I dialed the number and got a fax tone. I handwrote a message on a piece of paper and faxed it, asking if this number belonged to the Chudnovskys and, if so, would they be able to meet with me? There was no reply. Weeks passed. I gave up. But then one day my phone rang; it was David Chudnovsky. “Look, you are welcome,” he said. He had a genteel-sounding voice with a Russian accent.
On a cold winter day soon afterward, I rang the bell of Gregory’s apartment on 120th Street. I was carrying a little notebook and a mechanical pencil in my shirt pocket. David answered the door. He pulled the door open a few inches, and then it stopped. It was jammed against an empty cardboard box and a mass of hanging coats. He nudged the box out of the way with his foot. “Don’t worry,” he said. “Nothing unpleasant will happen to you here. We will not turn you into digits.” A Mini Maglite flashlight protruded from his shirt pocket.
We were standing in a long, dark hallway. The place was a swamp of heat. My face and armpits began to drip with sweat. The lights were off, and it was hard to see anything. This was the reason for David’s flashlight. The hall was lined on both sides with bookshelves supporting huge stacks of paper and books. The shelves took up most of the space, leaving a passage about two feet wide running down the length of the hallway. At the end of the hallway was a French door. Its mullioned glass panes were covered with translucent paper. The panes glowed.
We went along the hallway. We passed a bathroom and a bedroom door, which was closed. The bedroom belonged to Malka Benjaminovna Chudnovsky. We passed a sort of cave containing vast amounts of paper. This was Gregory’s bedroom, his junkyard. We passed a small kitchen, our feet rolling on computer cables. David opened the French door, and we entered the living room. This was the chamber of the supercomputer. A bare lightbulb burned in a ceiling fixture. The room contained seven display screens, two of which were filled with numbers; the other screens were turned off. The windows were closed and the shades were drawn. Gregory Chudnovsky sat on a chair facing the lit-up screens. He wore a tattered and patched lamb’s wool sweater, a starched white shirt, blue sweatpants, and the hand-stitched two-tone socks. From his toes trailed a pair of heelless leather slippers. His cane was hooked over his shoulder, hung there for convenience. “Right now, our goal is to compute pi,” he said. “For that we have to build our own computer.” He had a resonant voice and a Russian accent.
The Chudnovsky Mathematician: Gregory and David Chudnovsky in Gregory’s New York City apartment, 1992.
Irena Roman
“We are a full-service company,” David said. “Of course, you know what ‘full-service’ means in New York. It means ‘You want it? You do it yourself.’”
A steel frame stood in the center of the room, screwed together with bolts. It held split-open shells of personal computers—cheap PC clones, knocked wide open like cracked walnuts, their meat spilling all over the place. The brothers had crammed superfast logic boards inside the PCs. Red lights on the boards blinked. The floor was a quagmire of cables.
The brothers had also managed to fit into the room masses of empty cardboard boxes, and lots of books (Russian classics, with Cyrillic lettering on their spines), and screwdrivers, and data-storage tapes, and software manuals by the cubic yard, and stalagmites of obscure trade magazines, and a twenty-thousand-dollar engineering computer that they no longer used. “We use it as a place to stack paper,” Gregory explained. From an oval photograph on the wall, the face of Volf Chudnovsky, their late father, looked down on the scene. The walls and the French door were covered with sheets of drafting paper showing circuit diagrams. They resembled cities seen from the air. Various disk drives were scattered around the room. The drives were humming, and there was a continuous whir of fans. A strong warmth emanated from the equipment, as if a steam radiator were going in the room. The brothers were heating the apartment with silicon chips.
“MYASTHENIA GRAVIS is a funny thing,” Gregory Chudnovsky said one day from his bed in his bedroom, the junkyard. “In a sense, I’m very lucky, because I’m alive, and I’m alive after so many years. There is no standard prognosis. It sometimes strikes young women and older women. I wonder if it is some kind of sluggish virus.”
