The mountains of Pian article from the New Yorker, March 2, 1992.                                   [Image]                Gregory V. Chudnovsky and David V. ChudnovskyGREGORY VOLFOVICH CHUDNOVSKY recently built asupercomputer in his apart-ment from mail-order parts. Greg-ory Chudnovsky is a number theo-rist. His apartment is situated nearthe 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 dumpedat the end of the block. The world'smost powerful supercomputers in-clude the Cray Y-MP C90, theThinking Machines CM-5, theHitachi S-820/80, the nCube, theFujitsu parallel machine, theKendall Square Research parallelmachine, the NEC SX-3, theTouchstone Delta, and GregoryChudnovskv's apartment. Theapartment seems to be a kind of con-tainer for the supercomputer at leastas much as it is a container forpeople.Gregory Chudnovsky's partner inthe design and construction of thesupercomputer was his older brother,David Volfovich Chudnovsky, who isalso a mathematician, and who livesfive blocks away from Gregory. TheChudnovsky brothers call their ma-chine m zero. It occupies the formerliving 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 powerfulas a somewhat older Cray Y-MP, butit is not as fast as the latest of theY-MP machines, the C90, an ad-vanced supercomputer made by CrayResearch. A Cray Y-MP C90 costsmore than thirty million dollars. It isa black monolith, seven feet tall andeight feet across, in the shape of a squatcylinder, and is cooled by liquid freon.So far, the brothers have spent aroundseventy thousand` dollars on parts fortheir supercomputer, and much of themoney has come out of their wives'pockets.Gregory Chudnovsky is thirty-nineyears old, and he has a spare frameand a bony, handsome face. He has along beard, streaked with gray, anddark, unruly hair, a wide forehead,and wide-spaced brown eyes. He walksin a slow, dragging shuffle, leaning ona bentwood cane, while his brother,David, typically holds him under onearm, to prevent him from topplingover. 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 thetime," Gregory once told me. Hiscondition doesn't seem to be gettingbetter, and doesn't seem to be gettingworse. He developed the disease whenhe was twelve years old, in the city ofKiev, Ukraine, where he and Davidgrew up. He spends his days sitting orlying on a bed heaped with pillows, ina bedroom down the hall from theroom that houses the supercomputer.Gregory's bedroom is filled with pa-per; it contains at least a ton ofpaper. He calls the place his junk yard.The room faces east, and would befull of sunlight in the morning if heever raised the shades, but he keepsthem lowered, because light hurtshis eyes.You almost never meet one of theChudnovsky brothers without the other.You often find the brothers conjoined,like Siamese twins, David holdingGregory by the arm or under thearmpits. They complete each other'ssentences and interrupt each other,but they don't look alike. WhileGregory is thin and bearded, Davidhas a stout body and a plump,clean-shaven face. He is in hisearly forties. Black-and-gray curlyhair grows thickly on top of David'shead, and he has heavy-liddeddeep-blue eyes. He always wearsa 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 thousandwatts of power, and it runs dayand night. The brothers don't dareshut it down; if they did, it mightdie. At least twenty-five fans blowair through the machine to keep it cool;otherwise something might melt. Wasteheat permeates Gregory's apartment,and the room that contains m zeroclimbs to a hundred degrees Fahren-heit in summer. The brothers keep theapartment's lights turned off as muchas possible. If they switched on toomany lights while m zero was run-ning, they might blow the apartment'swiring. Gregory can't breathe city airwithout developing lung trouble, so hekeeps the apartment's windows closedall the time, with air-conditionersrunning in them during the summer,but that doesn't seem to reduce theheat, and as the temperature rises in-side the apartment the place can smellof cooking circuit boards, a sign thatm zero is not well. A steady streamof boxes arrives by Federal Express,and an opposing stream of boxes flowsback to mail-order houses, contain-ing parts that have bombed, alongwith letters from the brothers demand-ing an exchange or their money back.The building superintendent doesn'tknow that the Chudnovsky brothershave been using a supercomputer inGregory's apartment, and the broth-ers haven't expressed an eagerness totell him.The Chudnovskys, between them,have published a hundred and fifty-four papers and twelve books, mostlyin collaboration with each other, andmostly on the subject of number theoryor mathematical physics. They worktogether so closely that it is possible toargue 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 toFrance. The family lived there for sixmonths, then emigrated to the UnitedStates and settled in New York; theyhave become American citizens.The brothers enjoy an official rela-tionship with Columbia University:Columbia calls them senior researchscientists in the Department of Math-ematics, but they don't have tenure andthey don't teach students. They arereally lone inventors, operating out ofGregory's apartment in what you mightcall the old-fashioned Russo-Yankeestyle. Their wives are doing well.Gregory's wife, Christine Pardo Chud-novsky, is an attorney with a midtownlaw firm. David's wife, Nicole Lanne-grace, is a political-affairs officer atthe United Nations. It is their salariesthat help cover the funding needs ofthe brothers' supercomputing complexin Gregory and Christine's apart-ment. Malka Benjaminovna Chud-novsky, a retired engineer, who