At the time I first got to know the Chudnovsky brothers, they had built a powerful supercomputer out of mail-order parts. It filled the living room of Gregory’s apartment at the time, on 120th Street in Morningside Heights, near Columbia University. They said the machine didn’t really have a name, but they told me I could refer to it as “m zero,” in order to have something to call it. Gregory was then living in the apartment with his wife, Christine, who was an attorney at a midtown firm, and his mother, Malka Benjaminovna Chudnovsky, who was in declining health. Malka passed away in 2001. The Chudnovsky brothers were using their homemade supercomputer to calculate the number pi, or π, to beyond two billion decimal places. Pi is the ratio of the circumference of a circle to its diameter, and is one of the most mysterious numbers in mathematics. Expressed in digits, pi begins 3.14159…and runs on to an infinity of digits that never repeat. Though pi has been known for more than three thousand years, mathematicians have been unable to learn much about it. The digits show no predictable order or pattern. The Chudnovskys had been hoping, very faintly, that their supercomputer might see one. However, the pattern in pi may be too complex and subtle for the human mind to grasp or for any supercomputer to find. In any event, the supercomputer used a lot of electricity. In the summer, it heated Gregory’s apartment to above a hundred degrees Fahrenheit, so the brothers installed twenty-six fans around it to cool it down. The building superintendent had no idea that the brothers were investigating pi in Gregory’s apartment.
While this was going on, neither of the brothers had a permanent academic job. They were untenured senior research scientists at Columbia and were getting along on grants and consulting fees, and their wives were also contributing to the family income. Their employment problem was complex: they were a pair, yet they would need to fit into a math department as a single faculty member. In addition, they were using computers, an activity that some mathematicians regarded as unclean. And Gregory was unable to live anywhere except in a room where the air is purified with HEPA filters. (He suffered from allergies that could prove life-threatening.) He would require special care and arrangements from a math department, and it wasn’t clear how much teaching he’d be able to do.
One day the Chudnovskys were approached by a man named Jeffrey H. Lynford, who was the CEO of Wellsford Real Properties, a real estate investment firm. He and his wife, Tondra, had become fascinated with the Chudnovsky Problem and had become determined to try to solve it somehow—that is, to find jobs for the brothers. Jeff Lynford proposed trying to raise money to endow a chair of mathematics for the Chudnovskys at a university somewhere. In the end, after several years of fund-raising efforts, Jeff and Tondra Lynford gave $400,000 to Polytechnic University. This gift, along with others, including two gifts from a Dallas businessman named Morton H. Meyerson and a gift from the Wall Street investor John Mulheren, ended up being enough to partially endow the Institute for Mathematics and Advanced Supercomputing. (John Mulheren died in 2003.) Having an institute put the brothers on a more stable footing. Gregory and Christine moved to a specially modified apartment that had filtered air, in Forest Hills, and in 1999 they had a daughter, Marian.
At IMAS, the brothers set about building a new series of computers of Chudnovskian design. The latest of them was a powerful machine of a type called a cluster of nodes. It was similar in design to their original machine, m zero, but was very much more powerful. (The most powerful supercomputer today is tomorrow’s desktop machine.) The brothers ordered the parts for their new “supercomputer” through the mail. It sat inside a framework made of metal closet racks and white plastic plumbing pipes, and the structure was covered with window screens—those parts of the machine came from Home Depot. The brothers referred to their computer cluster modestly as “nothing.” Alternatively, they called it “the Home Depot thing.” “To be honest, we really call it It,” Gregory explained. “This is because It doesn’t exactly have a name.” They became interested in using It or the Home Depot thing to crack problems that had proved difficult, such as assembling large DNA sequences or making high-resolution 3-D images of works of art.
To try to make a perfect digital image of the Unicorn Tapestries sounded like an interesting problem to David and Gregory Chudnovsky. One day, David paid a visit to the Met. He left the Met carrying seventy CDs of the Unicorn Tapestries. He and Gregory planned to feed the data into the Home Depot thing and try to join the tiles together into seamless images of the tapestries. The images would be the largest and most complex digital photographs of any artwork ever made. “This will be easy,” David said to Barbara Bridgers as he left. He was wrong.
