family was in trouble, he persuaded Senator Henry M. Jackson, the powerful member of the Senate Armed Services Committee, to take up the Chudnovskys’ case. Jackson put pressure on the Soviets to let the family leave the country. Just before the K.G.B. attacked the parents, two members of a French parliamentary delegation that was in Kiev made an unofficial visit to the Chudnovskys to see what was going on. One of the visitors, a staff member of the delegation, was Nicole Lannegrace, who married David in 1983. Andrei Sakharov also helped to draw attention to the Chudnovskys’ increasingly desperate situation. Two months after the parents were attacked, the Soviet government unexpectedly let the family go. “That summer when I was getting killed by the K.G.B., I could never have imagined that the next year I would be in Paris or that I would wind up in New York, married to a beautiful Frenchwoman,” David said.__ The Chudnovsky family settled in New York, near Columbia University.

If pi is truly random? then at times pi will appear to be ordered. Therefore, if pi is random it contains accidental order. For example, somewhere in pi a sequence may run 07070707070707 for as many decimal places as there are, say, hydrogen atoms in the sun. It’s just an accident. Somewhere else the same sequence of zeros and sevens may appear, only this time interrupted by a single occurrence of the digit 3. Another accident. Those and all other “accidental” arrangements of digits almost certainly erupt in pi, but their presence 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 said.

If you were to assign letters of the alphabet to combinations of digits, and were to do this for all human alphabets, syllabaries, and ideograms, then you could fit any written character in any language to a combination of digits in pi. According to this system, pi could be turned into literature. Then, if you could look far enough into pi, you would probably find the expression “See the U.S.A. in a Chevrolet!” a billion times in a row. Elsewhere, you would find Christ’s Sermon on the Mount in His native Aramaic tongue, and you would find versions of the Sermon on the Mount that are pure blasphemy. Also, you would find a dictionary of Yanomamo curses. A guide to the pawnshops of Lubbock. The book about the sea which James Joyce supposedly declared he would write after he finished “Finnegans Wake.” The collected transcripts of “The Tonight Show” rendered into Etruscan. “Knowledge of All Existing Things,” by Ahmes the Egyptian scribe. Each occurrence of an apparently 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 be seen in pi, since it is infinitesimally unlikely that even as brief a text as an English sonnet will appear in the first 1077 digits of pi, which is the longest piece of pi that can be calculated in this universe.

Anything that can be produced by a simple method is by definition orderly. Pi can be produced by various simple methods of rational approximation, and those methods yield the same digits in a fixed order forever. Therefore, pi is orderly in the extreme. Pi may also be a powerful random-number generator, spinning out any and all possible combinations of digits. We see that the distinction between chance and fixity dissolves in pi. The deep connection between disorder and order, between cacophony and harmony, in the most famous ratio in mathematics fascinated Gregory and David Chudnovsky. They wondered if the digits of pi had a personality.

“We are looking for the appearance of some rules that will distinguish the digits of pi from other numbers,” Gregory explained. “It’s like studying writers by studying their use of words, their grammar. If you see a Russian sentence that extends for a whole page, with hardly a comma, it is definitely Tolstoy. If someone were to give you a million digits from somewhere in pi, could you tell it was from pi? We don’t really look for patterns; we look for rules. Think of games for children. If I give you the sequence one, two, three, four, five, can you tell me what the next digit is? Even a child can do it; the next digit is six. How about this game? Three, one, four, one, five, nine. Just by looking at that sequence, can you tell me the next digit? What if I gave you a sequence of a million digits from pi? Could you tell me the next digit just by looking at the sequence? Why does pi look like a totally unpredictable sequence with the highest complexity? We need to find out the rules that govern this game. For all we know, we may never find a rule in pi.”

Herbert Robbins, the co-author of “What Is Mathematics?,” is an emeritus professor of mathematical statistics at Columbia University. For the past six years, he has been teaching at Rutgers. The Chudnovskys call him once in a while to get his advice on how to use statistical tools to search for signs of order in pi. Robbins lives in a rectilinear house that has a lot of glass in it, in the woods on the outskirts of Princeton. Some of the twentieth century’s most creative and powerful discoveries in statistics and probability theory happened inside his head. Robbins is a tall, restless man in his seventies, with a loud voice furrowed cheeks, and penetrating eyes One recent day, he stretched himself out on a daybed in a garden room in his house and played with a rubber band, making a harp across his fingertips.

“It is a very difficult philosophical question, the question of what ‘random’ is,” he said. He plucked the rubber band with his thumb, boink, boink. “Everyone knows the famous remark of Albert Einstein, that God does not throw dice. Einstein just would not believe that there is an element of randomness in the construction of the world. The question of whether the universe is a random process or is determined in some way is a basic philosophical question that has nothing to do with mathematics. The question is important. People consider it when they decide what to do with their lives. It concerns religion. It is the question of whether our fate will be revealed or whether we live by blind chance. My God, how many people have been murdered over an answer to that 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 in the room, and sat on a couch, and went back to the table, and finally returned to the daybed. The man was in constant motion. It looked random to me, but it may have been systematic. It was the random walk of Herbert Robbins.

“Mathematics is broken into tiny specialties today, but Gregory Chudnovsky is a generalist who knows the whole of mathematics as well as anyone,” he said as he moved around. “You have to go back a hundred years, to David Hilbert, to find a mathematician as broadly knowledgeable as Gregory Chudnovsky. He’s like Mozart: he’s the last of his breed. I happen to think the brothers’ pi project is a will-o’-the-wisp, and is one of the least interesting things they’ve ever done. But what do I know? Gregory seems to be asking questions that can’t be answered. To ask for the system in pi is like asking ‘Is there life after death?’ When you die, you’ll find out. Most mathematicians are not interested in the digits of pi, because the question is of no practical importance. In order for a mathematician to become interested in a problem, there has to be a possibility of solving it. If you are an athlete, you ask yourself if you can jump thirty feet. Gregory likes to ask if he can jump around the world. He likes to do things that are impossible.”

At some point after the brothers settled in New York, it became obvious that Columbia University was not going to be able to invite them to become full-fledged members of the faculty. Since then, the brothers have always enjoyed cordial personal relationships with various members of the faculty, but as an institution the Mathematics Department has been unable to create permanent faculty positions for them. Robbins and a couple of fellow-mathematicians—Lipman Bers and the late Mark Kac—once tried to raise money from private sources for an endowed chair at Columbia to be shared by the brothers, but the effort failed. Then the John D. and Catherine T. MacArthur Foundation awarded Gregory Chudnovsky a “genius” fellowship; that happened in 1981, the first year the awards were given, as if to suggest that Gregory is a person for whom the MacArthur prize was invented. The brothers can exhibit other fashionable paper—a Prix Peccot-Vimont, a couple of Guggenheims, a Doctor of Science honoris causa from Bard College, the Moscow Mathematical Society Prize—but there is one defect in their résumé, which is the fact that Gregory has to lie in bed most of the day. The ugly truth is that Gregory Chudnovsky can’t get a permanent job at any American institution of higher learning because he is physically disabled. But there are other, more perplexing reasons that have led the Chudnovsky brothers to pursue their work in solitude, outside the normal academic hierarchy, since the day they arrived in the United States.

Columbia University has awarded each brother the title of senior research scientist in the Department of Mathematics. Their position at Columbia is ambiguous. The university officially considers them to be members of the faculty, but they don’t have tenure, and Columbia doesn’t spend its own funds to pay their salaries or to support their research. However, Columbia does give them health-insurance benefits and a housing subsidy.