In
1998, the Cloisters—the museum of medieval art in upper Manhattan—began
a renovation of the room where the seven tapestries known as “The Hunt
of the Unicorn” hang. The Unicorn tapestries are considered by many to
be the most beautiful tapestries in existence. They are also among the
great works of art of any kind. In the tapestries, richly dressed
noblemen, accompanied by hunters and hounds, pursue a unicorn through
forested landscapes. They find the animal, appear to kill it, and bring
it back to a castle; in the last and most famous panel, “The Unicorn in
Captivity,” the unicorn is shown bloody but alive, chained to a tree
surrounded by a circular fence, in a field of flowers. The tapestries
are twelve feet tall and up to fourteen feet wide (except for one,
which is in fragments). They were woven from threads of dyed wool and
silk, some of them gilded or wrapped in silver, around 1500, probably
in Brussels or Liège, for an unknown person or persons, and for an
unknown reason—possibly to honor a wedding. A monogram made from the
letters “A” and “E” is woven into the scenery in many places; no one
knows what it stands for. The tapestries’ meaning is mysterious: the
unicorn was a symbol of many things in the Middle Ages, including
Christianity, immortality, wisdom, lovers, marriage. For centuries, the
tapestries were in the possession of the La Rochefoucauld family of
France. In 1922, John D. Rockefeller, Jr., bought them for just over a
million dollars, and in 1937 he gave them to the Cloisters. Their
monetary value today is incalculable.
As the
construction work got under way, the tapestries were rolled up and
moved, in an unmarked vehicle and under conditions of high security, to
the Metropolitan Museum of Art, which owns the Cloisters. They ended up
in a windowless room in the museum’s textile department for cleaning
and repair. The room has white walls and a white tiled floor with a
drain running along one side. It is exceedingly clean, and looks like
an operating room. It is known as the wet lab, and is situated on a
basement level below the museum’s central staircase.
In
the wet lab, a team of textile conservators led by a woman named
Kathrin Colburn unpacked the tapestries and spread them out face down
on a large table, one by one. At some point, the backs of the
tapestries had been covered with linen. The backings, which protect the
tapestries and help to support them when they hang on a wall, were
turning brown and brittle, and had to be replaced. Using tweezers and
magnifying lenses, Colburn and her team delicately removed the threads
that held each backing in place. As the conservators lifted the backing
away, inch by inch, they felt a growing sense of awe. The backs were
almost perfect mirror images of the fronts, but the colors were
different. Compared with the fronts, they were unfaded: incredibly
bright, rich, and deep, more subtle and natural-looking. The backs of
the tapestries had, after all, been exposed to very little sunlight in
five hundred years. Nobody alive at the Met, it seems, had seen them
this way.
A tapestry is woven from lengths of colored
thread called the weft, which are passed around long, straight, strong
threads called the warp. The warp runs horizontally, and provides a
foundation for the delicate weft, which runs vertically. Medieval
tapestry weavers worked side by side, in teams, using their fingertips
and small tools to draw the weft around the warp. When they switched
from one color to the next, they cut off the ends of the weft threads
or wove them into the surface of the tapestry. The Unicorn weavers had
been compulsively neat. In less well-made tapestries, weavers left weft
threads dangling in a shaggy sort of mess, but the backs of these were
almost smooth. Kathrin Colburn recalls that as she and her associates
stared into the backs of the Unicorn tapestries it “felt like a great
exploration of the piece.” She said, “We simply got carried away,
seeing how the materials were used—how beautifully they were dyed and
prepared for weaving.” An expert medieval weaver might need an hour to
complete one square inch of a tapestry, which meant that in a good week
he might finish a patch maybe eight inches on a side. The weavers were
generally young men, and each Unicorn tapestry likely had a team of
between four and six working on it. They wove only by daylight, to
insure that the colors were consistent and not distorted by
candlelight. One tapestry would have taken a team at least a year to
complete.
The curator in charge of medieval art at the
Metropolitan and the Cloisters is a thoughtful man named Peter Barnet.
When he heard about the discovery, he hurried down to the wet lab for a
look. He got a shock. “The first of the tapestries—‘The Start of the
Hunt’—was lying in a clear, shallow pool of water,” Barnet said. The
lab is designed to function as a big tub, and had been filled about six
inches deep with purified water to bathe the tapestry. “Intellectually,
I knew the colors wouldn’t bleed, but the anxiety of seeing a Unicorn
tapestry underwater is something I’ll never forget,” he said. When
Barnet looked at the image through the water, he said, “the tapestry
seemed to be liquefied.” Once the room had been drained, it smelled
like a wet sweater.
Philippe de Montebello, the director
of the museum, declared that the Unicorn tapestries must be
photographed on both sides, to preserve a record of the colors and the
mirror images. Colburn and her associates would soon put new backing
material on them, made of cotton sateen. Once they were rehung at the
Cloisters, it might be a century or more before the true colors of the
tapestries would be seen again.
