Malka Benjaminovna improved slowly. When she got home from the hospital, the brothers settled her back in her room in Gregory’s apartment and hired a nurse to look after her. I visited them shortly after that, on a hot day in early summer. David answered the door. There were blue half circles under his eyes, and he had lost weight. He smiled weakly and greeted me by saying, “I believe it was Oliver Heaviside, the English physicist, who once said, ‘In order to know soup, it is not necessary to climb into a pot and be boiled.’ But look, my dear fellow, if you want to be boiled you are welcome to come in.” He led me down the dark hallway. Malka Benjaminovna was asleep in her bedroom, and the nurse was sitting beside her. Her room was lined with her late husband Volf’s bookshelves, and they were packed with paper. It was an overflow repository.
“Theoretically, the best way to cool a supercomputer is to submerge it in water,” Gregory said, from his bed in the junkyard.
“Then we could add goldfish,” David said.
“That would solve all our problems.”
“We are not good plumbers, Gregory. As long as I am alive, we will not cool a machine with water.”
“What is the temperature in there?” Gregory asked, nodding toward m zero’s room.
“It grows to above ninety Fahrenheit. This is not good. Things begin to fry.”
David took Gregory under the arm, and we passed through the French door into gloom and pestilential heat. The shades were drawn, the lights were off, and an air conditioner in a window ran in vain. Sweat immediately began to pour down my body. “I don’t like to go into this room,” Gregory said. The steel frame in the center of the room—the heart of m zero—seemed to have acquired more guts, and red lights blinked inside it. I could hear disk drives murmuring. The drives were copying and recopying huge segments of transcendental numbers, to check the digits for perfect accuracy. “If the machine makes an error in a single digit of pi, then every digit after that is nonsense. What comes out is not pi at all, it’s just some random number.” Thus they had to keep checking and rechecking the digits to make sure they were exactly pi to the last digit.
Gregory knelt on the floor, facing the steel frame.
David opened a cardboard box and removed an electronic board. He began to fit it into m zero. I noticed that his hands were marked with small cuts, which he had got from reaching into the machine.
“David, could you give me the flashlight?” Gregory asked.
David pulled the Mini Maglite from his shirt pocket and handed it to Gregory. The brothers knelt beside each other, Gregory shining the flashlight into the supercomputer. David reached inside with his fingers and palpated a logic board.
“Don’t!” Gregory said. “Okay, look. No! No!” They muttered to each other in Russian. “It’s too small,” Gregory said.
David adjusted an electric fan. “We bought it at a hardware store down the street,” he said to me. “We buy our fans in the winter. It saves money.” He pointed to a gauge that had a dial on it. “Here we have a meat thermometer.”
The brothers had thrust the meat thermometer between two circuit boards inside m zero, in order to look for hot spots. The thermometer’s dial was marked “Beef Rare—Ham—Beef Med—Pork.”
“You want to keep the machine below ‘Pork,’” Gregory remarked.
He lifted a keyboard out of a steel frame and typed something on it. Numbers filled a display screen. “The machine is checking its memory,” he said. A buzzer sounded. “It shut down!” he said. “It’s a disk-drive controller. The stupid thing obviously has problems.”
“It’s mentally deficient,” David commented. He went over to a bookshelf and picked up a hunting knife. I thought he was going to plunge it into the supercomputer, but he used it to rip open a cardboard box. “We’re going to ship the part back to the manufacturer,” he said to me. “You had better send it in the original box or you may not get your money back. Now you know the reason this apartment is full of empty boxes. Gregory, I wonder if you are tired.”
“If I stand up now, I will fall down,” Gregory said. “Therefore, I will sit in my center of gravity. Let me see, meanwhile, what is happening with this machine.” He typed something on his keyboard. “You won’t believe it, Dave, but the controller now seems to work.”
“We need to buy a new one anyway,” David said.
“Try Nevada.”
David dialed a computer-parts wholesaler in Nevada called Searchlight Compugear. He said loudly, in a Russian accent, “Hi, Searchlight. I need a fifteen-forty controller…. No! No! No! I don’t need anything else! Just a naked unit! How much you charge? What? Two hundred and fifty-seven dollars…?!”
Gregory glanced at his brother and shrugged. “Eh.”
