*an article from the New Yorker, March 2, 1992.*

- GREGORY VOLFOVICH CHUDNOVSKY recently built a
- supercomputer in his apart-
- ment from mail-order parts. Greg-
- ory Chudnovsky is a number theo-
- rist. His apartment is situated near
- the top floor of a run-down build-
- ing on the West Side of Manhat-
- tan, in a neighborhood near Co-
- lumbia University. Not long ago,
- a human corpse was found dumped
- at the end of the block. The world's
- most powerful supercomputers in-
- clude the Cray Y-MP C90, the
- Thinking Machines CM-5, the
- Hitachi S-820/80, the nCube, the
- Fujitsu parallel machine, the
- Kendall Square Research parallel
- machine, the NEC SX-3, the
- Touchstone Delta, and Gregory
- Chudnovskv's apartment. The
- apartment seems to be a kind of con-
- tainer for the supercomputer at least
- as much as it is a container for
- people.
- Gregory Chudnovsky's partner in
- the design and construction of the
- supercomputer was his older brother,
- David Volfovich Chudnovsky, who is
- also a mathematician, and who lives
- five blocks away from Gregory. The
- Chudnovsky brothers call their ma-
- chine m zero. It occupies the former
- living room of Gregory's-apartment,
- and its tentacles reach into other rooms.
- The brothers claim that m zero is a
- "true, general-purpose supercomputer,"
- and that it is as fast and powerful
- as a somewhat older Cray Y-MP, but
- it is not as fast as the latest of the
- Y-MP machines, the C90, an ad-
- vanced supercomputer made by Cray
- Research. A Cray Y-MP C90 costs
- more than thirty million dollars. It is
- a black monolith, seven feet tall and
- eight feet across, in the shape of a squat
- cylinder, and is cooled by liquid freon.
- So far, the brothers have spent around
- seventy thousand` dollars on parts for
- their supercomputer, and much of the
- money has come out of their wives'
- pockets.
- Gregory Chudnovsky is thirty-nine
- years old, and he has a spare frame
- and a bony, handsome face. He has a
- long beard, streaked with gray, and
- dark, unruly hair, a wide forehead,
- and wide-spaced brown eyes. He walks
- in a slow, dragging shuffle, leaning on
- a bentwood cane, while his brother,
- David, typically holds him under one
- arm, to prevent him from toppling
- over. He has severe myasthenia gravis,
- an auto-immune disorder of the muscles.
- The symptoms, in his case, are mus-
- cular weakness and difficulty in breath-
- ing. "I have to lie in bed most of the
- time," Gregory once told me. His
- condition doesn't seem to be getting
- better, and doesn't seem to be getting
- worse. He developed the disease when
- he was twelve years old, in the city of
- Kiev, Ukraine, where he and David
- grew up. He spends his days sitting or
- lying on a bed heaped with pillows, in
- a bedroom down the hall from the
- room that houses the supercomputer.
- Gregory's bedroom is filled with pa-
- per; it contains at least a ton of
- paper. He calls the place his junk yard.
- The room faces east, and would be
- full of sunlight in the morning if he
- ever raised the shades, but he keeps
- them lowered, because light hurts
- his eyes.
- You almost never meet one of the
- Chudnovsky brothers without the other.
- You often find the brothers conjoined,
- like Siamese twins, David holding
- Gregory by the arm or under the
- armpits. They complete each other's
- sentences and interrupt each other,
- but they don't look alike. While
- Gregory is thin and bearded, David
- has a stout body and a plump,
- clean-shaven face. He is in his
- early forties. Black-and-gray curly
- hair grows thickly on top of David's
- head, and he has heavy-lidded
- deep-blue eyes. He always wears
- a starched white shirt and, usu-
- ally, a gray silk necktie in a fou-
- lard print. His tie rests on a bulg-
- ing stomach.
- The Chudnovskian supercom-
- puter, m zero, burns two thousand
- watts of power, and it runs day
- and night. The brothers don't dare
- shut it down; if they did, it might
- die. At least twenty-five fans blow
- air through the machine to keep it cool;
- otherwise something might melt. Waste
- heat permeates Gregory's apartment,
- and the room that contains m zero
- climbs to a hundred degrees Fahren-
- heit in summer. The brothers keep the
- apartment's lights turned off as much
- as possible. If they switched on too
- many lights while m zero was run-
- ning, they might blow the apartment's
- wiring. Gregory can't breathe city air
- without developing lung trouble, so he
- keeps the apartment's windows closed
- all the time, with air-conditioners
- running in them during the summer,
- but that doesn't seem to reduce the
- heat, and as the temperature rises in-
- side the apartment the place can smell
- of cooking circuit boards, a sign that
- m zero is not well. A steady stream
- of boxes arrives by Federal Express,
- and an opposing stream of boxes flows
- back to mail-order houses, contain-
- ing parts that have bombed, along
- with letters from the brothers demand-
- ing an exchange or their money back.
- The building superintendent doesn't
- know that the Chudnovsky brothers
- have been using a supercomputer in
- Gregory's apartment, and the broth-
- ers haven't expressed an eagerness to
- tell him.
- The Chudnovskys, between them,
- have published a hundred and fifty-
- four papers and twelve books, mostly
- in collaboration with each other, and
- mostly on the subject of number theory
- or mathematical physics. They work
- together so closely that it is possible to
- argue that they are a single mathema-
- tician&emdash;anyway, it's what they claim.
- The brothers lived in Kiev until 1977,
- when they left the Soviet Union and,
- accompanied by their parents, went to
- France. The family lived there for six
- months, then emigrated to the United
- States and settled in New York; they
- have become American citizens.
- The brothers enjoy an official rela-
- tionship with Columbia University:
- Columbia calls them senior research
- scientists in the Department of Math-
- ematics, but they don't have tenure and
- they don't teach students. They are
- really lone inventors, operating out of
- Gregory's apartment in what you might
- call the old-fashioned Russo-Yankee
- style. Their wives are doing well.
- Gregory's wife, Christine Pardo Chud-
- novsky, is an attorney with a midtown
- law firm. David's wife, Nicole Lanne-
- grace, is a political-affairs officer at
- the United Nations. It is their salaries
- that help cover the funding needs of
- the brothers' supercomputing complex
- in Gregory and Christine's apart-
- ment. Malka Benjaminovna Chud-
- novsky, a retired engineer, who is
- Gregory and David's mother, lives in
- Gregory's apartment. David spends his
- days in Gregory's apartment, taking
- care of his brother, their mother, and
- m zero.
- When the Chudnovskys applied to
- leave the
- Soviet Union, the fact that
- they are Jewish and mathematical
- attracted at least a dozen K.G.B. agents
- to their case. The brothers' father, Volf
- Grigorevich Chudnovsky, who was
- severely beaten by the K.G.B. in 1977,
- died of heart failure in 1985. Volf
- Chudnovsky was a professor of civil
- engineering at the Kiev Architectural
- Institute, and he specialized in the
- structural stability of buildings, towers,
- and bridges. He died in America, and
- not long before he died he constructed
- in Gregory's apartment a maze of book-
- shelves, his last work of civil engineer-
- ing. The bookshelves extend into ev-
- ery corner of the apartment, and today
- they are packed with literature and
- computer books and books and papers
- on the subject of numbers. Since almost
- all numbers run to infinity (in digits)
- and are totally unexplored, an apart-
- mentful of thoughts about numbers
- holds hardly any thoughts at all, even
- with a supercomputer on the premises
- to advance the work.
- The brothers say that the "m" in
- "m zero" stands for "machine," and
- that they use a small letter to imply that
- the machine is a work in progress.
- They represent the name typographi-
- cally as "mO." The "zero" stands for
- success. It implies a dark history of
- failure&emdash;three duds (in Gregory's apart-
- ment) that the brothers now refer to
- as negative three, negative two, and
- negative one. The brothers broke up
- the negative machines for scrap, got on
- the telephone, and waited for Federal
- Express to bring more parts.
- M zero is a parallel supercomputer,
- a type of machine that has lately come
- to dominate the avant-garde in super-
- computer architecture, because the
- design offers succulent possibilities for
- speed in solving problems. In a par-
- allel machine, anywhere from half a
- dozen to thousands of processors work
- simultaneously on a problem, whereas
- in a so-called serial machine&emdash;a nor-
- mal computer&emdash;the problem is solved
- one step at a time. "A serial machine
- is bound to be very slow, because the
- speed of the machine will be limited
- by the slowest part of it," Gregory said.
- "In a parallel machine, many circuits
- take on many parts of the problem at
- the same time." As of last week, m zero
- contained sixteen parallel processors,
- which ruminate around the clock on
- the Chudnovskys' problems.
- The brothers' mail-order super-
- computer makes their lives more con-
- venient: m zero performs inhumanely
- difficult algebra, finding roots of gi-
- gantic systems of equations, and it has
- constructed colored images of the in-
- terior of Gregory Chudnovsky's body.
- According to the Chudnovskys, it could
- model the weather or make pictures of
- air flowing over a wing, if the brothers
- cared about weather or wings. What
- they care about is numbers. To them,
- numbers are more beautiful, more nearly
- perfect, possibly more complicated, and
- arguably more real than anything in
- the world of physical matter.
- The brothers have lately been using
- m zero to explore the number pi. Pi,
- which is denoted by the Greek letter
- Pi, is the most famous ratio in math-
- ematics, and is one of the most ancient
- numbers known to humanity. Pi is
- approximately 3.14&emdash;the number of
- times that a circle's diameter will fit
- around the circle. Here is a circle, with
- its diameter:
- Pi goes on forever, and can't
- be calculated to perfect
- precision:
- 3.1415926535897932384626433832
- 795028841971693993751.... This is
- known as the decimal expansion of pi.
- It is a bloody mess. No apparent pat-
- tern emerges in the succession of dig-
- its. The digits of pi march to infinity
- in a predestined yet unfathomable code:
- they do not repeat periodically, seem-
- ing to pop up by blind chance, lacking
- any perceivable order, rule, reason, or
- design&emdash;"random" integers, ad infini-
- tum. If a deep and beautiful design
- hides in the digits of pi, no one knows
- what it is, and no one has ever been
- able to see it by staring at the digits.
- Among mathematicians, there is a nearly
- universal feeling that it will never be
- possible, in principle, for an inhabitant
- of our finite 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 scrap of leftover pi.
- Before the Chudnovsky brothers built
- m zero, Gregory had to derive pi over
- the telephone network while lying in
- bed. It was inconvenient. Tapping at
- a small keyboard, which he sets on the
- blankets of his bed, he stares at a
- computer display screen on one of the
- bookshelves beside his bed. The key-
- board and the screen are connected to
- Internet, a network that leads Gregory
- through cyberspace into the heart of a
- Cray somewhere else in the United
- States. He calls up a Cray through
- Internet and programs the machine to
- make an approximation of pi. The job
- begins to run, the Cray trying to es-
- timate the number of times that the
- diameter of a circle goes around the
- periphery, and Gregory sits back on
- his pillows and waits, watching mes-
- sages from the Cray flow across his
- display screen. He eats dinner with his
- wife and his mother and then, back in
- bed, he takes up a legal pad and a red
- felt-tip pen and plays
- with number theory,
- trying to discover hid-
- den properties of num-
- bers. Meanwhile, the
- Cray is reaching toward
- pi at a rate of a hundred
- million operations per
- second. Gregory dozes
- beside his computer
- screen. Once in a while,
- he asks the Cray how
- things are going, and
- the Cray replies that the
- job is still active. Night
- passes, the Cray run-
- ning deep toward pi.