* * *
“WE THOUGHT TO OURSELVES that it would be just a bit of number crunching,” Gregory said.
But, David said, “it wasn’t trivial.”
The brothers had a fairly easy time setting up the tiles on the Home Depot thing. When they tried to fit the puzzle together, however, they found that the pieces wouldn’t join properly; the warp and weft threads didn’t run smoothly from one tile to the next. The differences were vast. It was as if a tapestry had not been the same object from one moment to the next as it was being photographed. Sutures were visible. The result was a sort of Frankenstein version of the Unicorn Tapestries. The Chudnovskys had no idea why.
David, in exasperation, called up Barbara Bridgers. “Somebody has been fooling around with these numbers,” he told her.
“I don’t think so, David. Nobody around here could do that.”
David informed her that he and Gregory would need to obtain the complete set of raw data from the Leica camera. The next day, he went to the museum and collected, from Bridgers, two large blue Metropolitan Museum shopping bags stuffed with more than two hundred CDs, containing every number the Leica had collected from the Unicorn Tapestries. There were at least a hundred billion numbers in the shopping bags.
David took the subway back to Brooklyn, stopping off at a supermarket to buy some fruit. In the lab, he put his things down, and Gregory began going through them. “Where are the rest of the CDs?” he asked. One of the Metropolitan Museum bags was missing.
“My God! I left it on the subway,” David said. Half the Unicorn Tapestries could have been anywhere between the Upper Bronx and Far Rockaway.
They began frantically calling the subway’s lost and found. “Naturally, there was no answer,” Gregory recalled. David retraced his route. He found the Met bag sitting under the lettuce bin at the supermarket. Apart from being slightly misted, the CDs were okay.
Then the brothers really began to dig into the numbers. Working with Tom Morgan, they created something called a vector field, and they used it to analyze the inconsistencies in the images.
The tapestries, they realized, had changed shape as they were lying on the floor and being photographed. They had been hanging vertically for centuries; when they were placed on the floor, the warp threads relaxed. The tapestries began to breathe, expanding, contracting, shifting. It was as if, when the conservators removed the backing, the tapestries woke up. The threads twisted and rotated restlessly. Tiny changes in temperature and humidity in the room caused the tapestries to shrink or expand from hour to hour, from minute to minute. The gold-and silver-wrapped threads changed shape at different speeds and in different ways from the wool and silk threads.
“We found out that a tapestry is a three-dimensional structure,” Gregory went on. “It’s made from interlocked loops of wool.”
“The loops move and change,” David said.
“The tapestry is like water,” Gregory said. “Water has no permanent shape.”
The photographers had placed a thin sheet of gray paper below the edge of the part of the tapestry they were shooting. Each time they moved the camera, they also moved the sheet of paper. Though the paper was smooth and thin, it tugged the tapestry slightly as it moved, creating ripples. It stretched the weft threads and rotated the warp threads—it resonated through the tapestry. All this made the tiles impossible
to join without the use of higher mathematics and the Home Depot thing.
A color digital photograph is composed of pixels. A pixel is the smallest picture element that contains color. The Unicorn Tapestries are themselves made up of the medieval equivalent of pixels—a single crossing of warp and weft is the smallest unit of color in the image. The woven pixels were maddening because they moved constantly. The brothers understood, at last, that it would be necessary to perform vast seas of calculations upon each individual pixel in order to make a complete image of a tapestry. Each pixel had to be calculated in its relationship to every other nearby pixel, a mathematical problem known as an N-problem, big enough to practically choke It. They decided to concentrate on just one of the tapestries, “The Unicorn in Captivity.” Gregory said, “This was a math problem similar to the analysis of DNA or speech recognition—”
“Look, my dear fellow, it was a real nightmare,” David said.