“Look, Searchlight, can you ship it to me Federal Express? For tomorrow morning. How much? …Thirty-nine dollars for FedEx? Come on! What about afternoon delivery?…Twenty-nine dollars before three P.M.? Relax. What is your name?…Bob. Fine. Okay. So, Bob, it is two hundred and fifty-seven dollars plus twenty-nine dollars for FedEx?”
“Twenty-nine dollars for Fed Ex!” Gregory burst out. “It should be fifteen.” He pulled a second keyboard out of the frame and tapped the keys. Another display screen came alive and filled with numbers.
“Tell me this,” David said to Bob in Nevada. “Do you have a thirty-day money-back guarantee?…No? Come on! Look, any device might not work.”
“Of course, a part might work,” Gregory muttered to his brother. “But usually it doesn’t.”
“Question Number Two: The FedEx should not cost twenty-nine bucks,” David said to Bob. “No, nothing! I’m just asking.” David hung up the phone. “I’m going to A.K.,” he said. “Hi, A.K., this is David Chudnovsky calling from New York. A.K., I need another controller, like the one you sent. Can you send it today FedEx?…How much you charge?…Naked! I want a naked unit! Not in a shoe box, nothing!”
A rhythmic clicking sound came from one of the disk drives. Gregory remarked to me, “We are calculating pi right now.”
“Do you want my MasterCard? Look, it’s really imperative that I get my unit tomorrow. Please, I really need my unit bad.” David hung up the telephone and sighed. “This is what has happened to a pure mathematician.”
“GREGORY AND DAVID are both extremely childlike, but I don’t mean childish at all,” Gregory’s wife, Christine Pardo Chudnovsky, said one muggy summer day, at the dining room table. “There is a certain amount of play in everything they do, a certain amount of fooling around between two brothers.” She was six years younger than Gregory; she had been an undergraduate at Barnard College when she first met him. “I fell in love with Gregory immediately. His illness came with the package.” She remained in love with him, even if at times they fought over the heaps of paper. (“I don’t have room to put my things down anywhere,” she told him.) As we talked, though, pyramids of boxes and stacks of paper leaned against the dining room windows, pressing against the glass and blocking daylight, and a smell of hot electrical gear crept through the room. “This house is an example of mathematics in family life,” she said. At night, she dreamed that she was dancing from room to room through an empty apartment that had parquet floors.
David brought his mother out of her bedroom, settled her at the table, and kissed her on the cheek. Malka Benjaminovna seemed frail but alert. She was a small, white-haired woman with a fresh face and clear blue eyes who spoke limited English. A mathematician once described Malka Benjaminovna as the glue that held the Chudnovsky family together. She’d been an engineer during the Second World War, when she designed buildings, laboratories, and proving grounds for testing the Katyusha rocket. Later, she taught engineering at schools around Kiev. Smiling, she handed me plates of roast chicken, kasha, pickles, cream cheese, brown bread, and little wedges of Laughing Cow cheese in foil. “Mother thinks you aren’t getting enough to eat,” Christine said. Malka Benjaminovna slid a jug of Gatorade across the table at me.
After we finished lunch, and were
fortified with Gatorade, the brothers and I went into the chamber of m zero, into a pool of thick heat. The room enveloped us like noon on the Amazon, and it teemed with hidden activity. The machine clicked, the red lights flashed, the air conditioner hummed, and you could hear dozens of whispering fans. Gregory leaned on his cane and stared into the machine. “Frankly, I don’t know what it’s doing right now. It’s doing some algebra, and I think it’s also backing up some pieces of pi.”
“Sit down, Gregory, or you will fall,” David said.
“What is it doing now, Dave?”
“It’s blinking.”
“It will die soon.”
“Gregory, I heard a funny noise.”
“You really heard it? Oh, God, it’s going to be like the last time.”
“That’s it!”
“We are dead! It crashed!”
“Sit down, Gregory, for God’s sake!”
Gregory sat on a stool and tugged at his beard. “What was I doing before the system crashed? With God’s help, I will remember.” He jotted a few notes in a notebook. David slashed open a cardboard box with his hunting knife and lifted something out of the box and plugged it into m zero. Gregory crawled under a table. “Oh, crap,” he said from beneath the table.
“Gregory! You killed the system again!”
“Dave, Dave, can you get me a flashlight?”