The manager of the
photography studio at the Met is a pleasant, lively woman named Barbara
Bridgers. Her goal is to make a high-resolution digital image of every
work of art in the Met’s collections. The job will take at least
twenty-five years; there are between two and two and a half million
catalogued objects in the Met—nobody knows the exact number. (One
difficulty is that there seems to be an endless quantity of scarab
beetles from Egypt.) But, when it’s done and backup files are stored in
an image repository somewhere else, then if an asteroid hits New York
the Metropolitan Museum may survive in a digital copy.
To
make a digital image of the Unicorn tapestries was one of the most
difficult assignments that Bridgers had ever had. She put together a
team to do it, bringing in two consultants, Scott Geffert and Howard
Goldstein, and two of the Met’s photographers, Joseph Coscia, Jr., and
Oi-Cheong Lee. They built a giant metal scaffolding inside the wet lab,
and mounted on it a Leica digital camera, which looked down at the
floor. The photographers were forbidden to touch the tapestries;
Kathrin Colburn and her team laid each one down, underneath the
scaffold, on a plastic sheet. Then the photographers began shooting.
The camera had a narrow view; it could photograph only one
three-by-three-foot section of tapestry at a time. The photographers
took overlapping pictures, moving the camera on skateboard wheels on
the scaffolding. Each photograph was a tile that would be used to make
a complete, seamless mosaic of each tapestry.
Joe Coscia
said that his experience with the Unicorn tapestries was incomparable:
“It was really quiet, and I was often alone with a tapestry. I really
got a sense that, for a short while, the tapestry belonged to me.” For
his part, Oi-Cheong Lee felt his sense of time dissolve. “The time we
spent with the tapestries was nothing—only a moment in the life of the
tapestries,” he said.
It took two weeks to photograph the
tapestries. When the job was done, every thread in every tile was
crystal-clear, and the individual twisted strands that made up
individual threads were often visible, too. The data for the digital
images, which consisted entirely of numbers, filled more than two
hundred CDs. With other, smaller works of art, Bridgers and her team
had been able to load digital tiles into a computer’s hard drives and
memory, and then manipulate them into a complete mosaic—into a seamless
image—using Adobe Photoshop software. But with the tapestries that
simply wouldn’t work. When they tried to assemble the tiles, they found
that the files were too large and too complex to manage. “We had to
lower the resolution of the images in order to fit them into the
computers we had, and it degraded the images so much that we just
didn’t think it was worth doing,” Bridgers said. Finally, they gave up.
Bridgers stored the CDs on a shelf and filed the project away as an
unsolved problem.
In
1992, I wrote in this magazine about two mathematicians named Gregory
and David Chudnovsky. The Chudnovskys, who are brothers, were born in
Kiev. They are number theorists—they investigate the properties of
numbers—and they design and work with supercomputers. The Chudnovsky
brothers insist that they are functionally one mathematician who
happens to occupy two human bodies. Currently, the Chudnovsky
Mathematician works at the Institute for Mathematics and Advanced
Supercomputing, or imas, which operates out of a laboratory room at Polytechnic University, in downtown Brooklyn. imasis essentially the Chudnovskys.
Gregory
Chudnovsky is a frail man in his early fifties, with longish hair and a
beard that are going gray, and sensitive, flickering brown eyes. His
health is uncertain. He has myasthenia gravis, a condition that he
developed in his teen-age years and that keeps him in bed or in a
wheelchair much of the time. David is five years older than Gregory. He
is a genial man, somewhat on the portly side, with a cultivated manner,
and he has curly graying hair and pale-blue eyes, which can have a look
of sadness in them.
At the time I wrote about 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, near Columbia University. Gregory was living
there with his wife, Christine, who is an attorney at a midtown firm,
and his mother, Malka Benjaminovna Chudnovsky. (She died in 2001.)
David lives on the Upper West Side with his wife, Nicole, who works for
the United Nations. The Chudnovsky brothers were using their homemade
supercomputer to calculate the number pi, or ?, to beyond two billion
decimal places. Pi is the ratio of the circumference of a circle to its
diameter. It is one of the most mysterious numbers in mathematics.
Expressed in digits, pi begins 3.14159 . . . , and it 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 were 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 are a pair, yet they would need to fit into a math
department as a single faculty member. In addition, they use computers,
which some mathematicians regard as unclean. And Gregory is unable to
live anywhere except in a room where the air is purified with hepa
filters. (He suffers from allergies that could prove life-threatening.)
He would require special care and arrangements by a math department,
and it wasn’t clear how much teaching he’d be able to do.