- Unfortunately, since the
- exact ratio of the circle's
- circumference to its di-
- ameter dwells at infin-
- ity, the Cray has not
- even begun to pinpoint
- pi. Abruptly, a message
- appears on Gregory's
- screen:
- LINE IS
- DISCONNECTED.
- "What the hell is
- going on?" Gregory ex-
- claims. It seems that
- the Cray has hung up
- the phone, and may
- have crashed. Once
- again, pi has demon-
- strated its ability to give
- a supercomputer a heart
- attack.
- MYASTHENIA GRA-
- VIS iS a funny
- thing Gregrory Chudnovsky said one day from his bed in
- the junk yard. "In a sense, I'm very
- lucky, because I'm alive, and I'm alive
- after so many years." He has a reso-
- nant voice and a Russian accent. "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. He lay against a
- heap of pillows, with his legs folded
- under him. He wore a tattered gray
- lamb's-wool sweater that had multiple
- patches on the elbows, and a starched
- white shirt, and baggy blue sweat pants,
- and a 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 the bookshelves, 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 also
- flooded two freestanding bookshelves
- (placed strategically around Gregory's
- bed), five chairs (three of them in a
- row beside his bed), two steamer trunks,
- and a folding cocktail table. I moved
- carefully around the room, fearful of
- triggering a paper slide, and sat on the
- room's one empty chair, facing the foot
- of Gregory's bed, my knees touching
- the blanket. The paper was piled in
- three-foot stacks on the chairs. It
- guarded his bed like the flanking tow-
- ers of a fortress, and his bed sat at the
- center of the keep. I sensed a profound
- happiness in Gregory Chudnovsky's
- bedroom. His happiness, it occurred to
- me later, sprang from the delicious
- melancholy of a life chained to a bed
- in a disordered world that breaks open
- through the portals of mathematics
- into vistas beyond time or decay.
- "The system of this paper is
- archeological," he said. "By
- looking at a slice, I know the
- year. This slice is 1986. Over
- here is some 1985. What you
- see in this room is our work-
- ing papers, as well as the papers
- we used as references for them.
- Some of the references we pull
- out once in a while to look at,
- and then we leave them some-
- where else, in another pile.
- Once, we had to make a Xerox
- copy of a book three times, and
- we put it in three different
- places in the piles, so we would
- be sure to find it when we
- needed it. Unfortunately, once
- we put a book into one of these
- piles we almost never go back
- to look for it. There are books
- in there by Kipling and Macau-
- lay. Actually, when we want
- to find a book it's easier to go
- back to the library. Eh. this
- place is a mess. Eventually, these papers
- or my wife will turn me out of the
- house."
- Much of the paper consists of legal
- pads covered with Gregory's hand-
- writing. His holograph is dense and
- careful, a flawless minuscule written
- with a red felt-tip pen&emdash;a mixture of
- theorems, calculations, proofs, and con-
- lectures concerning numbers. He uses
- a felt-tip pen because he doesn't have
- enough strength in his hand to press
- a pencil on paper. Mathematicians who
- have visited Gregory Chudnovsky's
- bedroom have come away dizzy, won-
- dering what secrets the scriptorium
- may hold. Some say he has published
- most of his work, while others wonder
- if his bedroom holds unpublished dis-
- coveries. He cautiously refers to his
- steamer trunks as valises. They are
- filled to the lids with compressed pa-
- per. When Gregory and David used
- to fly to Europe to speak at conferences,
- they took both "valises" with them, in
- case they needed to refer to a theorem,
- and the baggage particularly annoyed
- the Belgians. "The Belgians were
- always fining us for being overweight,"
- Gregory said.
- Pi is by no means the only unex-
- plored number in the Chudnovskys'
- inventory, but it is one that interests
- them very much. They wonder whether
- the digits contain a hidden rule, an as
- vet unseen architecture. close to the
- mind of God. A subtle and fantastic
- order may appear in the digits of pi
- way out there somewhere; no one
- knows. No one has ever proved, for
- example, that pi does not turn into
- nothing but nines and zeros, spattered
- to infinity in some peculiar arrange-
- ment. 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 may
- ramble forever in a hideous cacophony,
- which is a kind of absolute perfection
- to a mathematician like Gregory Chud-
- novsky. Pi looks "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&emdash;it's just the
- ratio of the circumference to the diam-
- eter&emdash;but the complexity of the se-
- quence 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 cor-
- ner of the room, his head and shoulders
- visible above towers of paper.
- Pi, or 1t, has had various names
- through the ages, and all of them are
- either words or abstract symbols, since
- pi is a number that can't be shown
- completely and exactly in any finite
- form of representation. Pi
- is a transcendental number. A transcendental
- number is a number that exists but
- can't be expressed in any finite series
- of either arithmetical or algebraic op-
- erations. For example, if you try to
- express pi as the solution to an equa-
- tion 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 never repeat
- periodically, as do the digits of a ra-
- tional 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 dis-
- play it in its totality. Ferdinand Lin-
- demann, a German mathematician,
- proved the transcendence of pi in 1882;
- he proved, in effect, that pi can't be
- written on a piece of paper, not even
- on a piece of paper as big as the
- universe. In a manner of speaking, pi
- is indescribable and can't be found.
- Pi possibly first entered human con-
- sciousness in Egypt. The earliest known
- reference to pi occurs in a Middle
- Kingdom papyrus scroll, written around
- 1650 B.C. by a scribe named Ahmes.
- Showing a restrained appreciation for
- his own work that is not uncommon
- in a mathematician, Ahmes began his
- scroll with the words "The Entrance
- Into the Knowledge of All Existing
- Things." He remarked in passing that
- he composed the scroll "in likeness to
- writings made of old," and then he led
- his readers through various mathemati-
- cal problems and their solutions, along
- several feet of papyrus, and toward the
- end of the scroll he found the area of
- a circle, using a rough sort of pi.
- Around 200 B.C., Archimedes of
- Syracuse found that pi is somewhere
- between 3 1O/7l and 3 1/7_that's about
- 3.14. (The Greeks didn't use deci-
- mals.) 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 the known value
- of pi to one part in a hundred. There
- knowledge of pi bogged down until the
- seventeenth century, when new for-
- mulas for approximating pi were dis-
- covered. Pi then came to be called the
- Ludolphian number, after Ludolph van
- Ceulen, a German mathematician who
- approximated it to thirty-five decimal
- places, or one part in a hundred mil-
- lion billion billion billion&emdash;a calcula-
- tion that took Ludolph most of his life
- to accomplish, and gave him such
- satisfaction that he had the digits en-
- graved on his tombstone, at the Ladies'
- Church in Leiden, in the Netherlands.
- Ludolph and his tombstone were later
- moved to Peter's Church in Leiden, to
- be installed in a special graveyard for
- professors, and from there the stone
- vanished, possibly to be turned into a
- sidewalk slab. Somewhere in Leiden,
- people may be walking over Ludolph's
- digits. The Germans still call pi the
- Ludolphian number. In the eighteenth
- century, Leonhard Euler, mathemati-
- cian to Catherine the Great, called it
- p or c. The first person to use the
- Greek letter Pi for the number was
- William Jones, an English mathema-
- tician, who coined it in 1706 for his
- book "A New Introduction to the Math-
- ematics." Euler read the book and
- switched to using the symbol Pi, and
- the number has remained Pi ever since.
- Jones probably meant Pi to stand for the
- English word "periphery."
- Physicists have noted 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 ripples and spectra of all
- kinds, and therefore pi occurs in colors
- and music. Pi has lately turned up in
- superstrings, the hypothetical loops of
- energy vibrating inside subatomic par-
- ticles. 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
- necessarily has to be so.
- Eugene Wigner, the physi-
- cist, once said, "The mir-
- acle of the appropriateness
- of the language of math-
- ematics for the formula-
- tion of the laws of physics
- is a wonderful gift which
- we neither understand nor
- deserve." We may not un-
- derstand pi or deserve it,
- but nature at least seems to
- be aware of it, as Captain
- 0. C. Fox learned while
- he was recovering in a
- hospital from a wound sus-
- tained in the American
- Civil War. Having noth-
- in 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 elev-
- en hundred times, Captain Fox was
- able to derive pi to two places after the
- decimal point, to 3.14. If he had had
- a thousand years to recover from his
- wound, he might have derived pi to
- perhaps another decimal place. To go
- deeper into pi, you need a powerful
- machine.
- The race toward pi happens in
- cyberspace, inside supercomputers. In
- 1949, George Reitwiesner, at the Bal-
- listic Research Laboratory, in Mary-
- land, derived pi to two thousand and
- thirty-seven decimal places with the
- ENIAC, the first general-purpose elec-
- tronic digital computer. Working at
- the same laboratory, John von Neu-
- mann (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 I.B.M. 7090
- mainframe computer, and saw noth-
- ing. The race continued desultorily,
- through hundreds of thousands of digits,
- until 1981, when Yasumasa Kanada,
- the head of a team of computer scien-
- tists at Tokyo University, used a 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, noticing nothing remarkable. A
- year later, William Gosper, a math-
- ematician and distinguished hacker em-
- ployed at Symbolics, Inc., in Sunny-
- vale, California, computed pi
- to seventeen and a half mil-
- lion decimal places with a
- Symbolics workstation, beat-
- ing Kanada's team by a mil-
- lion digits. Gosper saw noth-
- ing of interest.
- The next year, David H.
- Bailey, at the National Aeronautics
- and Space Administration, used a Cray 2
- supercomputer and a formula discov-
- ered by two brothers, Jonathan and
- Peter Borwein, to scoop twenty-nine
- million digits of pi. Bailey found noth-
- ing unusual. A year after that, in 1987,
- Yasumasa Kanada and his team got a
- hundred and thirty-four million digits
- of pi, using a NEC SX-2 supercom-
- puter. They saw nothing of interest.
- In 1988, Kanada kept going, past two
- hundred million digits, and saw fur-
- ther amounts of nothing. Then, in the
- spring of 1989, the Chudnovsky broth-
- ers (who had not previously been known
- to have any interest in calculating pi)
- suddenly announced that they had
- obtained four hundred and eighty mil-
- lion digits of pi&emdash;a world record&emdash;
- using supercomputers at two sites in
- the United States, and had seen noth-
- ing. Kanada and his team were a little
- surprised to learn of unknown compe-
- tition operating in American cyberspace,
- and they got on a Hitachi supercom-
- puter and ripped through five hundred
- and thirty-six million digits, beating
- the Chudnovksys, setting a new world
- record, and seeing nothing. The broth-
- ers kept calculating and soon cracked
- a billion digits, but Kanada's restless
- boys and their Hitachi then nosed into
- a little more than a billion digits. The
- Chudnovskys pressed onward, too, and
- by the fall of 1989 they had squeaked
- past Kanada again, having computed
- pi to one billion one hundred and
- thirty million one hundred and sixty
- thousand six hundred and sixty-four
- decimal places, without finding any-
- thing special. It was another world
- record. At that point, the brothers gave
- up, out of boredom.
- If a billion decimals of pi were
- printed in ordinary type, they would
- stretch from New York City to the
- middle of Kansas. This notion raises
- the question: What is the point of
- computing pi from New York to Kan-
- sas? The question has indeed been
- asked among mathematicians, since an
- expansion of pi to only forty-seven
- decimal places would be sufficiently
- precise to inscribe a circle around the
- visible universe that doesn't
- deviate from perfect circu-
- larity by more than the dis-
- tance across a single proton.