Two of the tiles on the front of “The Unicorn in Captivity” had an eerie green tinge. While the photographers were shooting them, someone had opened a door leading to the next room, where a fluorescent light was on, causing a subtle flare. The Chudnovskys corrected the lighting by using the color on the back threads as a reference. “It took us three months of computation,” Gregory said. “We should have just dropped it.”
The final assembly of the image took twenty-four hours inside the nodes of It, the Home Depot thing. Gregory and David stayed up all night and ran It from their respective apartments. In the preceding months, each pixel in “The Unicorn in Captivity” had been crunched through many billions of calculations. That last night, there were billions more calculations. By sunrise, the machine had recaptured “The Unicorn in Captivity” in its entirety. The image was flawless.
* * *
ONE DAY IN AUTUMN, my wife and our three children and I went to Brooklyn and paid a visit to the Chudnovskys at IMAS, which is in Rogers Hall, on the Polytechnic campus. David met us in the lobby. He wore a starched white shirt, dark slacks, and Hush Puppies. We were joined by Tom Morgan, a quiet man in his fifties with blue eyes, gold-rimmed spectacles, and a ponytail. He handed us disposable booties, of the kind worn by medical people in operating rooms. Then we went in.
The IMAS lab was a large, loftlike industrial room, with computer-controlled shades and lights, and filtered air. The lights were dim. The walls were concrete and painted white. The brothers projected images on the walls, and they also used the walls as a whiteboard to perform calculations with erasable markers. The walls were covered with scribbles—work in progress. Most of the floor consisted of a vast digital image, in color, showing 115 different equations arranged in a vast spiral that breaks up into waves near the walls—a whirlpool of mathematics.
The equations were a type known as a hypergeometric series. Among other things, they rapidly produce the digits of pi. The Chudnovskys discovered most of them; others were found by the great Indian mathematician Srinivasa Ramanujan, in the early twentieth century, and by Leonhard Euler, in the eighteenth century. On one corner of the floor there was a huge digital image of Albrecht Dürer’s engraving Melencolia I. In it, Melancholy is sitting lost in thought surrounded by various strange objects, including a magic square and a polyhedron with eight sides, called Dürer’s solid. The Chudnovskys suspected that Dürer’s solid is more curious mathematically than meets the eye.
Gregory Chudnovsky was half lying on the couch in his stocking feet, his body extended, facing the figure of Melancholy. His shoes, which were tucked inside surgical booties, had been left on the floor. He wore jeans and a soft leather jacket, and he seemed relaxed. Christine and Marian, who was five, were there. Marian was chattering and running around the lab happily. The effect of the child circling over her father’s swirling equations was slightly vertiginous.
“At first, we were going to cover the entire floor with Melencolia, but it made people dizzy,” Gregory said. “It made us dizzy, too. So we shrank it and moved it near the couch.”
Close to the windows stood the cluster of bare computers, sitting inside the frame of plumbing pipes and covered with window screens—It. There was a sound of many small whirring fans running inside It, keeping It cool. (I associate this sound with any room professionally occupied by the Chudnovskys.)
David and Gregory Chudnovsky in their laboratory at the Institute for Mathematics and Advanced Supercomputing (IMAS). The shadow profile behind them is that of Tom Morgan, their collaborator.
Dudley Reed
My daughter Marguerite, who was fifteen at the time, wanted to know which of the many equations in the floor was the one that the brothers had used to calculate pi with their previous supercomputer.
“Walk this way,” David said to her. “Now you are standing on the equation.”
She looked down. The equation swooped for a yard under her feet.
Because the Chudnovsky equation for pi is the most powerful and accurate formula for pi that’s ever been found, it is also the most nearly perfect representation of pi known to humanity, other than a symbol. Whether some hidden order exists in pi is still something unknown to humanity.
At the far end of the room hung two thirteen-foot-tall sheets of cloth, mounted at right angles to each other, that displayed perfect digital images of, respectively, the front and back of “The Unicorn in Captivity.” We walked up to the two pictures of the unicorn. First I looked at the front. I could see each thread clearly. The unicorn is spattered with droplets of red liquid that seems to be blood, although it may be pomegranate juice dripping from fruit in the tree. The threads in the droplets of blood are so deftly woven that they create an illusion that the blood is semitransparent. The white coat of the unicorn shines through.