Shortly
after my article was published, the Chudnovskys were approached by a
man named Jeffrey H. Lynford, who is the C.E.O. of Wellsford Real
Properties, a real-estate investment firm. Lynford proposed trying to
raise money to endow a chair of mathematics for the Chudnovskys at a
university. In the end, after several years of trying, Lynford and his
wife, Tondra, gave four hundred thousand dollars to Polytechnic
University, and this gift, along with others, was enough to partially
endow imas. The job put the brothers on
a more stable footing. Gregory and Christine moved to a specially
modified apartment that has 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 these is a powerful machine of a
type called a cluster of nodes. The brothers ordered the parts through
the mail. It sits inside a framework made of metal closet racks and
white plastic plumbing pipes, and the structure is covered with window
screens—those parts of the machine came from Home Depot. The brothers
refer to their computer cluster modestly as “nothing.” Alternatively,
they call 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 to crack problems that had
proved difficult, such as assembling large DNA sequences or making
high-resolution 3-D images of works of art.
One day in
the spring of 2003, David and Nicole Chudnovsky were having lunch at
the Bedford Hills estate of Errol Rudman, a hedge-fund manager and a
patron of the Metropolitan Museum, and his wife, Diana. Walter Liedtke,
the curator of European paintings at the Met, was there with his wife,
Nancy, who is a math teacher. David began talking about digital
imagery. Walter Liedtke, who is a Rembrandt scholar, felt a little out
of his depth—“I had the illusion that I actually understood it,” he
said. “But this was pearls before swine.” Liedtke decided to put David
in touch with the Met’s photographers. Not long afterward, David, along
with Tom Morgan, a Ph.D. candidate who works with the Chudnovskys,
visited Barbara Bridgers in the Met’s photography studio. Bridgers told
them, “I have a real-world problem for you.”
David left
the Met carrying seventy of the CDs of the Unicorn tapestries. He and
Gregory planned to feed the data into It 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 art work ever
made, for the time. “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 It. When they
tried to fit the puzzle pieces together, however, they 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 said to her.
“I don’t think so, David. Nobody around here could do that.”
David
informed her that the brothers 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
that 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 down his things, and Gregory began going
through them. “Where are the rest of the CDs?” he asked David. 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 on the B.M.T. 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 O.K.
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 had woken up. The threads twisted and rotated restlessly.
Tiny changes in temperature and humidity in the room had 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 It.
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.
“This
is like forensics,” Gregory explained. “If the photographers had
touched it, we would have seen it in the numbers. The camera was also
moving vertically and horizontally a little bit. This made the sizes of
the weaves not quite right from place to place. The camera lens itself
distorted it a little bit.”
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 apparently 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. 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 last fall, 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. David
said, “The booties are for the sake of protecting the floor,” and he
explained that the floor of imas consists of digital images embedded in a soft plastic material. Then we went in.
The imas
lab is a large, loftlike industrial room, with computer-controlled
shades and lights, and filtered air. The lights were dim. The walls are
concrete and painted white. The brothers project images on the walls,
and they also use 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 a hundred and fifteen different equations arranged in a
vast spiral that breaks up into waves near the walls—a whirlpool of
mathematics.
The equations are 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 is 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 an unknown number of sides, called
Dürer’s solid. The Chudnovskys suspect 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
is 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.)
My
daughter Marguerite, who is fifteen, 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.
At
the far end of the room hung two thirteen-foot-tall sheets of cloth,
mounted at right angles to each other, which 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, which 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 semi-transparent. 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.” Fifty microns is 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
catalogue the brushstrokes in the sequence they occurred, as they were
laid down on top of one another.”
Mathematicians, when
they work, engage in intensely serious play. They follow their
curiosity into problems that interest them and toward the smell of a
solution. After playing with the unicorn, the Chudnovskys moved on.
“What are you doing now?” I asked.
David
told me that they were working with I.B.M. to design what may be the
world’s most powerful supercomputer. The machine, code-named C64, is
being built for a United States government agency. It’s rather like It,
multiplied many times over, though nothing in C64 will come from Home
Depot. When the machine is finished, it will contain two million
processors and fourteen thousand hard drives. It will use two and a
half million watts of electricity—enough to power a few thousand homes.
Two thousand gallons of water per minute will flow through the core of
C64 to keep it cool. If the pumps fail, it will 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
full of velvety pools and shimmering surfaces, alive with color and
detail. 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 is a tall,
polished man in his late fifties, and has 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.
“That
happens on Mondays, when the Cloisters is closed,” he said. He spends
anywhere from a minute to an hour with the tapestries. “It can be an
exceedingly frustrating experience. One ponders so many questions about
the tapestries for which there are no more answers today than there
were when I was in graduate school.” In some of the scenes, the unicorn
may represent Christ. Alive and chained to the tree, after its apparent
death in the hunt, it may speak of the immortality of the soul. Or the
drops of blood may represent the pains of love. The truth is that the
modern world has lost touch with the meanings in the Unicorn
tapestries. “Sometimes I come in here and try to pretend I have never
read anything about them, never heard anything about them, and I just
try to look at them,” Husband said. “But it’s not easy to shed that
baggage, is it? And my other reaction, sometimes, is just to say, ‘To
hell with it, someday someone will figure them out.’ And then there is
a solace in their beauty, and one can stare at them in pure amazement.”