- A billion decimals of pi go so
- far beyond that kind of pre-
- cision, into such a lunacy of
- exactitude, that physicists will
- never need to use the quantity in any
- experiment&emdash;at least, not for any phys-
- ics we know of today&emdash;and the thought
- of a billion decimals of pi oppresses
- even some mathematicians, who de-
- clare the Chudnovskys' effort trivial. I
- once asked Gregory if a certain im-
- pression I had of mathematicians was
- true, that they spent immoderate
- amounts of time declaring each other's
- work trivial. "It is true," he admitted.
- "There is actually a reason for this.
- Because once you know the solution to
- a problem it usually is trivial."
- Gregory did the calculation from his
- bed in New York, working through
- cyberspace on a Cray 2 at the Minne-
- sota Supercomputer Center, in Minne-
- apolis, and on an I.B.M. 3090-VF
- supercomputer at the I.B.M. Thomas J.
- Watson Research Center, in York-
- town Heights, New York. The calcu-
- lation triggered some dramatic crashes,
- and took half a year, because the broth-
- ers could get time on the supercomputers
- only in bits and pieces, usually during
- holidays and in the dead-of night. It
- was also quite expensive&emdash;the use of
- the Cray cost them seven hundred and
- fifty dollars an hour, and the money
- came from the National Science Foun-
- dation. By the time of this agony, the
- brothers had concluded that it would
- be cheaper and more convenient to
- build a supercomputer in Gregory's
- apartment. Then they could crash their
- own machine all they wanted, while
- they opened doors in the house of
- numbers. The brothers planned to
- compute two billion digits of pi on their
- new machine&emdash;to try to double their
- old world record. They thought it
- would be a good way to test their
- supercomputer: a maiden voyage into
- pi would put a terrible strain on their
- machine, might blow it up. Presuming
- that their machine wouldn't overheat
- or strangle on digits, they planned to
- search the huge resulting string of pi
- for signs of hidden order. If what the
- Chudnovsky brothers have seen in the
- Ludolphian number is a message from
- God, the brothers aren't sure what
- God is trying to say.
- ON a cold winter day, when the
- Chudnovskys were about to be-
- gin their two-billion-digit expedition
- into pi, I rang the bell of Gregory
- Chudnovsky's apartment, and David
- answered the door. He pulled the door
- open a few inches, and then it stopped,
- jammed against an empty cardboard
- box and a wad of hanging coats. He
- nudged the box out of the way with his
- foot. "Look, don't worry," he said.
- "Nothing unpleasant will happen to
- you. We will not turn you into digits."
- A Mini Mag-Lite flashlight protruded
- from his shirt pocket.
- We were standing in a long, dark
- hallway. The lights were off, and it
- was hard to see anything. To try to
- find something in Gregory's apartment
- is like spelunking; that was the reason
- for David's flashlight. The hall is
- lined on both sides with bookshelves,
- and they hold a mixture of paper and
- books. The shelves leave a passage
- about two feet wide down the length
- of the hallway. At the end of the
- hallway is a French door, its mul-
- lioned glass covered with translucent
- paper, and it glowed.
- The apartment's rooms are strung
- out along the hallway. We passed a
- bathroom and a bedroom. The bed-
- room belonged to Malka Benjaminovna
- Chudnovsky. We passed a cave of
- paper, Gregory's junk yard. We passed
- a small kitchen, our feet rolling on
- computer cables. David opened the
- French door, and we entered the room
- of the supercomputer. A bare light bulb
- burned in a ceiling fixture. The room
- contained seven display screens: two of
- them were filled with numbers; the
- others were turned off. The windows
- were closed and the shades were drawn.
- Gregory Chudnovsky sat on a chair
- facing the screens. He wore the usual
- outfit&emdash;a tattered and patched lamb's-
- wool sweater, a starched white shirt,
- blue sweat pants, 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. I shook
- his hand. "Our first goal is to compute
- pi," he said. "For that we have to build
- our own computer."
- "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 shells of mail-
- order personal computers&emdash;
- cheap P.C. clones, knocked
- wide open, like cracked wal-
- nuts, their meat spilling all
- over the place. The brothers
- had crammed special logic
- boards inside the personal
- computers. 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 letter-
- ing on their spines), and screwdrivers,
- and data-storage tapes, and software
- manuals by the cubic yard, and stalag-
- mites of obscure trade magazines, and
- a twenty-thousand-dollar computer
- workstation that the brothers no longer
- used. ("We use it as a place to stack
- paper," Gregory said.) From an oval
- photograph on the wall, the face of
- their late father&emdash;a robust man, squint-
- ing thoughtfully&emdash;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: the
- brothers had big plans for m zero.
- Computer disk drives stood around the
- room. The drives hummed, and there
- was a continuous whirr of fans, and
- a strong warmth emanated from the
- equipment, as if a steam radiator were
- going in the room. The brothers heat
- their apartment largely with chips.
- Gregory said, "Our knowledge of pi
- was barely in the millions of digits&emdash;"
- "We need many billions of digits,"
- David said. "Even a billion digits is a
- drop in the bucket. Would you like a
- Coca-Cola?" He went into the kitchen,
- and there was a horrible crash. "Never
- mind, I broke a glass," he called.
- "Look, it's not a problem." He came
- out of the kitchen carrying a glass of
- Coca-Cola on a tray, with a paper
- napkin under the glass, and as he
- handed it to me he urged me to hold
- it tightly, because a Coca-Cola spilled
- into&emdash;He didn't want to think about
- it; it would set back the project by
- months. He said, "Galileo had to build
- his telescope&emdash;"
- "Because he couldn't afford the Dutch
- model," Gregory said.
- "And we have to build our machine,
- because we have&emdash;"
- "No money," Gregory said. "When
- people let us use their computer, it's
- alwavs done as a kindness." He grinned
- and pinched his finger and
- thumb together. "They say,
- 'You can use it as long as
- nobody complains."'
- I asked the brothers
- when they planned to build
- their supercomputer.
- They burst out laughing.
- "You are sitting inside it!"
- David roared.
- "Tell us how a super-
- computer should look," Gregory said.
- I started to describe a Cray to the
- brothers.
- David turned to his brother and
- said, "The interviewer answers our
- questions. It's Pirandello! The inter-
- viewer becomes a person in the story."
- David turned to me and said, "The
- problem is, you should change your
- thinking. If I were to put inside this
- Cray a chopped-meat machine, you
- wouldn't know it was a meat chopper."
- "Unless you saw chopped meat
- coming out of it. Then you'd suspect
- it wasn't a Cray," Gregory said, and
- the brothers cackled.
- "In ten years, a Cray will fit in your
- pocket," David said.
- Supercomputers are evolving incred-
- ibly fast. The notion of what a super-
- computer is and what it can do changes
- from year to year, if not from month
- to month, as new machines arise. The
- definition of a supercomputer is simply
- this: one of the fastest and most pow-
- erful scientific computers in the world,
- for its time. The power of a super-
- computer is revealed, generally speak-
- ing, in its ability to solve tough prob-
- lems. A Cray Y-MP8, running at its
- peak working speed, can perform more
- than two billion floating-point opera-
- tions per second. Floating-point opera-
- tions&emdash;or flops, as they are called&emdash;are
- a standard measure of speed. Since a
- Cray Y-MP8 can hit two and a half
- billion flops, it is considered to be a
- gigaflop supercomputer. Giga (from
- the Greek for "giant") means a bil-
- lion. Like all supercomputers, a Cray
- often cruises along significantly below
- its peak working speed. (There is a
- heated controversy in the supercom-
- puter industry over how to measure the
- typical working performance of any
- given supercomputer, and there are
- many claims and counterclaims.) A
- Cray is a so-called vector-processing
- machine, but that design is going out
- of fashion. Cray Research has an-
- nounced that next year it will begin
- selling an even more powerful parallel
- machine.
- "Our machine is a gigaflop super-
- computer," David Chudnovsky told
- me. "The working speed of our ma-
- chine is from two hundred million flops
- to two gigaflops&emdash;roughly in the range
- of a Cray Y-MP8. We can probably go
- faster than a Y-MP8, but we don't
- want to get too specific about it."
- M zero is not the only ultrapowerful
- silicon engine to gleam in the Chud-
- novskian Ïuvre. The brothers recently
- fielded a supercomputer named Little
- Fermat, which they designed with
- Monty Denneau, an I.B.M. super-
- computer architect, and Saed Younis,
- a graduate student at the Massachu-
- setts Institute of Technology. Younis
- did the grunt work: he mapped out
- circuits containing more than fifteen
- thousand connections and personally
- plugged in some five thousand chips.
- Little Fermat is seven feet tall, and sits
- inside a steel frame in a laboratory at
- M.I.T., where it considers numbers.
- What m zero consists of is a group
- of high-speed processors linked by cables
- (which cover the floor of the room).
- The cables form a network of connec-
- tions among the processors&emdash;a web.
- Gregory sketched on a piece of paper
- the layout of the machine. He drew a
- box and put an "x" through it, to show
- the web, or network, and he attached
- some processors to the web:
- "Each processor is connected to a
- high-speed switching network that
- connects it to all the others," he said.
- "It's like a telephone network&emdash;every-
- body is talking to everybody else. As
- far as I know, no one except us has
- built a machine that has this type of
- web. In other parallel machines, the
- processors are connected only to near
- neighbors, while they have to talk to
- more distant processors through inter-
- vening processors. Think of a phone
- system: it wouldn't be very pleasant if
- you had to talk to distant people by
- sending them messages through your
- neighbors. But the truth is that nobody
- really knows how the hell parallel
- machines should perform, or the best
- design for them. Right now we have
- eight processors. We plan to have
- two hundred and fifty-six processors.
- We will be able to fit them into the
- apartment."
- He said that each processor had its
- own memory attached to it, so that
- each processor was in fact a separate
- computer. After a processor was fed
- some data and had got a result, it could
- send the result through the web to
- another processor. The brothers wrote
- the machine's application software in
- FORTRAN, a programming language
- that is "a dinosaur from the late fifties,"
- Gregory said, adding, "There is al-
- ways new life in this dinosaur." The
- software can break a problem into
- pieces, sending the pieces to the ma-
- chine's different processors. "It's the
- principle of divide and conquer," Greg-
- ory said. He said that it was very hard
- to know what exactly was happening
- in the web when the machine was
- running&emdash;that the web seemed to have
- a life of its own.
- "Our machine is mostly made of
- connections," David said. "About ninety
- per cent of its volume is cables. Your
- brain is the same way. It is mostly
- made of connections. If I may say so,
- your brain is a liquid-cooled parallel
- supercomputer." He pointed to his nose
- "This is the fan."
- The design of the web is the key
- element in the Chudnovskian architec-
- ture. Behind the web hide several new
- r findings in number theory, which the
- Chudnovskys have not yet published
- The brothers would not disclose to me
- the exact shape of the web, or the
- discoveries behind it, claiming that
- they needed to protect their competitive
- edge in a worldwide race to develop
- faster supercomputers. "Anyone with a
- hundred million dollars and brains
- could be our competitor," David said
- dryly.
- The Chudnovskys have formidable
- competitors. Thinking Machines Cor-
- poration, in Cambridge, Massachu-
- setts, sells massively parallel super-
- computers. The price of the latest model,
- the CM-S, starts at one million four
- hundred thousand dollars and goes up
- from there. If you had a hundred
- mil1ion dollars, you could order a CM-S
- that would be an array of black mono-
- liths the size of a Burger King, and
- it would burn enough electricity to
- light up a neighborhood. Seymour Cray
- is another competitor of the brothers,
- as it were. He invented the original
- Cray series of supercomputers, and is
- now the head of the Cray Computer
- Corporation, a spinoff from Cray Re-
- search. Seymour Cray has been work-
- ing to develop his Cray 3 for several
- years. His company's effort has re-
- cently been troubled by engineering
- delays and defections of potential cus-
- tomers, but if the machine ever is
- released to customers it may be an
- octagon about four feet tall and four
- feet across, and it will burn more than
- two hundred thousand watts. It would
- melt instantly if its cooling system
- were to fail.