David handed his Mini Maglite under the table. Gregory joined some cables together and stood up. “Whoo! Very uncomfortable. David, boot it up.”
“Sit down for a moment.”
Gregory slumped in a chair.
“This monster is going on the blink,” David said, tapping a keyboard.
“It will be all right.”
On a screen, m zero declared, “The system is ready.”
“Ah,” David said.
The machine began to click, while its processors silently multiplied and joined huge numbers, heading ever deeper into pi. Gregory went off to bed, David holding him by the arm.
In his junkyard, his nest, his chamber of memory and imagination, Gregory kicked off his gentleman’s slippers, lay down on his bed, and brought into his mind’s eye the shapes of computing machines yet un-built.
IN THE NINETEENTH CENTURY, mathematicians attacked pi with the help of human computers. The most powerful of these was a man named Johann Martin Zacharias Dase. A prodigy from Hamburg, Dase could multiply large numbers in his head. He made a living exhibiting himself to crowds and hiring himself out as a computer for use by mathematicians. A mathematician once asked Dase to multiply 79,532,853 by 93,758,479, and Dase gave the right answer in fifty-four seconds. Dase extracted the square root of a hundred-digit number in fifty-two minutes, and he was able to multiply a couple of hundred-digit numbers in his head in slightly less than nine hours. Dase could do this kind of thing for weeks on end, even as he went about his daily business. He would break off a calculation at bedtime, store everything in his memory for the night, and resume the calculation in the morning. Occasionally, Dase had a system crash. In 1845, he crashed while he was trying to demonstrate his powers to an astronomer named Heinrich Christian Schumacher—he got wrong every multiplication he tried. He explained to Schumacher that he had a headache. Schumacher also noticed that Dase did not in the least understand mathematics. A mathematician named Julius Petersen once tried in vain for six weeks to teach Dase the rudiments of geometry—such things as an equilateral triangle and a circle—but they absolutely baffled Dase. He had no problem with large numbers. In 1844, L. K. Schulz von Strassnitsky hired him to compute pi. Dase ran the job in his brain for almost two months. At the end of that time he wrote down pi correctly to the first two hundred decimal places—then a world record.
To many mathematicians, mathematical objects such as a circle seem to exist in an external, objective reality. Numbers, as well, seem to have a reality apart from time or the world. Numbers seem to transcend the universe. Numbers might even exist if the universe did not. I suspect that in their hearts most working mathematicians are Platonists, in that they accept the notion that mathematical reality stands apart from the world, and is at least as real as the world, and possibly gives shape to the world, as Plato suggested. Most mathematicians would probably agree that the ratio of the circle to its diameter exists luminously and eternally in the nature beyond nature, and would exist even if the human mind was not aware of it. Pi might exist even if God had not bothered to create it. One could imagine that pi existed before the universe came into being and will exist after the universe is gone. Pi may even exist apart from God. This is in the opinion of some mathematicians, anyway, who would argue that while there is at least some reason to doubt the existence of God, there is no good reason to doubt the existence of the circle.
“To an extent, pi is more real than the machine that is computing it,” Gregory remarked to me one day. “Plato was right. I am a Platonist. Since pi is there, it exists. What we are doing is really close to experimental physics—we are ‘observing pi.’ Observing pi is easier than studying physical phenomena, because you can prove things in mathematics, whereas you can’t prove anything in physics. And, unfortunately, the laws of physics change once every generation.”
“Is mathematics a form of art?” I asked.
“Mathematics is partially an art, even though it is a natural science,” he said. “Everything in mathematics does exist now. It’s a matter of naming it. The thing doesn’t arrive from God in a fixed form; it’s a matter of representing it with symbols. You put it through your mind to make sense of it.”
Pi is elusive and can be approached only through approximations. There is no equation built from whole numbers that will give an exact value for pi. If equations are trains threading the landscape of numbers, no train stops at pi. A formula that heads toward pi will never get there, though it can get ever closer to pi. It will consist of a chain of operations that never ends. It is an infinite series. In 1674, Gottfried Wilhelm Leibniz (the coinventor of calculus, along with Isaac Newton) discovered an extraordinary pattern of numbers buried in the circle. This string of numbers—the Leibniz series for pi—has been called one of the most beautiful mathematical discoveries of the seventeenth century:
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 a
pparent 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.