isGregory and David's mother, lives inGregory's apartment. David spends hisdays in Gregory's apartment, takingcare of his brother, their mother, andm zero.When the Chudnovskys applied toleave theSoviet Union, the fact thatthey are Jewish and mathematicalattracted at least a dozen K.G.B. agentsto their case. The brothers' father, VolfGrigorevich Chudnovsky, who wasseverely beaten by the K.G.B. in 1977,died of heart failure in 1985. VolfChudnovsky was a professor of civilengineering at the Kiev ArchitecturalInstitute, and he specialized in thestructural stability of buildings, towers,and bridges. He died in America, andnot long before he died he constructedin 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 todaythey are packed with literature andcomputer books and books and paperson the subject of numbers. Since almostall numbers run to infinity (in digits)and are totally unexplored, an apart-mentful of thoughts about numbersholds hardly any thoughts at all, evenwith a supercomputer on the premisesto advance the work.The brothers say that the "m" in"m zero" stands for "machine," andthat they use a small letter to imply thatthe machine is a work in progress.They represent the name typographi-cally as "mO." The "zero" stands forsuccess. It implies a dark history offailure&emdash;three duds (in Gregory's apart-ment) that the brothers now refer toas negative three, negative two, andnegative one. The brothers broke upthe negative machines for scrap, got onthe telephone, and waited for FederalExpress to bring more parts.M zero is a parallel supercomputer,a type of machine that has lately cometo dominate the avant-garde in super-computer architecture, because thedesign offers succulent possibilities forspeed in solving problems. In a par-allel machine, anywhere from half adozen to thousands of processors worksimultaneously on a problem, whereasin a so-called serial machine&emdash;a nor-mal computer&emdash;the problem is solvedone step at a time. "A serial machineis bound to be very slow, because thespeed of the machine will be limitedby the slowest part of it," Gregory said."In a parallel machine, many circuitstake on many parts of the problem atthe same time." As of last week, m zerocontained sixteen parallel processors,which ruminate around the clock onthe Chudnovskys' problems.The brothers' mail-order super-computer makes their lives more con-venient: m zero performs inhumanelydifficult algebra, finding roots of gi-gantic systems of equations, and it hasconstructed colored images of the in-terior of Gregory Chudnovsky's body.According to the Chudnovskys, it couldmodel the weather or make pictures ofair flowing over a wing, if the brotherscared about weather or wings. Whatthey care about is numbers. To them,numbers are more beautiful, more nearlyperfect, possibly more complicated, andarguably more real than anything inthe world of physical matter.The brothers have lately been usingm zero to explore the number pi. Pi,which is denoted by the Greek letterPi, is the most famous ratio in math-ematics, and is one of the most ancientnumbers known to humanity. Pi isapproximately 3.14&emdash;the number oftimes that a circle's diameter will fitaround the circle. Here is a circle, withits diameter:Pi goes on forever, and can'tbe calculated to perfectprecision:3.1415926535897932384626433832795028841971693993751.... This isknown 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 infinityin a predestined yet unfathomable code:they do not repeat periodically, seem-ing to pop up by blind chance, lackingany perceivable order, rule, reason, ordesign&emdash;"random" integers, ad infini-tum. If a deep and beautiful designhides in the digits of pi, no one knowswhat it is, and no one has ever beenable to see it by staring at the digits.Among mathematicians, there is a nearlyuniversal feeling that it will never bepossible, in principle, for an inhabitantof our finite universe to discover thesystem in the digits of pi. But for thepresent, if you want to attempt it, youneed a supercomputer to probe theendless scrap of leftover pi.Before the Chudnovsky brothers builtm zero, Gregory had to derive pi overthe telephone network while lying inbed. It was inconvenient. Tapping ata small keyboard, which he sets on theblankets of his bed, he stares at acomputer display screen on one of thebookshelves beside his bed. The key-board and the screen are connected toInternet, a network that leads Gregorythrough cyberspace into the heart of aCray somewhere else in the UnitedStates. He calls up a Cray throughInternet and programs the machine tomake an approximation of pi. The jobbegins to run, the Cray trying to es-timate the number of times that thediameter of a circle goes around theperiphery, and Gregory sits back onhis pillows and waits, watching mes-sages from the Cray flow across hisdisplay screen. He eats dinner with hiswife and his mother and then, back inbed, he takes up a legal pad and a redfelt-tip pen and playswith number theory,trying to discover hid-den properties of num-bers. Meanwhile, theCray is reaching towardpi at a rate of a hundredmillion operations persecond. Gregory dozesbeside his computerscreen. Once in a while,he asks the Cray howthings are going, andthe Cray replies that thejob is still active. Nightpasses, the Cray run-ning deep toward pi.Unfortunately, since theexact ratio of the circle'scircumference to its di-ameter dwells at infin-ity, the Cray has noteven begun to pinpointpi. Abruptly, a messageappears on Gregory'sscreen:LINE ISDISCONNECTED."What the hell isgoing on?" Gregory ex-claims. It seems thatthe Cray has hung upthe phone, and mayhave