- Then, there's the Intel Corporation.
- Intel, together with a consortium of
- federal agencies, has invested more
- than twenty-seven million dollars in
- the Touchstone Delta, a five-foot-high,
- fifteen-foot-long parallel supercomputer
- that sits in a computer room at Caltech.
- The machine consumes twenty-five
- thousand watts of power, and is kept
- from overheating by chilled air flow-
- ing through its core. One day, I called
- Paul Messina, a Caltech research sci-
- entist, who is the head of the Touch-
- stone Delta project, to get his opinion
- of the Chudnovsky brothers. It turned
- out that Messina hadn't heard of
- them. As for their claim to have built
- a pi-computing gigaflop supercomputer
- out of mail-order parts for around
- seventy thousand dollars, he flatly
- believed it. "It can be done, definitely,"
- Messina said. "Of course, seventy
- thousand dollars is just the cost of the
- components. The Chudnovskys are
- counting very little of their human
- time."
- Yasumasa Kanada, the brothers' pi
- rival at Tokyo University, uses a Hitachi
- S-820/80 supercomputer that is be-
- lieved to be considerably faster than a
- Cray Y-MP8, and it burns close to
- half a million watts&emdash;half a megawatt,
- practically enough power to melt steel.
- The Chudnovsky brothers particularly
- hoped to leave Kanada and his Hita-
- chi in the dust with their mail-order
- funny car.
- "We want to test our hardware,"
- Gregory said.
- "Pi is the best stress test for a su-
- percomputer," David said.
- "We also want to find out what
- makes pi different from other num-
- bers. It's a business."
- "Galileo saw the moons of Jupiter
- through his telescope, and he tried to
- figure out the laws of gravity by look-
- ing at the moons, but he couldn't,"
- David said. "With pi, we are at the
- stage of looking at the moons of Ju-
- piter." He pulled his Mini Mag-Lite
- flashlight out of his pocket and shone
- it into a bookshelf, rooted through
- some file folders, and handed me a
- color photograph of pi. "This is a pi-
- scape," he said. The photograph showed
- a mountain range in cyberspace: bony
- peaks and ridges cut by valleys. The
- mountains and valleys were splashed
- with colors&emdash;yellow, green, orange,
- violet, and blue. It was the first eight
- million digits of pi, mapped as a fractal
- landscape by an I.B.M. GF-l 1 super-
- computer at Yorktown Heights, which
- Gregory had programmed from his
- bed. Apart from its vivid colors, pi
- looks like the Himalayas.
- Gregory thought that the mountains
- of pi seemed to contain structure. "I
- see something systematic in this land-
- scape, but it may be just an attempt by
- the brain to translate some random
- visual pattern into order," he said. As
- he gazed into the nature beyond na-
- ture, he wondered if he stood close to
- a revelation about the circle and its
- diameter. "Any very high hill in this
- picture, or any flat plateau, or deep
- valley, would be a sign of something in
- pi," he said. "There are slight varia-
- tions from randomness in this land-
- scape. There are fewer peaks and
- valleys than you would expect if pi
- were truly random, and the peaks and
- valleys tend to stay high or low a little
- longer than you'd expect." In a man-
- ner of speaking, the mountains of pi
- looked to him as if they'd been molded
- by the hand of the Nameless One,
- Deus absconditus (the hidden God), but
- he couldn't really express in words
- what he thought he saw and, to his
- great frustration, he couldn't express it
- in the language of mathematics, either.
- "Exploring pi is like exploring the
- universe," David remarked.
- "It's more like exploring underwa-
- ter," Gregory said. "You are in the
- mud, and everything looks the same.
- You need a flashlight. Our computer
- is the flashlight."
- David said, "Gregory&emdash;I think,
- really&emdash;you are getting tired."
- A fax machine in a corner beeped
- and emitted paper. It was a message
- from a hardware dealer in Atlanta.
- David tore off the paper and stared at
- it. "They didn't ship it! I'm going to
- kill them! This a service economy. Of
- course, you know what that means&emdash;
- the service is terrible."
- "We collect price quotes by fax,"
- Gregory said.
- "It's a horrible thing. Window-
- shopping in supercomputerland. We
- can't buy everything&emdash;"
- "Because everything won't exist,"
- Gregory said.
- "We only want to build a ma-
- chine to compute a few transcendental
- numbers&emdash;"
- "Because we are not licensed for
- transcendental meditation," Gregory
- said.
- "Look, we are getting nutty," David
- said.
- "We are not the only ones," Greg-
- ory said. "We are getting an average
- of one letter a month from someone or
- other who is trying to prove Fermat's
- Last Theorem."
- I asked the brothers if they had
- published any of their digits of pi in
- a book.
- Gregory said that he didn't know
- how many trees you would have to
- grind up in order to publish a billion
- digits of pi in a book. The brothers' pi
- had been published on fifteen hundred
- microfiche cards stored somewhere in
- Gregory's apartment. The cards held
- three hundred thousand pages of data,
- a slug of information much bigger
- than the Encylopaedia Britannica, and
- containing but one entry, "Pi." David
- offered to find the cards for me; they
- had to be around here somewhere. He
- switched on the lights in the hallway
- and began to shift boxes. Gregory
- rifled bookshelves.
- "Please sit down, Gregory," David
- said. Finally, the brothers confessed
- that they had temporarily lost their pi.
- "Look, it's not a problem," David said.
- "We keep it in different places." He
- reached inside m zero and pulled out
- a metal box. It was a naked hard-disk
- drive, studded with chips. He handed
- me the object. "There's pi stored on
- this drive." It hummed gently. "You
- are holding some pi in your hand. It
- weighs six pounds."
- MONTHS passed before I visited
- the Chudnovskys again. The
- brothers had been tinkering with their
- machine and getting it ready to go for
- two billion digits of pi, when Gregory
- developed an abnormality related to
- one of his kidneys. He went to the
- hospital and had some CAT scans made
- of his torso, to see what things looked
- like, but the brothers were disappointed
- in the pictures, and persuaded the doctors
- to give them the CAT data on a mag-
- netic tape. They took the tape home,
- processed it in m zero, and got spec-
- tacular color images of Gregory's torso.
- The images showed cross-sectional
- slices of his body, viewed through
- different angles, and they were far
- more detailed than any image from a
- CAT scanner. Gregory wrote the im-
- aging software. It took him a few
- weeks. "There's a lot of interesting
- mathematics in the problem of imag-
- ing a body," he remarked. For the
- moment, it was more interesting than
- pi, and it delayed the brothers' probe
- into the Ludolphian number.
- Spring came, and Federal Express
- was active at the Chudnovskys' build-
- ing. Then the brothers began to cal-
- culate pi, slowly at first, more intensely
- as they gained confidence in their
- machine, but in May the weather
- warmed up and Con Edison betrayed
- the brothers. A heat wave caused a
- brownout in New York City, and as
- it struck, m zero automatically shut
- itself down, to protect its circuits, and
- died. Afterward, the brothers couldn't
- get electricity running properly through
- the machine. They spent two weeks
- restarting it, piece by piece.
- Then, on Memorial Day weekend,
- as the calculation was beginning to
- progress, Malka Benjaminovna suffered
- a heart attack. Gregory was alone with
- his mother in the apartment. He gave
- her chest compressions and breathed
- air into her lungs, although David
- later couldn't understand how his
- brother didn't kill himself saving her.
- An ambulance rushed her to St. Luke's
- Hospital. The brothers were terrified
- that they would lose her, and
- the strain almost killed David. ;¥
- One day, he fainted in his
- mother's hospital room and
- threw up blood. He had devel-
- oped a bleeding ulcer. "Look,
- it's not a problem," he said
- later. After Malka Benjaminovna had
- been moved out of intensive care,
- Gregory rented a laptop computer,
- plugged it into the telephone line in
- her hospital room, and talked to m zero
- at night through cyberspace, driving
- the supercomputer toward pi and
- watching his mother's blood pressure
- at the same time.
- Malka Benjaminovna improved
- slowly. When St. Luke's released her,
- the brothers settled her 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 sum-
- mer. 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, if you want to be boiled you are
- welcome to come inside." He led me
- down the dark hallway. Malka Benja-
- minovna was asleep in her bedroom,
- and the nurse was sitting beside her.
- Her room was lined with bookshelves,
- packed with paper&emdash;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 junk yard.
- "Then we could add goldfish," David
- said.
- "That would solve all our problems."
- "We are not good plumbers, Greg-
- ory. 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 thirty-four degrees Cel-
- sius. 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 immedi-
- ately began to pour down my body. "I
- don't like to go into this room," Greg-
- ory said. The steel frame in the center
- of the room&emdash;the heart of
- m zero&emdash;had acquired more
- logic boards, and more red lights
- blinked inside the machine. I
- could hear disk drives murmur-
- ing. The drives were copying
- and recopying segments of tran-
- scendental numbers, to check the digits
- for perfect accuracy. Gregory knelt on
- the floor, facing the steel frame.
- David opened a cardboard box and
- removed an electronic board. He be-
- gan to fit it into m zero. I noticed that
- his hands were marked with small
- cuts, which he had got from reaching
- inside the machine.
- "David, could you give me the
- flashlight?" Gregory said.
- David pulled the Mini Mag-Lite
- from his shirt pocket and handed it to
- Gregory. The brothers knelt beside
- each other, Gregory shining the flash-
- light into the supercomputer. David
- reached inside with his fingers and
- palpated a logic board.
- "Don't!" Gregory said. "O.K., 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 ther-
- mometer between two circuit boards in
- order to look for hot spots inside m zero.
- The thermometer's dial was marked
- "Beef Rare&emdash;Ham&emdash;Beef Med&emdash;
- Pork."
- "You want to keep the machine
- below 'Pork,' " Gregory remarked. He
- lifted a keyboard out of the steel frame
- and typed something on it, staring at
- a display screen. Numbers filled the
- 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 book-
- shelf 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 apart-
- ment is full of empty boxes. We have
- to save them. Gregory, I wonder if you
- are tired."
- "If I stand up now, I will fall
- down," Gregory said, from the floor.
- "Therefore, I will sit in my center of
- gravity. I will maintain 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 control-
- ler now seems to work."
- "We need to buy a new one," David
- said.
- "Try Nevada."
- David dialled a mail-order house in
- Nevada that here will be called Search-
- light Computers. He said loudly, in a
- thick Russian accent, "Hi, Searchlight.
- I need a fifteen-forty controller....
- No! No! No! I don't need any-
- thing else! Just the controller! Just a
- naked unit! Naked! How much you
- charge? . . . 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 Fed Ex? Come on! What
- about afternoon delivery? . . . Twenty-
- nine dollars before 3 P.M.? Relax. What
- is your name? . . . Bob. Fine. O.K. So
- it's two hundred and fifty-seven dol-
- lars plus twenty-nine dollars for Fed-
- eral Express?"
- "Twenty-nine dollars for Fed Ex!"
- Gregory burst out. "It should be fifteen."
- He pulled a second keyboard out of the
- steel frame and tapped the keys. An-
- other display screen came alive and
- filled with numbers.
- "Tell me this," David said to Bob
- in Nevada. "Do you have 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
- it usually doesn't."
- "Question Number Two: The Fed
- Ex 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 call A.K.," he said. "Hi,
- A.K., this is David Chudnovsky, call-
- ing from New York. A.K., I need
- another controller, like the one you
- sent. Can you send it today Fed
- Ex? . . . How much you charge? . . .
