Panic in Level 4

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