crashed. Onceagain, pi has demon-strated its ability to givea supercomputer a heartattack.MYASTHENIA GRA-VIS iS a funnything Gregrory Chudnovsky said one day from his bed inthe junk yard. "In a sense, I'm verylucky, because I'm alive, and I'm aliveafter so many years." He has a reso-nant voice and a Russian accent. "Thereis no standard prognosis. It sometimesstrikes young women and older women.I wonder if it is some kind of sluggishvirus."It was a cold afternoon, and rainpelted the windows; the shades weredrawn, as always. He lay against aheap of pillows, with his legs foldedunder him. He wore a tattered graylamb's-wool sweater that had multiplepatches on the elbows, and a starchedwhite shirt, and baggy blue sweat pants,and a pair of handmade socks. I hadnever seen socks like Gregory's. Theywere two-tone socks, wrinkled andfloppy, hand-sewn from pieces of dark-blue and pale-blue cloth, and theylooked comfortable. They were thework of Malka Benjaminovna, hismother. Lines of computer code flickeredon the screen beside his bed.This was an apartment built forlong voyages. The paper in the roomwas jammed into the bookshelves, fromfloor to ceiling. The brothers hadwedged the paper, sheet by sheet, intomanila folders, until the folders hadgrown as fat as melons. The paper alsoflooded two freestanding bookshelves(placed strategically around Gregory'sbed), five chairs (three of them in arow beside his bed), two steamer trunks,and a folding cocktail table. I movedcarefully around the room, fearful oftriggering a paper slide, and sat on theroom's one empty chair, facing the footof Gregory's bed, my knees touchingthe blanket. The paper was piled inthree-foot stacks on the chairs. Itguarded his bed like the flanking tow-ers of a fortress, and his bed sat at thecenter of the keep. I sensed a profoundhappiness in Gregory Chudnovsky'sbedroom. His happiness, it occurred tome later, sprang from the deliciousmelancholy of a life chained to a bedin a disordered world that breaks openthrough the portals of mathematicsinto vistas beyond time or decay."The system of this paper isarcheological," he said. "Bylooking at a slice, I know theyear. This slice is 1986. Overhere is some 1985. What yousee in this room is our work-ing papers, as well as the paperswe used as references for them.Some of the references we pullout once in a while to look at,and then we leave them some-where else, in another pile.Once, we had to make a Xeroxcopy of a book three times, andwe put it in three differentplaces in the piles, so we wouldbe sure to find it when weneeded it. Unfortunately, oncewe put a book into one of thesepiles we almost never go backto look for it. There are booksin there by Kipling and Macau-lay. Actually, when we wantto find a book it's easier to goback to the library. Eh. thisplace is a mess. Eventually, these papersor my wife will turn me out of thehouse."Much of the paper consists of legalpads covered with Gregory's hand-writing. His holograph is dense andcareful, a flawless minuscule writtenwith a red felt-tip pen&emdash;a mixture oftheorems, calculations, proofs, and con-lectures concerning numbers. He usesa felt-tip pen because he doesn't haveenough strength in his hand to pressa pencil on paper. Mathematicians whohave visited Gregory Chudnovsky'sbedroom have come away dizzy, won-dering what secrets the scriptoriummay hold. Some say he has publishedmost of his work, while others wonderif his bedroom holds unpublished dis-coveries. He cautiously refers to hissteamer trunks as valises. They arefilled to the lids with compressed pa-per. When Gregory and David usedto fly to Europe to speak at conferences,they took both "valises" with them, incase they needed to refer to a theorem,and the baggage particularly annoyedthe Belgians. "The Belgians werealways 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 intereststhem very much. They wonder whetherthe digits contain a hidden rule, an asvet unseen architecture. close to themind of God. A subtle and fantasticorder may appear in the digits of piway out there somewhere; no oneknows. No one has ever proved, forexample, that pi does not turn intonothing but nines and zeros, spatteredto infinity in some peculiar arrange-ment. If we were to explore the digitsof pi far enough, they might resolveinto a breathtaking numerical pattern,as knotty as "The Book of Kells," andit might mean something. It might bea small but interesting message fromGod, hidden in the crypt of the circle,awaiting notice by a mathematician.On the other hand, the digits of pi mayramble forever in a hideous cacophony,which is a kind of absolute perfectionto a mathematician like Gregory Chud-novsky. Pi looks "monstrous" to him."We know absolutely nothing aboutpi," he declared from his bed. "Whatthe hell does it mean? The definitionof pi is really very simple&emdash;it's just theratio of the circumference to the diam-eter&emdash;but the complexity of the se-quence it spits out in digits is reallyunbelievable. We have a sequence ofdigits that looks like gibberish.""Maybe in the eyes of God pi looksperfect," David said, standing in a cor-ner of the room, his head and shouldersvisible above towers of paper.Pi, or 1t, has had various namesthrough the ages, and all of them areeither words or abstract symbols, sincepi is a number that can't be showncompletely and exactly in any finiteform of representation. Piis a transcendental number. A transcendentalnumber is a number that exists butcan't be expressed in any finite seriesof either arithmetical or algebraic op-erations. For example, if you try toexpress pi as the solution to an equa-tion you will find that the equationgoes