- Naked! I want a naked unit! Not in
- a shoebox, 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. A.K., 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 ex-
- tremely 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
- 52
- brothers." She is six years younger
- than Gregory; she was an undergradu-
- ate at Barnard College when she first
- met him. "I fell in love with Gregory
- immediately. His illness came with the
- package." She is still in love with him,
- even if at times they fight over his
- heaps of paper. ("I don't have room
- to put my things down," she says to
- 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
- dreams that she is dancing from room
- to room through an empty apartment
- that has parquet floors.
- David brought his mother out of her
- bedroom, settled her at the table, and
- kissed her on the cheek. Malka Ben-
- jaminovna seemed frail but alert. She
- is a small, white-haired woman with
- a fresh face and clear blue eyes, who
- speaks limited English. A mathemati-
- cian once described Malka Ben-
- jaminovna as the glue that holds the
- Chudnovsky family together. She was
- an engineer during the Second World
- War, when she designed buildings,
- laboratories, and proving grounds in
- the Urals for testing the Katyusha
- rocket; later, she taught engineering at
- schools around Kiev. She handed me
- plates of roast chicken, kasha, pickles,
- cream cheese, brown bread, and little
- wedges of The 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 lunch, and 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 disk drives
- clicked, the red lights flashed, the air-
- conditioner hummed, and you could
- hear dozens of whispering fans. Greg-
- ory leaned on his cane and contem-
- plated the machine. "It's doing many
- jobs at the moment," he said. "Frankly,
- I don't know what it's doing. 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&emdash;"
- "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 laboratory notebook. David
- slashed open a cardboard box with his
- hunting knife and lifted out a board
- studded with chips, for making color
- images on a display screen, and plugged
- it into m zero. Gregory crawled under
- a table. "Oh, shit," he said, from
- beneath the table.
- "Gregory, you killed the system
- again!"
- "Dave, Dave, can you get me a
- flashlight?"
- David handed his Mini Mag-Lite
- 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 into 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 drives began to click, and the
- parallel processors silently multiplied
- and conjoined huge numbers. Gregory
- headed for bed, David holding him by
- the arm.
- In the junk yard, his nest, his paper-
- lined oubliette, Gregory kicked off his
- gentleman's slippers, lay down on the
- bed, and predicted the future. He said,
- "The gigaflop supercomputers of today
- are almost useless. What is needed is
- a teraflop machine. That's a machine
- that can run at a trillion flops, a trillion
- floating-point operations per second, or
- roughly a thousand times as fast as a
- Cray Y-MP8. One such design for a
- teraflop machine, by Monty Denneau,
- at I.B.M., will be a parallel super-
- computer in the form of a twelve-foot-
- wide box. You want to have at least
- MARCH 2, 1992
- sixty-four thousand processors in the
- machine, each of which has the power
- of a Cray. And the processors will be
- joined by a network that has the total
- switching capacity of the entire tele-
- phone network in the United States.
- I think a teraflop machine will exist
- by 1993. Now, a better machine is a
- petaflop machine. A petaflop is a qua-
- drillion flops, a quadrillion floating-
- point operations per second, so a petaflop
- machine is a thousand times as fast as
- a teraflop machine, or a million times
- as fast as a Cray Y-MP8. The petaflop
- machine will exist by the year 2000,
- or soon afterward. It will fit into a
- sphere less than a hundred feet in
- diameter. It will use light and mir-
- rors&emdash;the machine's network will con-
- sist of optical cables rather than copper
- wires. By that time, a gigaflop 'super-
- computer' will be a single chip. I think
- that the petaflop machine will be used
- mainly to simulate machines like itself,
- so that we can begin to design some
- real machines."
- I N the nineteenth century, math-
- ematicians aKacked pi with the help
- of human computers. The most pow-
- erful of these was Johann Martin
- Zacharias Dase, a prodigy from Ham-
- burg. Dase could multiply large num-
- bers in his head, and he made a living
- exhibiting himself to crowds in Ger-
- many, Denmark, and England, and
- hiring himself out to 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 num-
- bers in his head during a period of
- eight and three-quarters hours. Dase
- could do this kind of thing for weeks
- on end, running as an unattended
- supercomputer. He would break off a
- calculation at bedtime, store every-
- thing in his memory for the night, and
- resume calculation in the morning.
- Occasionally, Dase had a system crash.
- In 1845, he bombed while trying to
- demonstrate his powers to a mathema-
- tician and astronomer named Hein-
- rich Christian Schumacher, reckoning
- wrongly every multiplication that he
- attempted. He explained to Schumacher
- that he had a headache. Schumacher
- also noted that Dase did not in the least
- understand theoretical mathematics. A
- mathematician named Julius Petersen
- once tried in vain for six weeks to
- teach Dase the rudiments of Euclidean
- geometry, but they absolutely baffled
- Dase. Large numbers Dase could
- handle, and in 1844 L. K. Schulz
- von Strassnitsky hired him to compute
- pi. Dase ran the job for almost two
- months iri his brain, and at the end of
- the time he wrote down pi correctly to
- the first two hundred decimal places&emdash;
- then a world record.
- To many mathematicians, math-
- ematical objects such as the number pi
- seem to exist in an external,
- objective reality. Numbers seem
- to exist apart from time or the
- world; numbers seem to tran-
- scend the universe; numbers
- might exist even if the uni-
- verse did not. I suspect that in
- their hearts most working
- mathematicians are Platonists,
- in that they take it as a matter of
- unassailable if unprovable fact 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 math-
- ematicians would probably agree that
- the ratio of the circle to its diameter
- exists brilliantly in the nature beyond
- nature, and would exist even if the
- human mind was not aware of it, and
- might exist even if God had not both-
- ered 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, in the opinion of some
- mathematicians, for while there is
- reason to doubt the existence of Gd,
- by their way of thinking 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. Of
- course pi is a natural object. Since pi
- is there, it exists. What we are doing
- is really close to experimental phys-
- ics&emdash;we are 'observing pi.' Since we
- can observe pi, I prefer to think of pi
- as a natural object. Observing pi is
- easier than studying physical phenom-
- ena, because you can prove things in
- mathematics, whereas you can't prove
- anything in physics. And, unfortu-
- nately, 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 represent-
- ing it with symbols. You put it through
- your mind in order to make sense
- of it."
- Mathematicians have sorted num-
- bers into classes in order to make sense
- of them. One class of numbers is that
- of the rational numbers. A rational
- number is a fraction composed of
- inte¥ers (whole numbers):
- l/l, T/3, 3/5, 1°/7l, and so on.
- Every rational number, when
- it is expressed in decimal form,
- repeats periodically: I/3, for
- example, becomes .333....
- Next, we come to the irratio-
- nal numbers. An irrational
- number can't be expressed as
- a fraction composed of whole numbers,
- and, furthermore, its digits go to in-
- finity without repeating periodically.
- The square root of two (¥12) is an
- irrational number. There is simply no
- way to represent any irrational number
- as the ratio of two whole numbers; it
- can't be done. Hippasus of Metapontum
- supposedly made this discovery in the
- fifth century B.C., while travelling in
- a boat with some mathematicians who
- were followers of Pythagoras. The
- Pythagoreans believed that everything
- in nature could be reduced to a ratio
- of two whole numbers, and they threw
- Hippasus overboard for his discovery,
- since he had wrecked their universe.
- Expanded as a decimal, the square root
- of two begins 1.41421 . . . and runs in
- "random" digits forever. It looks ex-
- actly like pi in its decimal expan-
- sion; it is a hopeless jumble, show-
- ing no obvious system or design. The
- square root of two is not a transcen-
- dental number, because it can be found
- with an equation. It is the solution
- (root) of an equation. The equation is
- x2 = 2, and a solution is the square
- root of two. Such numbers are called
- algebraic.
- While pi is indeed an irrational
- number&emdash;it can't be expressed as a
- fraction made of whole numbers&emdash;
- more important, it can't be expressed
- with finite algebra. Pi is therefore said
- to be a transcendental number, because
- it transcends algebra. Simply and gen-
- erally speaking, a transcendental num-
- ber can't be pinpointed through an
- equation built from a finite number of
- integers. There is no finite algebraic
- 54
- equation built from whole numbers
- that will give an exact value for pi.
- The statement can be turned around
- this way: pi is not the solution to any
- equation built from a less than infinite
- series of whole numbers. If equations
- are trains threading the landscape of
- numbers, then no train stops at pi.
- Pi is elusive, and can be approached
- only through rational approximations.
- The approximations hover around the
- number, closing in on it, but do not
- touch it. Any formula that heads to-
- ward pi will consist of a chain of
- operations that never ends. It is an
- infinite series. In 1674, Gottfried
- Wilhelm Leibniz (the co-inventor of
- the calculus, along with Isaac New-
- ton) noticed an extraordinary pattern
- of numbers buried in the circle. The
- Leibniz series for pi has been called
- one of the most beautiful mathematical
- discoveries of the seventeenth century:
- In English: pi over four equals one
- minus a third plus a fifth minus a
- seventh plus a ninth&emdash;and so on. You
- follow the 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 mys-
- terious that a chain of discrete rational
- numbers can connect so easily to ge-
- ometry, to the smooth and continuous
- circle.
- As an experiment in "observing pi,"
- as Gregory Chudnovsky puts it, I
- computed the Leibniz series on a pocket
- calculator. It was easy, and I got
- results that did seem to wander slowly
- toward pi. As the series progresses, the
- answers touch on 2.66, 3.46, 2.89, and
- 3.34, in that order. The answers land
- higher than pi and lower than pi,
- skipping back and forth across pi, and
- gradually closing in on pi. A math-
- ematician would say that the series
- "converges on pi." It converges on pi
- forever, playing hopscotch over pi but
- never landing on pi.
- You can take the Leibniz series out
- a long distance&emdash;you can even dra-
- matically speed up its movement to-
- ward pi by adding a few corrections to
- it&emdash;but no matter how far you take
- the Leibniz series, and no matter
- how many corrections you hammer into
- it, when you stop the operation and
- sum the terms, you will get a rational
- number that is somewhere around pi
- but is not pi, and you will be damned
- if you can put your hands on pi.
- Transcendental numbers continue
- forever, as an endless non-repeating
- string, in whatever rational form you
- choose to display them, whether as
- digits or as 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. Look-
- ing 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. Leibniz wasn't the first
- mathematician to discover the Leibniz
- series. Nilakantha, an astronomer,
- grammarian, and mathematician who
- lived on the Kerala coast of India,
- described the formula in Sanskrit po-
- etry around the year 1500.
- It is worth thinking about what a
- decimal place means. Each decimal
- place of pi is a range that shows the
- approxima¥e location of pi to an accu-
- racy 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 apparent 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 wobbles around pi. There
- is no such thing as a formula that is
- steady enough or 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 in 1984, and it attacks pi
- with great ferocity and elegance. The
- Chudnovsky formula is the fastest series
- for pi ever found which uses rational
- numbers. Various other series for pi,
- which use irrational numbers, have
- also been found, and they converge on
- pi faster than the Chudnovsky for-
- mula, but in practice they run more
- slowly on a computer, because irratio-
- nal numbers are harder to compute.
- The Chudnovsky formula for pi is
- thought to be "extremely beautiful" by
- persons who have a good feel for
- numbers, and it is based on a torus (a
- doughnut), rather than on a circle. It
- uses large assemblages of whole num-
- bers to hunt for pi, and it owes much
- to an earlier formula for pi worked out
- in 1914 by Srinivasa Ramanujan, a
- mathematician from Madras, who was
- a number theorist of unsurpassed ge-
- nius. Gregory says that the Chudnovsky
- formula "is in the style of Ramanujan,"
- and that it "is really very simple, and
- can be programmed into a computer
- with a few lines of code."