It was a cold afternoon, and rain pelted the windows; the shades were drawn, as always, and the room was stiflingly warm. He lay against a heap of pillows with his legs folded under him. His bed was surrounded by freestanding bookshelves packed and piled with ramparts of stacked paper. That day, he wore the same tattered wool sweater, a starched white shirt, blue sweatpants, and another pair of handmade socks. I had never seen socks like Gregory’s. They were two-tone socks, wrinkled and floppy, hand-sewn from pieces of dark blue and pale blue cloth, and they looked comfortable. They were the work of Malka Benjaminovna, his mother. Lines of computer code flickered on the screen beside his bed.
This was an apartment built for long voyages. The paper in the room was jammed into bookshelves along the wall, too, from floor to ceiling. The brothers had wedged the paper, sheet by sheet, into manila folders, until the folders had grown as fat as melons. The paper was also stacked chest-high to chin-high on five chairs (three of them in a row beside his bed). It was heaped on top of and filled two steamer trunks that sat beneath the window, and the paper had accumulated in a sort of unstable-looking lava flow on a small folding cocktail table. I moved carefully around the room, fearful of triggering a paperslide, and I sat down on the room’s one empty chair, facing the foot of Gregory’s bed, my knees touching the blanket. The paper surrounded his bed like the walls of a fortress, and his bed sat at the center of the keep. I sensed a profound happiness in Gregory Chudnovsky. His happiness, it occurred to me later, sprang from the delicious melancholy of a life spent largely in bed while he explored a more perfect world that opened through the portals of mathematics into vistas beyond time or decay.
“The system of this paper is archaeological,” he said. “By looking at a slice, I know the date. This slice is from five years ago. Over here is some paper from four years ago. What you see in this room is our working papers, as well as the papers we used as references. Some of the references we pull out once in a while to look at, and then we leave them in another pile. Once, we had to make a Xerox copy of the same book three times, and we put it in three different piles, so we could be sure to find it when we needed it. There are books in there by Kipling and Macaulay. Eh, this place is a mess. Actually, when we want to find a book it’s easier to go to the library.”
Much of the paper consisted of legal pads covered with Gregory’s handwriting. His handwriting was dense and careful, a flawless minuscule written with a felt-tipped pen. The writing contained a mixture of theorems, calculations, proofs, and conjectures concerning numbers. He used a felt-tip pen because he didn’t have enough strength in his hand to press a pencil on paper. Mathematicians who had visited Gregory Chudnovsky’s bedroom had come away dizzy, wondering what secrets the scriptorium might hold. He cautiously referred to the steamer trunks beneath the window as valises. They were filled to the lids with compressed paper. When Gregory and David flew to Europe to speak at conferences on the subject of numbers, they took both “valises” with them, in case they needed to refer to a proof or a theorem. Their baggage particularly annoyed Belgian officials. “The Belgians were always fining us for being overweight,” Gregory said.
The brothers’ mail-order supercomputer made their lives more convenient. It performed inhumanly difficult algebra, finding roots of gigantic systems of equations, and it constructed colored images of the interior of Gregory Chudnovsky’s body. They used the supercomputer to analyze and predict fluctuations in the stock market. They had been working with a well-known Wall Street investor named John Mulheren, helping him get a profitable edge in computerized trades on the stock market. One day I called John Mulheren to find out what the brothers had been doing for him. “Gregory and David have certainly made us money,” Mulheren said, but he wouldn’t give any details on what the brothers had done. Mulheren had been
paying the Chudnovskys out of his trading profits; they used the money to help fund their research into numbers. To them, numbers were more beautiful, more nearly perfect, possibly more complicated, and arguably more real than anything in the world of physical matter.
THE NUMBER PI, or , is the most famous ratio in mathematics. It is also one of the most ancient numbers known to humanity. Nobody knows when pi first came to the awareness of the human species. Pi may very well have been known to the builders of Stonehenge, around 2,600 B.C.E. Certainly it was known to the ancient Egyptians. Pi is approximately 3.14—it is the number of times that a circle’s diameter will fit around a circle. On the following page is a circle with its diameter.
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.
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 was William Jones, an English mathematician, who coined it in 1706. Jones probably meant to stand for “periphery.”