on forever. Expressed in digits, piextends into the distance as far as theeye can see, and the digits never repeatperiodically, as do the digits of a ra-tional number. Pi slips away from allrational methods used to locate it. Piis a transcendental number because ittranscends 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 bewritten on a piece of paper, not evenon a piece of paper as big as theuniverse. In a manner of speaking, piis indescribable and can't be found.Pi possibly first entered human con-sciousness in Egypt. The earliest knownreference to pi occurs in a MiddleKingdom papyrus scroll, written around1650 B.C. by a scribe named Ahmes.Showing a restrained appreciation forhis own work that is not uncommonin a mathematician, Ahmes began hisscroll with the words "The EntranceInto the Knowledge of All ExistingThings." He remarked in passing thathe composed the scroll "in likeness towritings made of old," and then he ledhis readers through various mathemati-cal problems and their solutions, alongseveral feet of papyrus, and toward theend of the scroll he found the area ofa circle, using a rough sort of pi.Around 200 B.C., Archimedes ofSyracuse found that pi is somewherebetween 3 1O/7l and 3 1/7_that's about3.14. (The Greeks didn't use deci-mals.) Archimedes had no special termfor pi, calling it "the perimeter to thediameter." By in effect approximatingpi to two places after the decimal point,Archimedes narrowed the known valueof pi to one part in a hundred. Thereknowledge of pi bogged down until theseventeenth century, when new for-mulas for approximating pi were dis-covered. Pi then came to be called theLudolphian number, after Ludolph vanCeulen, a German mathematician whoapproximated it to thirty-five decimalplaces, or one part in a hundred mil-lion billion billion billion&emdash;a calcula-tion that took Ludolph most of his lifeto accomplish, and gave him suchsatisfaction that he had the digits en-graved on his tombstone, at the Ladies'Church in Leiden, in the Netherlands.Ludolph and his tombstone were latermoved to Peter's Church in Leiden, tobe installed in a special graveyard forprofessors, and from there the stonevanished, possibly to be turned into asidewalk slab. Somewhere in Leiden,people may be walking over Ludolph'sdigits. The Germans still call pi theLudolphian number. In the eighteenthcentury, Leonhard Euler, mathemati-cian to Catherine the Great, called itp or c. The first person to use theGreek letter Pi for the number wasWilliam Jones, an English mathema-tician, who coined it in 1706 for hisbook "A New Introduction to the Math-ematics." Euler read the book andswitched to using the symbol Pi, andthe number has remained Pi ever since.Jones probably meant Pi to stand for theEnglish word "periphery."Physicists have noted the ubiquity ofpi in nature. Pi is obvious in the disksof the moon and the sun. The doublehelix of DNA revolves around pi. Pihides in the rainbow, and sits in thepupil of the eye, and when a raindropfalls into water pi emerges in thespreading rings. Pi can be found inwaves and ripples and spectra of allkinds, and therefore pi occurs in colorsand music. Pi has lately turned up insuperstrings, the hypothetical loops ofenergy vibrating inside subatomic par-ticles. Pi occurs naturally in tables ofdeath in what is known as a Gaussiandistribution of deaths in apopulation; that is, when aperson dies, the event"feels" the Ludolphiannumber.It is one of the greatmysteries why nature seemsto know mathematics. Noone can suggest why thisnecessarily has to be so.Eugene Wigner, the physi-cist, once said, "The mir-acle of the appropriatenessof the language of math-ematics for the formula-tion of the laws of physicsis a wonderful gift whichwe neither understand nordeserve." We may not un-derstand pi or deserve it,but nature at least seems tobe aware of it, as Captain0. C. Fox learned whilehe was recovering in ahospital from a wound sus-tained in the AmericanCivil War. Having noth-in better to do with histime than lie in bed and derive pi,Captain Fox spent a few weeks tossingpieces of fine steel wire onto a woodenboard ruled with parallel lines. Thewires fell randomly across the lines insuch a way that pi emerged in thestatistics. After throwing his wires elev-en hundred times, Captain Fox wasable to derive pi to two places after thedecimal point, to 3.14. If he had hada thousand years to recover from hiswound, he might have derived pi toperhaps another decimal place. To godeeper into pi, you need a powerfulmachine.The race toward pi happens incyberspace, inside supercomputers. In1949, George Reitwiesner, at the Bal-listic Research Laboratory, in Mary-land, derived pi to two thousand andthirty-seven decimal places with theENIAC, the first general-purpose elec-tronic digital computer. Working atthe same laboratory, John von Neu-mann (one of the inventors of theENIAC) searched those digits for signs oforder, but found nothing he could puthis finger on. A decade later, DanielShanks and John W. Wrench, Jr.,approximated pi to a hundred thousanddecimal places with an I.B.M. 7090mainframe 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 NECsupercomputer, a Japanese machine, tocompute two million digits of pi. Peoplewere astonished that anyone wouldbother to do it, but that was only thebeginning of the affair. In 1984, Kanadaand his team got sixteen million digitsof pi, noticing nothing remarkable. Ayear later, William Gosper, a math-ematician and distinguished hacker em-ployed at Symbolics, Inc., in Sunny-vale, California, computed pito seventeen and a half mil-lion decimal places with aSymbolics 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 