- In 1873, Georg Cantor, a Russian-
- born mathematician who was one of
- the towering intellectual figures of the
- nineteenth century, proved that the set
- of transcendental numbers is infinitely
- more extensive than the set of alge-
- braic numbers. That is, finite algebra
- can't find or describe most numbers. To
- put it another way, most numbers are
- infinitely long and non-repeating in
- any rational form of representation. In
- this respect, most numbers are like pi.
- Cantor's proof was a disturbing piece
- of news, for at that time very few
- transcendental numbers were actually
- known. ( Not until nearly a decade
- later did Ferdinand Lindemann finally
- prove the transcendence of pi; before
- that, mathematicians had only conjec-
- tured that pi was transcendental.) Per-
- haps even more disturbing, Cantor
- offered no clue, in his proof, to what
- a transcendental number might look
- like, or how to construct such a beast.
- Cantor's celebrated proof of the exis-
- tence of uncountable multitudes of tran-
- scendental numbers resembled a proof
- that the world is packed with micro-
- scopic angels&emdash;a proof, however, that
- does not tell us what the angels look
- like or where they can be found; it
- merely proves that they exist in un-
- countable multitudes. While Cantor's
- proof lacked any specific description of
- a transcendental number, it showed
- that algebraic numbers (such as the
- square root of two) are few and far
- between: they poke up like marker
- buoys through the sea of transcenden-
- tal numbers.
- Cantor's proof disturbed some math-
- ematicians because, in the first place,
- it suggested that they had not yet
- discovered most numbers, which were
- transcendentals, and in the second
- place that they lacked any tools or
- methods that would determine whether
- a given number was transcendental
- or not. Leopold Kronecker, an influ-
- ential older mathematician, rejected
- Cantor's proof, and resisted the whole
- notion of "discovering" a number. (He
- once said, in a famous remark, "God
- made the integers, all else is the work
- of man.") Cantor's proof has with-
- stood such attacks, and today the de-
- bate over whether transcendental num-
- bers are a work of God or man has
- subsided, mathematicians having de-
- cided to work with transcendental
- numbers no matter who made them.
- The Chudnovsky brothers claim that
- the digits of pi form the most nearly
- perfect random sequence of digits that
- has ever been discovered. They say
- that nothing known to humanity ap-
- pears to be more deeply unpredictable
- than the succession of digits in pi,
- except, perhaps, the haphazard clicks
- of a Geiger counter as it detects the
- decay of radioactive nuclei. But pi is
- not random. The fact that pi can be
- produced by a relatively simple for-
- mula means that pi is orderly. Pi looks
- random only because the pattern in the
- digits is fantastically complex. The
- Ludolphian number is fixed in eter-
- nity&emdash;not a digit out of place, all char-
- acters 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. Vari-
- ous simple methods of approximation
- will always yield the same succession
- of digits in the same order. If a single
- digit in pi were to be changed any-
- where between here and infinity, the
- resulting number would no longer be
- pi; it would be "garbage," in David's
- word, because to change a single digit
- in pi is to throw all the following digits
- out of whack and miles from pi.
- "Pi is a damned good fake of a
- random number," Gregory said. "I
- just wish it were not as good a fake.
- It would make our lives a lot easier."
- Around the three-hundred-millionth
- decimal place of pi, the digits go
- 88888888&emdash;eight eights pop up in a
- row. Does this mean anything? It
- appears to be random noise. Later,
- ten sixes erupt: 6666666666. What
- does this mean? Apparently nothing,
- only more noise. Somewhere past the
- half-billion mark appears the string
- 123456789. It's an accident, as it were.
- "We do not have a good, clear, crys-
- tallized idea of randomness," Gregory
- said. "It cannot be that pi is truly
- random. Actually, a truly random se-
- quence of numbers has not yet been
- discovered."
- No one knows what happens to the
- digits of pi in the deeper regions, as the
- number is resolved toward infinity. Do
- the digits turn into nothing but eights
- and fives, say? Do they show a pre-
- dominance of sevens? Similarly, no
- one knows if a digit stops appearing in
- pi. This conjecture says that after a
- certain point in the sequence a digit
- drops out completely. For example, no
- more fives appear in pi&emdash;something
- like that. Almost certainly, pi does not
- do such things, Gregory Chudnovsky
- thinks, because it would be stupid, and
- nature isn't stupid. Nevertheless, no
- one has ever been able to prove or
- disprove a certain basic conjecture about
- pi: that every digit has an equal chance
- of appearing in pi. This is known as
- the normality conjecture for pi. The
- normality conjecture says that, on
- average, there is no more or less of any
- digit in pi: for example, there is no
- excess of sevens in pi. If all digits do
- appear with the same average fre-
- quency in pi, then pi is a "normal"
- number&emdash;"normal" by the narrow
- mathematical definition of the word.
- "This is the simplest possible conjec-
- ture about pi," Gregory said. "There
- is absolutely no doubt that pi is a
- 'normal' number. Yet we can't prove
- it. We don't even know how to try to
- prove it. We know very little about
- transcendental numbers, and, what is
- worse, the number of conjectures about
- them isn't growing." No one knows
- even how to tell the difference between
- the square root of two and pi merely
- by looking at long strings of their
- digits, though the two numbers have
- completely distinct mathematical prop-
- erties, one being algebraic and the
- other transcendental.
- Even if the brothers couldn't prove
- anything about the digits of pi, they felt
- that by looking at them through the
- window of their machine they migh¥
- at least see something that could lead
- to an important conjecture about pi or
- about transcendental numbers as a class.
- You can learn a lot about all cats by
- looking closely at one of them. So if
- you wanted to look closely at pi how
- much of it could you see with a very
- large supercomputer? What if you
- turned the universe into a supercom-
- puter? What then? How much pi could
- you see? Naturally, the brothers had
- considered this project. They had
- imagined a computer built from the
- universe. Here's how they estimated
- the machine's size. It has been calcu-
- lated that there are about 1079 electrons
- and protons in the observable universe;
- this is the so-called Eddington number
- of the universe. (Sir Arthur Stanley
- Eddington¥ the astrophysicist, first came
- up with the number.) The Edding-
- ton number is the digit 1 followed by
- seventy-nine zeros: 10,000,000,000,000,
- 000,000,000,000,000,000,000,000,000,
- ooo,ooo,ooo,ooojooo,ooo,ooo,ooo,ooo,
- 000,000,000,000. Ten vigintsextillion.
- The Eddington number. It declares
- the power of the Eddington machine.
- The Eddington machine would be
- the universal supercomputer. It would
- be made of all the atoms in the uni-
- verse. The Eddington machine would
- contain ten vigintsextillion parts, and
- if the Chudnovsky brothers could figure
- out how to program it with FORTRAN
- they might make it churn toward pi.
- "In order to study the sequence of pi,
- you have to store it in the Eddington
- machine's memory," Gregory said. To
- be realistic, the brothers thought that
- a practical Eddington machine wouldn't
- be able to store pi much beyond 1077
- digits&emdash;a number that is only a hun-
- dredth of the Eddington number. Now,
- what if the digits of pi only begin to
- show regularity beyond 1077 digits?
- Suppose, for example, that pi manifests
- a regularity starting at 101°° decimal
- places? That number is known as a
- googol. If the design in pi appears only
- after a googol of digits, then not even
- the Eddington machine will see any
- system in pi; pi will look totally dis-
- ordered to the universe, even if pi
- contains a slow, vast, delicate struc-
- ture. A mere googol of pi might be only
- the first knot at the corner of a kind
- of limitless Persian rug, which is woven
- into increasingly elaborate diamonds,
- cross-stars, gardens, and cosmogonies.
- It may never be possible, in principle,
- to see the order in the digits of pi. Not
- even nature itself may know the nature
- of pi.
- "If pi doesn't show systematic be-
- havior until more than ten to the
- seventy-seven decimal places, it would
- MARCH 2, 1992
- really be a disaster," Gregory said. "It
- would be actually horrifying."
- "I wouldn't give up," David said.
- "There might be some other way of
- leaping over the barrier&emdash;"
- "And of attacking the son of a bitch,"
- Gregory said.
- THE brothers first came in contact
- with the membrane that divides
- the dreamlike earth from mathemati-
- cal reality when they were boys, grow-
- ing up in Kiev, and their father gave
- David a book entitled "What Is Math-
- ematics?," by two mathematicians named
- Richard Courant and Herbert Rob-
- bins. The book is a classic&emdash;millions
- of copies of it have been printed in
- unauthorized Russian and Chinese edi-
- tions alone&emdash;and after the brothers
- finished reading "Robbins," as the book
- is called in Russia, David decided to
- become a mathematician, and Gregory
- soon followed his brother's footsteps
- into the nature beyond nature. Gregory's
- first publication, in the journal Soviet
- Mathematics&emdash;Doklady, came when he
- was sixteen years old: "Some Results
- in the Theory of Infinitely Long Ex-
- pressions." Already you can see where
- he was headed. David, sensing his
- younger brother's power, encouraged
- him to grapple with central problems
- in mathematics. Gregory made his first
- major discovery at the age of seven-
- teen, when he solved Hilbert's Tenth
- Problem. (It was one of twenty-three
- great problems posed by David Hilbert
- in 1900.) To solve a Hilbert problem
- would be an achievement for a life-
- time; Gregory was a high-school stu-
- dent who had read a few books on
- mathematics. Strangely, a young Rus-
- sian mathematician named Yuri Matya-
- sevich had just solved Hilbert's Tenth
- Problem, and the brothers hadn't heard
- the news. Matyasevich has recently
- said that the Chudnovsky method is
- the preferred way to solve Hilbert's
- Tenth Problem.
- The brothers enrolled at Kiev State
- University, and both graduated summa
- cum laude. They took their Ph.D.s at
- the Institute of Mathematics at the
- Ukrainian Academy of Sciences. At
- first, they published their papers sepa-
- rately, but by the mid-nineteen-seventies
- they were collaborating on much of
- their work. They lived with their parents
- in Kiev until the family decided to try
- to take Gregory abroad for treatment,
- and in 1976 Volf and Malka Chud-
- novsky applied to the government to
- emigrate. Volf was immediately fired
- from his job.
- The K.G.B. began tailing the broth-
- ers. "Gregory would not believe me
- until it became totally obvious," David
- said. "I had twelve K.G.B. agents on
- my tail. No, look, I'm not kidding!
- They shadowed me around the clock
- in two cars, six agents in each car.
- Three in the front seat and three in
- the back seat. That was how the K.G. B.
- operated." One day, in 1976, David
- was walking down the street when
- K.G.B. officers attacked him, breaking
- his skull. He went home and nearly
- died, but didn't go to the hospital. "If
- I had gone to the hospital, I would
- have died for sure," he told me. "The
- hospital is run by the state. I would
- forget to breathe."
- On July 22, 1977, plainclothesmen
- from the K.G.B. accosted Volf and
- Malka on a street in Kiev and beat
- them up. They broke Malka's arm and
- fractured her skull. David took his
- mother to the hospital. "The doctor in
- the emergency room said there was no
- fracture," David said.
- Gregory, at home in bed, was not
- so vulnerable. Also, he was conspicu-
- ous in the West. Edwin Hewitt, a
- mathematician at the University of
- Washington, in Seattle, had visited
- Kiev in 1976 and collaborated with
- Gregory on a paper, and later, when
- Hewitt learned that the Chudnovsky
- family was in trouble, he persuaded
- Senator Henry M. Jackson, the pow-
- erful member of the Senate Armed
- Services Committee, to take up the
- Chudnovskys' case. Jackson put pres-
- sure on the Soviets to let the family
- leave the country. Just before the K.G.B.
