Richard Preston - Panic in Level 4

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] More precisely: a transcendental number cannot be expressed as the exact solution to any polynomial equation that has a finite number of terms with integer coefficients. 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 3 10/71 and 3 1/7. 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.