Aeronauticsand Space Administration, used a Cray 2supercomputer and a formula discov-ered by two brothers, Jonathan andPeter Borwein, to scoop twenty-ninemillion digits of pi. Bailey found noth-ing unusual. A year after that, in 1987,Yasumasa Kanada and his team got ahundred and thirty-four million digitsof pi, using a NEC SX-2 supercom-puter. They saw nothing of interest.In 1988, Kanada kept going, past twohundred million digits, and saw fur-ther amounts of nothing. Then, in thespring of 1989, the Chudnovsky broth-ers (who had not previously been knownto have any interest in calculating pi)suddenly announced that they hadobtained four hundred and eighty mil-lion digits of pi&emdash;a world record&emdash;using supercomputers at two sites inthe United States, and had seen noth-ing. Kanada and his team were a littlesurprised to learn of unknown compe-tition operating in American cyberspace,and they got on a Hitachi supercom-puter and ripped through five hundredand thirty-six million digits, beatingthe Chudnovksys, setting a new worldrecord, and seeing nothing. The broth-ers kept calculating and soon crackeda billion digits, but Kanada's restlessboys and their Hitachi then nosed intoa little more than a billion digits. TheChudnovskys pressed onward, too, andby the fall of 1989 they had squeakedpast Kanada again, having computedpi to one billion one hundred andthirty million one hundred and sixtythousand six hundred and sixty-fourdecimal places, without finding any-thing special. It was another worldrecord. At that point, the brothers gaveup, out of boredom.If a billion decimals of pi wereprinted in ordinary type, they wouldstretch from New York City to themiddle of Kansas. This notion raisesthe question: What is the point ofcomputing pi from New York to Kan-sas? The question has indeed beenasked among mathematicians, since anexpansion of pi to only forty-sevendecimal places would be sufficientlyprecise to inscribe a circle around thevisible universe that doesn'tdeviate from perfect circu-larity by more than the dis-tance across a single proton.A billion decimals of pi go sofar beyond that kind of pre-cision, into such a lunacy ofexactitude, that physicists willnever need to use the quantity in anyexperiment&emdash;at least, not for any phys-ics we know of today&emdash;and the thoughtof a billion decimals of pi oppresseseven some mathematicians, who de-clare the Chudnovskys' effort trivial. Ionce asked Gregory if a certain im-pression I had of mathematicians wastrue, that they spent immoderateamounts of time declaring each other'swork trivial. "It is true," he admitted."There is actually a reason for this.Because once you know the solution toa problem it usually is trivial."Gregory did the calculation from hisbed in New York, working throughcyberspace on a Cray 2 at the Minne-sota Supercomputer Center, in Minne-apolis, and on an I.B.M. 3090-VFsupercomputer 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 supercomputersonly in bits and pieces, usually duringholidays and in the dead-of night. Itwas also quite expensive&emdash;the use ofthe Cray cost them seven hundred andfifty dollars an hour, and the moneycame from the National Science Foun-dation. By the time of this agony, thebrothers had concluded that it wouldbe cheaper and more convenient tobuild a supercomputer in Gregory'sapartment. Then they could crash theirown machine all they wanted, whilethey opened doors in the house ofnumbers. The brothers planned tocompute two billion digits of pi on theirnew machine&emdash;to try to double theirold world record. They thought itwould be a good way to test theirsupercomputer: a maiden voyage intopi would put a terrible strain on theirmachine, might blow it up. Presumingthat their machine wouldn't overheator strangle on digits, they planned tosearch the huge resulting string of pifor signs of hidden order. If what theChudnovsky brothers have seen in theLudolphian number is a message fromGod, the brothers aren't sure whatGod is trying to say.ON a cold winter day, when theChudnovskys were about to be-gin their two-billion-digit expeditioninto pi, I rang the bell of GregoryChudnovsky's apartment, and Davidanswered the door. He pulled the dooropen a few inches, and then it stopped,jammed against an empty cardboardbox and a wad of hanging coats. Henudged the box out of the way with hisfoot. "Look, don't worry," he said."Nothing unpleasant will happen toyou. We will not turn you into digits."A Mini Mag-Lite flashlight protrudedfrom his shirt pocket.We were standing in a long, darkhallway. The lights were off, and itwas hard to see anything. To try tofind something in Gregory's apartmentis like spelunking; that was the reasonfor David's flashlight. The hall islined on both sides with bookshelves,and they hold a mixture of paper andbooks. The shelves leave a passageabout two feet wide down the lengthof the hallway. At the end of thehallway is a French door, its mul-lioned glass covered with translucentpaper, and it glowed.The apartment's rooms are strungout along the hallway. We passed abathroom and a bedroom. The bed-room belonged to Malka BenjaminovnaChudnovsky. We passed a cave ofpaper, Gregory's junk yard. We passeda small kitchen, our feet rolling oncomputer cables. David opened theFrench door, and we entered the roomof the supercomputer. A bare light bulbburned in a ceiling fixture. The roomcontained seven display screens: two ofthem were filled with numbers; theothers were turned off. The windowswere closed and the shades were drawn.Gregory Chudnovsky sat on a chairfacing the screens. He wore the usualoutfit&emdash;a tattered