- attacked the parents, two members of
- a French parliamentary delegation that
- was in Kiev made an unofficial visit to
- the Chudnovskys to see what was going
- on. One of the visitors, a staff member
- of the delegation, was Nicole Lanne-
- grace, who married David in 1983.
- Andrei Sakharov also helped to draw
- attention to the Chudnovskys' increas-
- ingly desperate situation. Two months
- after the parents were attacked, the
- Soviet government unexpectedly let the
- family go. "That summer when I was
- geKing killed by the K.G.B., I could
- never have imagined that the next year
- I would be in Paris or that I would
- wind up in New York, married to a
- beautiful Frenchwoman," David said.
- The Chudnovsky family settled in New
- York, near Columbia University.
- I F pi is truly random, then at
- times pi will appear to be ordered.
- Therefore, if pi is random it contains
- accidental order. For example, some-
- where in pi a sequence may run
- 07070707070707 for as many decima]
- places as there are, say, hydrogen at-
- oms in the sun. It's just an accident.
- Somewhere else the same sequence of
- zeros and sevens may appear, only this
- time interrupted by a single occurrence
- of the digit 3. Another accident. Those
- and all other "accidental" arrange-
- ments of digits almost certainly erupl
- in pi, but their presence has never been
- proved. "Even if pi is not truly ran-
- dom, you can still assume that you
- get every string of digits in pi," Greg-
- ory said.
- If you were to assign letters of the
- alphabet to combinations of digits, and
- were to do this for all human alpha-
- bets, syllabaries, and ideograms, then
- you could fit any written character in
- any language to a combination of digits
- in pi. According to this system, pi could
- be turned into literature. Then, if you
- could look far enough into pi, you
- would probably find the expression
- "See the U.S.A. in a Chevrolet!" a
- billion times in a row. Elsewhere, you
- would find Christ's Sermon on the
- Mount in His native Aramaic tongue,
- and you would find versions of the
- Sermon on the Mount that are pure
- blasphemy. Also, you would find a
- dictionary of Yanomamo curses. A
- guide to the pawnshops of Lubbock.
- The book about the sea which James
- Joyce supposedly declared he would
- write after he finished "Finnegans
- Wake." The collected transcripts of
- "The Tonight Show" rendered into
- Etruscan. "Knowledge of All Existing
- Things," by Ahmes the Egyptian scribe.
- Each occurrence of an apparently- or-
- dered string in pi, such as the words
- t "Ruin hath taught me thus to rumi-
- nate/That Time will come and take
- my love away," is followed by unimag-
- inable deserts of babble. No book and
- none but the shortest poems will ever
- be seen in pi, since it is infinitesimally
- unlikely that even as brief a text as
- an English sonnet will appear in the
- first 1077 digits of pi, which is the
- longest piece of pi that can be calcu-
- lated in this universe.
- Anything that can be produced by a
- simple method is by definition orderly.
- Pi can be produced by various simple
- methods of rational approximation, and
- those methods yield the same digits in
- a fixed order forever. Therefore, pi is
- - orderly in the extreme. Pi may also be
- a powerful random-number generator,
- spinning out any and all possible com-
- binations of digits. We see that the
- distinction between chance and fixity
- dissolves in pi. The deep connection
- between disorder and order, between
- cacophony and harmony, in the most
- famous ratio in mathematics fascinated
- Gregory and David Chudnovsky. They
- wondered if the digits of pi had a
- personality.
- "We are looking for the appearance
- of some rules that will distinguish the
- digits of pi from other numbers,"
- Gregory explained. "It's like studying
- writers by studying their use of words,
- their grammar. If you see a Russian
- sentence that extends for a whole page,
- with hardly a comma, it is definitely
- Tolstoy. If someone were to give you
- a million digits from somewhere in pi,
- could you tell it was from pi? We don't
- really look for patterns; we look for
- rules. Think of games for children. If
- I give you the sequence one, two,
- three, four, five, can you tell me what
- the next digit is? Even a child can do
- it; the next digit is six. How about this
- game? Three, one, four, one, five,
- nine. Just by looking at that sequence,
- can you tell me the next digit? What
- if I gave you a sequence of a million
- digits from pi? Could you tell me the
- next digit just by looking at the se-
- quence? Why does pi look like a totally
- unpredictable sequence with the high-
- est complexity? We need to find out the
- rules that govern this game. For all we
- know, we may never find a rule in pi."
- HERBERT ROBBINS, the co-author of
- "What Is Mathematics?," is an
- emeritus professor of mathematical
- statistics at Columbia University. For
- the past six years, he has been teaching
- at Rutgers. The Chudnovskys call him
- once in a while to get his advice on
- how to use statistical tools to search for
- signs of order in pi. Robbins lives in
- a rectilinear house that has a lot of
- glass in it, in the woods on the out-
- skirts of Princeton. Some of the twen-
- tieth century's most creative and pow-
- erful discoveries in statistics and
- probability theory happened inside his
- head. Robbins is a tall, restless man
- in his seventies, with a loud voice,
- furrowed cheeks, and penetrating eyes.
- One recent day, he stretched himself
- out on a daybed in a garden room
- in his house and played with a rub-
- ber band, making a harp across his
- fingertips.
- "It is a very difficult philosophical
- question, the question of what 'ran-
- dom' is," he said. He plucked the
- rubber band with his thumb, boink,
- boink. "Everyone knows the famous
- remark of Albert Einstein, that God
- does not throw dice. Einstein just would
- not believe that there is an element of
- randomness in the construction of the
- world. The question of whether the
- universe is a random process or is
- determined in some way is a basic
- philosophical question that has nothing
- to do with mathematics. The question
- is important. People consider it when
- they decide what to do with their lives.
- It concerns religion. It is the question
- of whether our fate will be revealed or
- whether we live by blind chance. My
- God, how many people have been
- murdered over an answer to that ques-
- tion! Mathematics is a lesser activity
- than religion in the sense that we've
- agreed not to kill each other but to
- discuss things."
- Robbins got up from the daybed and
- sat in an armchair. Then he stood up
- and paced the room, and sat at a table
- in the room, and sat on a couch, and
- went back to the table, and finally
- returned to the daybed. The man was
- in constant motion. It looked random
- to me, but it may have been systematic.
- It was the random walk of Herbert
- Robbins.
- "Mathematics is broken into tiny
- specialties today, but Gregory Chud-
- novsky is a generalist who knows the
- whole of mathematics as well as any-
- one," he said as he moved around.
- "You have to go back a hundred years,
- to David Hilbert, to find a mathema-
- tician as broadly knowledgeable as
- Gregory Chudnovsky. He's like Mozart:
- he's the last of his breed. I happen to
- think the brothers' pi project is a will-
- o'-the-wisp, and is one of the least
- interesting things they've ever done.
- But what do I know? Gregory seems
- to be asking questions that can't be
- answered. To ask for the system in pi
- is like asking 'Is there life after death?'
- When you die, you'll find out. Most
- mathematicians are not interested in
- the digits of pi, because the question is
- of no practical importance. In order for
- a mathematician to become interested
- in a problem, there has to be a possi-
- bility of solving it. If you are an
- athlete, you ask yourself if you can
- jump thirty feet. Gregory likes to ask
- if he can jump around the world. He
- likes to do things that are impossible."
- At some point after the brothers
- settled in New York, it became obvi-
- ous that Columbia University was not
- going to be able to invite them to
- become full-fledged members of the
- faculty. Since then, the brothers have
- always enjoyed cordial personal rela-
- tionships with various members of the
- faculty, but as an institution the Math-
- ematics Department has been unable
- to create permanent faculty positions
- for them. Robbins and a couple of
- fellow-mathematicians&emdash;Lipman Bers
- and the late Mark Kac&emdash;once tried to
- raise money from private sources for
- an endowed chair at Columbia to be
- shared by the brothers, but the effort
- failed. Then the John D. and Cath-
- erine T. MacArthur Foundation award-
- ed Gregory Chudnovsky a "genius"
- fellowship; that happened in 1981, the
- first year the awards were given, as if
- to suggest that Gregory is a person
- for whom the MacArthur prize was
- invented. The brothers can exhibit
- other fashionable paper&emdash;a Prix Peccot-
- Vimont, a couple of Guggenheims, a
- Doctor of Science honor¥s causa from
- Bard College, the Moscow Mathemati-
- cal Society Prize&emdash;but there is one
- defect in their résumé, which is the fact
- that Gregory has to lie in bed most of
- the day. The ugly truth is that Gregory
- Chudnovsky can't get a permanent job
- at any American institution of higher
- learning because he is physically dis-
- abled. But there are other, more per-
- plexing reasons that have led the Chud-
- novsky brothers to pursue their work
- in solitude, outside the normal aca-
- demic hierarchy, since the day they
- arrived in the United States.
- Columbia University has awarded
- each brother the title of senior research
- scientist in the Department of Math-
- ematics. Their position at Columbia is
- ambiguous. The university officially
- considers them to be members of the
- faculty, but they don't have tenure, and
- Columbia doesn't spend its own funds
- to pay their salaries or to support their
- research. However, Columbia does give
- them heakh-insurance benefits and a
- housing subsidy.
- The brothers have been living on
- modest grants from the National Sci-
- ence Foundation and various other
- research agencies, which are funnelled
- through Columbia and have to be
- applied for regularly. Nicole Lanne-
- grace and Christine Chudnovsky fi-
- nanced m zero out of their paychecks.
- Christine's father, Gonzalo Pardo, who
- is a professor of dentistry at the State
- University of New York at Stony Brook,
- built the steel frame for m zero in his
- basement during a few weekends, using
- a wrench and a hacksaw.
- The brothers' mode of existence has
- come to be known among mathema-
- ticians as the Chudnovsky Problem.
- Herbert Robbins eventually decided
- that it was time to ask the entire
- American mathematics profession why
- it could not solve the Chudnovsky
- Problem. Robbins is a member of the
- National Academy of Sciences, and in
- 1986 he sent a letter to all of the
- mathematicians in the academy:
- I fear that unless a decent and honorable
- position in the American educational and re-
- search system is found for the brothers soon,
- a personal and scientific tragedy will take
- place for which all American mathematicians
- will share responsibility....
- I have asked many of my colleagues why
- this situation exists, and what can be done to
- put an end to what I regard as a national
- disgrace. I have never received an answer
- that satisfies me.... I am asking you, then,
- as one of the leaders of American mathemat-
- ics, to tell me what you are prepared to do to
- acquaint yourselves with their present cir-
- cumstances, and if you are convinced of the
- merits of their case, to find a suitable position
- somewhere in the country for them as a pair.
- There wasn't much of a response.
- Robbins says that he received three
- written replies to his letter. One, from
- a faculty member at a well-known
- East Coast university, complained about
- David Chudnovsky's personality. He
- remarked that "when David learns to
- be less overbearing" the brothers might
- have better luck. He also did not fully
- understand the tone of Rob- .
- bins' letter: while he agreed
- that some resolution to the
- Chudnovsky Problem must
- be found, he thought that
- Herb Robbins ought to ap-
- proach the subject realisti
- cally and with more candor.
- ("More candor? How could I have
- been more candid?" Robbins asked.)
- Academic administrator. I'm sorry I have
- nothing more effective to propose."