Then I turned to the back of the tapestry. Here the droplets were a more intense red, with clearer highlights, and they seemed to jump out at the eye. The leaves of the flowers were a vibrant, plantlike green. (There are as many as twenty species of flowers in this tapestry. They are depicted with great scientific accuracy—greater than in any of the botany textbooks of the time. They include English bluebells, oxlip, bistort, cuckoopint, and Madonna lily. Botanists haven’t been able to identify a few; it’s possible that they are flowers that have gone extinct since 1500.) On the front, in contrast, the yellow dye in the green leaves has faded a bit, leaving them looking slightly bluish gray.
Gregory got up from the couch. David warned him to be careful, and he put his arm around Gregory’s waist, while Gregory leaned on David and put his arm over David’s shoulders. Then the Chudnovsky Mathematician moved slowly across the floor, until the brothers were standing (rather precariously) beside It. David explained that their image of the tapestry was a first step toward making even finer digital images of works of art. He said, “It’s simple to take a picture of a Vermeer, but what you really want is an image of the painting in 3-D, with a resolution better than fifty microns”—that’s about half the thickness of a human hair. “Then you can see the brushstrokes,” he went on, raising his voice over the whirring of the fans inside It. “You can catalog the brushstrokes in the sequence they occurred, as they were laid down on top of one another.”
The eye of the unicorn, in reverse, on the back side of the tapestry. Detail from “The Unicorn in Captivity,” South Netherlandish, ca. 1495–1505. Wool warp, wool, silk, silver, and gilt wefts; 12 ft. 1 in. × 99 in. (368 cm. × 251.5 cm.). The Metropolitan Museum of Art, gift of John D. Rockefeller, Jr., 1937 (37.80.6).
Image © Metropolitan Museum of Art
When mathematicians work, they engage in intensely serious play. They follow their curiosity into problems that interest them. After playing with the unicorn, the Chudnovskys moved on.
“What are you doing now?” I asked.
David told me that they were working with IBM to design what may be the world’s most powerful supercomputer. The machine, code-named C64, was being built for a United States government agency.
Among their projects, the Chudnov
skys seemed to be involved with something called dark mathematics. Dark mathematics is classified mathematics. It’s mathematics with national security implications that is done for government agencies. It isn’t published in journals. Dark mathematics involves things like codes.
The superpowerful Chudnovskian C64 was rather like It, multiplied many times over, though nothing in C64 would come from Home Depot. When the government machine was finished, it would contain two million processors and fourteen thousand hard drives. It would use two and a half million watts of electricity—enough to power a few thousand homes. Two thousand gallons of water per minute would flow through the core of C64 to keep it cool. If the pumps failed, it would melt down in less than ten seconds.
* * *
ONE DAY, I went to see the Unicorn Tapestries in the physical universe, as distinct from the universe of numbers. It was a quiet winter afternoon at the Cloisters. The gallery where the tapestries hang was almost deserted. When I looked at them, each flower and plant, each animal, each human face took on a character of its own. The tapestries were alive with color and detail, full of velvety pools and shimmering surfaces. In the fence that surrounds the captive unicorn, tarnished silver, mixed with gold, gleamed in the grain of the wood. In comparison, the digital images, good and accurate as they were, had seemed flat. They had not captured the translucent landscape of the Unicorn Tapestries, as the weft threads dive around the warp, or the way they seemed to open into a world beyond the walls of the room.
Timothy Husband, the curator of the Cloisters, walked in. He was a tall, polished man in his late fifties, and had been at the Cloisters for thirty-five years. We sat down in one of the window seats facing the tapestries. “There is a luminosity and depth in them,” he said quietly. “It didn’t come about by chance on the part of the weavers.”
I asked Husband how he felt when he was alone with the tapestries.
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