and patched lamb's-wool sweater, a starched white shirt,blue sweat pants, and the hand-stitchedtwo-tone socks. From his toes traileda pair of heelless leather slippers. Hiscane was hooked over his shoulder,hung there for convenience. I shookhis hand. "Our first goal is to computepi," he said. "For that we have to buildour own computer.""We are a full-service company,"David said. "Of course, you knowwhat 'full-service' means in New York.It means 'You want it? You do ityourself."'A steel frame stood in the center ofthe room, screwed together with bolts.It held split shells of mail-order personal computers&emdash;cheap P.C. clones, knockedwide open, like cracked wal-nuts, their meat spilling allover the place. The brothershad crammed special logicboards inside the personalcomputers. Red lights on theboards blinked. The floorwas a quagmire of cables.The brothers had also managed tofit into the room masses of emptycardboard boxes, and lots of books(Russian classics, with Cyrillic letter-ing on their spines), and screwdrivers,and data-storage tapes, and softwaremanuals by the cubic yard, and stalag-mites of obscure trade magazines, anda twenty-thousand-dollar computerworkstation that the brothers no longerused. ("We use it as a place to stackpaper," Gregory said.) From an ovalphotograph on the wall, the face oftheir late father&emdash;a robust man, squint-ing thoughtfully&emdash;looked down on thescene. The walls and the French doorwere covered with sheets of draftingpaper showing circuit diagrams. Theyresembled cities seen from the air: thebrothers had big plans for m zero.Computer disk drives stood around theroom. The drives hummed, and therewas a continuous whirr of fans, anda strong warmth emanated from theequipment, as if a steam radiator weregoing in the room. The brothers heattheir apartment largely with chips.Gregory said, "Our knowledge of piwas barely in the millions of digits&emdash;""We need many billions of digits,"David said. "Even a billion digits is adrop in the bucket. Would you like aCoca-Cola?" He went into the kitchen,and there was a horrible crash. "Nevermind, I broke a glass," he called."Look, it's not a problem." He cameout of the kitchen carrying a glass ofCoca-Cola on a tray, with a papernapkin under the glass, and as hehanded it to me he urged me to holdit tightly, because a Coca-Cola spilledinto&emdash;He didn't want to think aboutit; it would set back the project bymonths. He said, "Galileo had to buildhis telescope&emdash;""Because he couldn't afford the Dutchmodel," Gregory said."And we have to build our machine,because we have&emdash;""No money," Gregory said. "Whenpeople let us use their computer, it'salwavs done as a kindness." He grinnedand pinched his finger andthumb together. "They say,'You can use it as long asnobody complains."'I asked the brotherswhen they planned to buildtheir 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 thebrothers.David turned to his brother andsaid, "The interviewer answers ourquestions. It's Pirandello! The inter-viewer becomes a person in the story."David turned to me and said, "Theproblem is, you should change yourthinking. If I were to put inside thisCray a chopped-meat machine, youwouldn't know it was a meat chopper.""Unless you saw chopped meatcoming out of it. Then you'd suspectit wasn't a Cray," Gregory said, andthe brothers cackled."In ten years, a Cray will fit in yourpocket," David said.Supercomputers are evolving incred-ibly fast. The notion of what a super-computer is and what it can do changesfrom year to year, if not from monthto month, as new machines arise. Thedefinition of a supercomputer is simplythis: 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 itspeak working speed, can perform morethan two billion floating-point opera-tions per second. Floating-point opera-tions&emdash;or flops, as they are called&emdash;area standard measure of speed. Since aCray Y-MP8 can hit two and a halfbillion flops, it is considered to be agigaflop supercomputer. Giga (fromthe Greek for "giant") means a bil-lion. Like all supercomputers, a Crayoften cruises along significantly belowits peak working speed. (There is aheated controversy in the supercom-puter industry over how to measure thetypical working performance of anygiven supercomputer, and there aremany claims and counterclaims.) ACray is a so-called vector-processingmachine, but that design is going outof fashion. Cray Research has an-nounced that next year it will beginselling an even more powerful parallelmachine."Our machine is a gigaflop super-computer," David Chudnovsky toldme. "The working speed of our ma-chine is from two hundred million flopsto two gigaflops&emdash;roughly in the rangeof a Cray Y-MP8. We can probably gofaster than a Y-MP8, but we don'twant to get too specific about it."M zero is not the only ultrapowerfulsilicon engine to gleam in the Chud-novskian |uvre. The brothers recentlyfielded a supercomputer named LittleFermat, which they designed withMonty Denneau, an I.B.M. super-computer architect, and Saed Younis,a graduate student at the Massachu-setts Institute of Technology. Younisdid the grunt work: he mapped outcircuits containing more than fifteenthousand connections and personallyplugged in some five thousand chips.Little Fermat is seven feet tall, and sitsinside a steel frame in a laboratory atM.I.T., where it considers numbers.What m zero consists of is a groupof 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 paperthe layout of the machine. He drew abox and put an "x" through it, to showthe web, or network, and he attachedsome processors to the web:"Each processor is connected to ahigh-speed