- An emotional reaction to Robbins'
- campaign on behalf of the Chudnovskys
- came a bit later from Edwin HewiK,
- the mathematician who had helped get
- the family out of the Soviet Union, and
- one of the few Americans who has
- ever worked with Gregory Chudnovsky.
- Hewitt wrote to colleagues, "I have
- collaborated with many excelIent math-
- ematicians . . . but with no one else
- have I witnessed an outpouring of
- mathematics like that from Gregory.
- He simply KNOWS what is true and
- what is not." In another letter, Hewitt
- wrote:
- The Chudnovsky situation is a national
- disgrace. Everyone says, "Oh, what a crying
- shame" & then suggests that they be placed at
- somebody else's institution. No one seems to
- want the admittedly burdensome task of car-
- ing for the Chudnovsky family. I imagine it
- would be a full-time, if not an impossible,
- job. We may remember that both Mozart and
- Beethoven were disagreeable people, to say
- nothing of Gauss.
- The brothers would have to be hired
- as a pair. Gregory won't take any job
- unless David gets one, and vice versa.
- Physically and intellectually com-
- mingled, like two trees that have grown
- together at the root and bole, the broth-
- ers claim that they can't be separated
- without becoming deadfalls and crash-
- ing to the ground. To hire the Chud-
- novsky pair, a department would have
- to create a joint opening for them.
- Gregory can't teach classes in the normal
- way, because he is more or less confined to bed.
- It would require a small degree of
- flexibility in a department to
- allow Gregory to concen
- trate on research, while David
- handled the teaching. The
- A problem is that Gregory might
- Another letter came from a faculty
- member at Princeton University, who
- offered to put in a good word with the
- National Science Foundation to help
- the brothers get their grants, but did
- not mention a job at Princeton or
- anywhere else. The most thoughtful
- response came from a faculty member
- at M.I.T., who remarked, "It does
- seem odd that they have not been more
- sought after." He wondered if in some
- part this might be a consequence of
- their breadth. "A specialist appears as
- a safer investment to a cautious aca-
- working with a few brilliant graduate
- students&emdash;a privilege that might not go
- down well in an American academic
- department.
- "They are prototypical Russians,"
- Robbins said. "They combine a rather
- grandiose vision of themselves with an
- ability to live on scraps rather than
- compromise their principles. These are
- people the world is not able to cope
- with, and they are not making it any
- easier for the world. I don't see that
- the world is particularly trying to keep
- Gregory Chudnovsky alive. The trag-
- edy&emdash;the disgrace, so to speak&emdash;is that
- the American scientific and educational
- establishment is not benefitting from
- the Chudnovskys' assistance. Thirteen
- years have gone by since the Chud-
- novskys arrived here, and where are
- all the graduate students who would
- have worked with the brothers? How
- many truly great mathematicians have
- you ever heard of who couldn't get a
- job? I think the Chudnovskys are the
- only example in history. This vast
- educational system of ours has poured
- the Chudnovskys out on the sand, to
- waste. Yet Gregory is one of the re-
- markable personalities of our time.
- When I go up to that apartment and
- sit by his bed, I think, My God, when
- I was a student at Harvard I was in
- contact with people far less interesting
- than this. What happens to me in
- Gregory's room is like that line in the
- Gerard Manley Hopkins poem: 'Mar-
- garet, are you grieving/Over Golden-
- grove unleaving?' I'm grieving, and I
- guess it's me I'm grieving for."
- T" WO billion digits of pi? Where
- do they keep them?" Samuel
- Eilenberg said to me. Eilenberg is a
- gifted and distinguished topologist, and
- an emeritus professor of mathematics
- at Columbia University. He was the
- chairman of the department when the
- question of hiring the brothers first
- became troublesome to Columbia.
- "There is an element of fatigue in the
- Chudnovsky Problem," he said. "In
- the academic world, we have to be
- careful who our colleagues are. David
- is a pain in the neck. He interrupts
- people, and he is not interested in
- anything except what concerns him
- and his brother. He is a nudnick!
- Gregory is certainly unusual, but he is
- not great. You can spend all your life
- computing digits. What for? You know
- in advance that you can't see any
- regularity in pi. It's about as interest-
- ing as going to the beach and counting
- sand. I wouldn't be caught dead doing
- that kind of work! Most mathemati-
- cians probably feel this way. An im-
- portant ingredient in mathematics is
- taste. Mathematics is mostly about giving
- pleasure. The ultimate criterion of
- mathematics is aesthetic, and to calcu-
- late the two-billionth digit of pi is to
- me abhorrent."
- "Abhorrent&emdash;yes, most mathemati-
- cians would probably agree with that,"
- said Dale Brownawell, a respected
- number theorist at Penn State. "Tastes
- change, though. If something were to
- begin to show up in the digits of pi,
- it would boggle everyone's mind."
- Brownawell met the Chudnovskys at
- the Vienna airport when they escaped
- from the Soviet Union. "They didn't
- bring much with them, just a pile ol
- bags and boxes. David would walk
- through a wall to do what is right for
- MARCH 2, 1992
- his brother. In the situation they are
- in, how else can they survive? To see
- the Chudnovskys carrying on science
- at such a high level with such meagre
- support is awe-inspiring."
- Richard Askey, a prominent math-
- ematician at the University of Wis-
- consin at Madison, occasionally flies to
- New York to sit at the foot of Gregory
- Chudnovsky's bed and learn about
- mathematics. "David Chudnovsky is a
- very good mathematician," Askey said
- to me. "Gregory is one of the few great
- mathematicians of our time. Gregory
- is so much beKer than I am that it is
- impossible for me to say how good he
- really is. Is he the best in the world,
- or one of the three best? I feel uncom-
- fortable evaluating people at that level.
- The brothers' pi stuff is just a small
- part of their work. They are really
- trying to find out what the word 'ran-
- dom' means. I've heard some people
- say that the brothers are wasting their
- time with that machine, but Gregory
- Chudnovsky is a very intelligent man,
- who has his head screwed on straight,
- and I wouldn't begin to question his
- priorities. The tragedy is that Gregory
- has had hardly any students. If he dies
- without having passed on not only his
- knowledge but his whole way of think-
- ing, then it will be a great tragedy.
- Rather than blame Columbia Univer-
- sity, I would prefer to say that the
- blame lies with all American math-
- ematicians. Gregory Chudnovsky is a
- national problem."
- I" T looks like kvetching," Gregory
- said from his bed. "It looks cheap,
- and it is cheap. We are here in the
- United States by our own choice. I
- don't think we were somehow wronged.
- I really can't teach. So what does one
- want to do about it? Attempts to change
- the system are very expensive and
- time-consuming and largely a waste of
- time. We barely have time to do the
- things we want to do."
- "To reform the system?" David said,
- playing his flashlight across the ceil-
- ing. "In this country? Look. Come on.
- It's much easier to reform a totalitarian
- system."
- "Yes, you just make a decree,"
- Gregory said. "Anyway, this sort of
- talk moves into philosophical ques-
- tions. What is life, and where does the
- money come from?" He shrugged.
- F Toward the end of the summer of
- 1991, the brothers halted their probe
- into pi. They had surveyed pi to two
- billion two hundred and sixty million
- three hundred and twenty-one thou-
- sand three hundred and thirty-six dig-
- its. It was a world record, doubling the
- record that the Chudnovskys had set in
- 1989. If the digits were printed in
- ordinary type, they would stretch from
- New York to Southern California. The
- brothers had temporarily ditched their
- chief competitor, Yasumasa Kanada&emdash;
- a pleasing development when the broth-
- ers considered that Kanada had access
- to a half-megawatt Hitachi monster
- that was supposed to be faster than a
- Cray. Kanada reacted gracefully to the
- Chudnovskys' achievement, and he told
- Science News that he might be able to
- get at least a billion and a half digits
- of pi if he could obtain enough time
- on a Japanese supercomputer.
- "You see the advantage to being
- truly poor. We had to build our ma-
- chine, but now we get to use it, too,"
- Gregory said.
- The Chudnovskys' machine had
- spent its time both calculating pi and
- checking the result. The job had taken
- about two hundred and fifty hours on
- m zero. The machine had spent most
- of its time checking the answer, to
- make sure each digit was correct, rather
- than doing the fundamental computa-
- tion of pi.
- "We have done our tests for pat-
- terns, and there is nothing," Gregory
- said. "It would be rather stupid if there
- were something in a few billion digits.
- There are the usual things. The digit
- three is repeated nine times in a row,
- and we didn't see that before. Unfor-
- tunately, we still don't have enough
- computer power to see anything in pi."
- Such was their scientific conclusion,
- and yet the brothers felt that they may
- have noticed something in pi. It hov-
- ered out of reach, but it seemed a little
- closer now. It was a slight sign of
- order&emdash;a possible sign&emdash;and it had to
- do with the running average of tht
- digits. You can take an average of any
- string of digits in pi. It is like getting
- a batting average, an average height,
- an average weight. The average of the
- digits in pi should be 4.5. That's the
- average of the decimal digits zero
- through nine. The brothers noticed
- that the average seems to be slightly
- skewed. It stays a little high through
- most of the first billion digits, and then
- it stays a little low through the next
- billion digits. The running average of
- pi looks like a tide that rises and
- retreats through two billion digits, as
- if a distant moon were passing over a
- sea of digits, pulling them up and
- down. It may or may not be a hint of
- a rule in pi. "It's unfortunately not
- statistically significant yet," Gregory
- said. "It's close to the edge of signifi-
- cance." The brothers may have glimpsed
- only their desire for order. The tide
- that seems to flow through pi may be
- nothing but aimless gabble, but what
- if it is a wave rippling through pi?
- What if the wave begins to show a
- weird and complicated pulsation as you
- go deeper in pi? You could become
- obsessive thinking about things like
- this. You might have to build more
- machines. "We need a trillion digits,"
- David said. A trillion digits printed in
- ordinary type would stretch from here
- to the moon and back, twice. The
- brothers thought that if they didn't get
- bored with pi and move on to other
- problems they would easily collect a
- trillion digits in a few years, with the
- L help of increasingly powerful super-
- computing equipment. They would orbit
- the moon in digits, and head for ALpha
- Centauri, and if they lived and their
- machines held, perhaps someday they
- would begin to see the true nature of pi.
- Gregory is lying in bed in the junk
- yard, a keyboard on his lap. He offers
- to show me a few digits of pi, and taps
- at the keys.
- On the screen beside his bed, m zero
- responds: "Please, give the beginning
- of the decimal digit to look."
- Gregory types a command, and
- suddenly the whole screen fills with
- the raw Ludolphian number, moving
- like Niagara Falls. We observe pi in
- silence for quite a while, until it ends
- with:
- . . . 18820 54573 01261 27678 17413 87779
- 66981 15311 24707 34258 41235 99801 92693
- 52561 92393 53870 24377 10069 16106 22971
- 02523 30027 49528 06378 64067 12852 77857
- 42344 28836 88521 72435 85924 57786 36741
- 32845 66266 96498 68308 59920 06168 63376
- 85976 35341 52906 04621 44710 52106 99079
- 33563 54625 71001 37490 77872 43403 57690
- 01699 82447 20059 93533 82919 46119 87044
- 02125 12329 11964 10087 41341 42633 88249
- 48948 31198 27787 03802 08989 05316 75375
- 43242 20100 43326 74069 33751 86349 40467
- 52687 79749 68922 29914 46047 47109 31678
- 05219 48702 00877 32383 87446 91871 49136
- 90837 88525 51575 35790 83982 20710 59298
- 41193 81740 92975 31.
- "It showed the last digits we've
- found," Gregory says. "The last shall
- be first."
- "Thanks for asking," m zero re-
- marks, on the screen.
- &emdash;RICHARD PRESTON