Panic in Level 4: Cannibals, Killer Viruses, and Other Journeys to the Edge of Science (11 page)

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 10
⁷⁹
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 10
⁷⁷
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
10
⁷⁷
digits? Suppose, for example, that pi were only to begin manifesting a regularity starting at 10
¹⁰⁰
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.

T
HE 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.