switching network thatconnects it to all the others," he said."It's like a telephone network&emdash;every-body is talking to everybody else. Asfar as I know, no one except us hasbuilt a machine that has this type ofweb. In other parallel machines, theprocessors are connected only to nearneighbors, while they have to talk tomore distant processors through inter-vening processors. Think of a phonesystem: it wouldn't be very pleasant ifyou had to talk to distant people bysending them messages through yourneighbors. But the truth is that nobodyreally knows how the hell parallelmachines should perform, or the bestdesign for them. Right now we haveeight processors. We plan to havetwo hundred and fifty-six processors.We will be able to fit them into theapartment."He said that each processor had itsown memory attached to it, so thateach processor was in fact a separatecomputer. After a processor was fedsome data and had got a result, it couldsend the result through the web toanother processor. The brothers wrotethe machine's application software inFORTRAN, a programming languagethat is "a dinosaur from the late fifties,"Gregory said, adding, "There is al-ways new life in this dinosaur." Thesoftware can break a problem intopieces, sending the pieces to the ma-chine's different processors. "It's theprinciple of divide and conquer," Greg-ory said. He said that it was very hardto know what exactly was happeningin the web when the machine wasrunning&emdash;that the web seemed to havea life of its own."Our machine is mostly made ofconnections," David said. "About ninetyper cent of its volume is cables. Yourbrain is the same way. It is mostlymade of connections. If I may say so,your brain is a liquid-cooled parallelsupercomputer." He pointed to his nose"This is the fan."The design of the web is the keyelement in the Chudnovskian architec-ture. Behind the web hide several newr findings in number theory, which theChudnovskys have not yet publishedThe brothers would not disclose to methe exact shape of the web, or thediscoveries behind it, claiming thatthey needed to protect their competitiveedge in a worldwide race to developfaster supercomputers. "Anyone with ahundred million dollars and brainscould be our competitor," David saiddryly.The Chudnovskys have formidablecompetitors. 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 fourhundred thousand dollars and goes upfrom there. If you had a hundredmil1ion dollars, you could order a CM-Sthat would be an array of black mono-liths the size of a Burger King, andit would burn enough electricity tolight up a neighborhood. Seymour Crayis another competitor of the brothers,as it were. He invented the originalCray series of supercomputers, and isnow the head of the Cray ComputerCorporation, a spinoff from Cray Re-search. Seymour Cray has been work-ing to develop his Cray 3 for severalyears. His company's effort has re-cently been troubled by engineeringdelays and defections of potential cus-tomers, but if the machine ever isreleased to customers it may be anoctagon about four feet tall and fourfeet across, and it will burn more thantwo hundred thousand watts. It wouldmelt instantly if its cooling systemwere to fail.Then, there's the Intel Corporation.Intel, together with a consortium offederal agencies, has invested morethan twenty-seven million dollars inthe Touchstone Delta, a five-foot-high,fifteen-foot-long parallel supercomputerthat sits in a computer room at Caltech.The machine consumes twenty-fivethousand watts of power, and is keptfrom overheating by chilled air flow-ing through its core. One day, I calledPaul Messina, a Caltech research sci-entist, who is the head of the Touch-stone Delta project, to get his opinionof the Chudnovsky brothers. It turnedout that Messina hadn't heard ofthem. As for their claim to have builta pi-computing gigaflop supercomputerout of mail-order parts for aroundseventy thousand dollars, he flatlybelieved it. "It can be done, definitely,"Messina said. "Of course, seventythousand dollars is just the cost of thecomponents. The Chudnovskys arecounting very little of their humantime."Yasumasa Kanada, the brothers' pirival at Tokyo University, uses a HitachiS-820/80 supercomputer that is be-lieved to be considerably faster than aCray Y-MP8, and it burns close tohalf a million watts&emdash;half a megawatt,practically enough power to melt steel.The Chudnovsky brothers particularlyhoped to leave Kanada and his Hita-chi in the dust with their mail-orderfunny 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 whatmakes pi different from other num-bers. It's a business.""Galileo saw the moons of Jupiterthrough his telescope, and he tried tofigure out the laws of gravity by look-ing at the moons, but he couldn't,"David said. "With pi, we are at thestage of looking at the moons of Ju-piter." He pulled his Mini Mag-Liteflashlight out of his pocket and shoneit into a bookshelf, rooted throughsome file folders, and handed me acolor photograph of pi. "This is a pi-scape," he said. The photograph showeda mountain range in cyberspace: bonypeaks and ridges cut by valleys. Themountains and valleys were splashedwith colors&emdash;yellow, green, orange,violet, and blue. It was the first eightmillion digits of pi, mapped as a fractallandscape by an I.B.M. GF-l 1 super-computer at Yorktown Heights, whichGregory had programmed from hisbed. Apart from its vivid colors, pilooks like the Himalayas.Gregory thought that the mountainsof pi seemed to contain structure. "Isee something systematic in this land-scape, but it may be just an attempt bythe brain to translate some randomvisual pattern into order," he