Panic in Level 4

^{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.