said. Ashe gazed into the nature beyond na-ture, he wondered if he stood close toa revelation about the circle and itsdiameter. "Any very high hill in thispicture, or any flat plateau, or deepvalley, would be a sign of something inpi," he said. "There are slight varia-tions from randomness in this land-scape. There are fewer peaks andvalleys than you would expect if piwere truly random, and the peaks andvalleys tend to stay high or low a littlelonger than you'd expect." In a man-ner of speaking, the mountains of pilooked to him as if they'd been moldedby the hand of the Nameless One,Deus absconditus (the hidden God), buthe couldn't really express in wordswhat he thought he saw and, to hisgreat frustration, he couldn't express itin the language of mathematics, either."Exploring pi is like exploring theuniverse," David remarked."It's more like exploring underwa-ter," Gregory said. "You are in themud, and everything looks the same.You need a flashlight. Our computeris the flashlight."David said, "Gregory&emdash;I think,really&emdash;you are getting tired."A fax machine in a corner beepedand emitted paper. It was a messagefrom a hardware dealer in Atlanta.David tore off the paper and stared atit. "They didn't ship it! I'm going tokill them! This a service economy. Ofcourse, 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. Wecan't buy everything&emdash;""Because everything won't exist,"Gregory said."We only want to build a ma-chine to compute a few transcendentalnumbers&emdash;""Because we are not licensed fortranscendental meditation," Gregorysaid."Look, we are getting nutty," Davidsaid."We are not the only ones," Greg-ory said. "We are getting an averageof one letter a month from someone orother who is trying to prove Fermat'sLast Theorem."I asked the brothers if they hadpublished any of their digits of pi ina book.Gregory said that he didn't knowhow many trees you would have togrind up in order to publish a billiondigits of pi in a book. The brothers' pihad been published on fifteen hundredmicrofiche cards stored somewhere inGregory's apartment. The cards heldthree hundred thousand pages of data,a slug of information much biggerthan the Encylopaedia Britannica, andcontaining but one entry, "Pi." Davidoffered to find the cards for me; theyhad to be around here somewhere. Heswitched on the lights in the hallwayand began to shift boxes. Gregoryrifled bookshelves."Please sit down, Gregory," Davidsaid. Finally, the brothers confessedthat they had temporarily lost their pi."Look, it's not a problem," David said."We keep it in different places." Hereached inside m zero and pulled outa metal box. It was a naked hard-diskdrive, studded with chips. He handedme the object. "There's pi stored onthis drive." It hummed gently. "Youare holding some pi in your hand. Itweighs six pounds."MONTHS passed before I visitedthe Chudnovskys again. Thebrothers had been tinkering with theirmachine and getting it ready to go fortwo billion digits of pi, when Gregorydeveloped an abnormality related toone of his kidneys. He went to thehospital and had some CAT scans madeof his torso, to see what things lookedlike, but the brothers were disappointedin the pictures, and persuaded the doctorsto 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-sectionalslices of his body, viewed throughdifferent angles, and they were farmore detailed than any image from aCAT scanner. Gregory wrote the im-aging software. It took him a fewweeks. "There's a lot of interestingmathematics in the problem of imag-ing a body," he remarked. For themoment, it was more interesting thanpi, and it delayed the brothers' probeinto the Ludolphian number.Spring came, and Federal Expresswas active at the Chudnovskys' build-ing. Then the brothers began to cal-culate pi, slowly at first, more intenselyas they gained confidence in theirmachine, but in May the weatherwarmed up and Con Edison betrayedthe brothers. A heat wave caused abrownout in New York City, and asit struck, m zero automatically shutitself down, to protect its circuits, anddied. Afterward, the brothers couldn'tget electricity running properly throughthe machine. They spent two weeksrestarting it, piece by piece.Then, on Memorial Day weekend,as the calculation was beginning toprogress, Malka Benjaminovna suffereda heart attack. Gregory was alone withhis mother in the apartment. He gaveher chest compressions and breathedair into her lungs, although Davidlater couldn't understand how hisbrother didn't kill himself saving her.An ambulance rushed her to St. Luke'sHospital. The brothers were terrifiedthat they would lose her, andthe strain almost killed David. ;ÛOne day, he fainted in hismother's hospital room andthrew up blood. He had devel-oped a bleeding ulcer. "Look,it's not a problem," he saidlater. After Malka Benjaminovna hadbeen moved out of intensive care,Gregory rented a laptop computer,plugged it into the telephone line inher hospital room, and talked to m zeroat night through cyberspace, drivingthe supercomputer toward pi andwatching his mother's blood pressureat the same time.Malka Benjaminovna improvedslowly. When St. Luke's released her,the brothers settled her in her room inGregory's apartment and hired a nurseto look after her. I visited them shortlyafter that, on a hot day in early sum-mer. David answered the door. Therewere blue half circles under his eyes,and he had lost weight. He smiledweakly and greeted me by saying, "Ibelieve it was Oliver Heaviside, theEnglish physicist, who once said, 'Inorder to know soup, it is not necessaryto climb into a pot and be boiled.' But,look, if you want to be boiled you arewelcome to come inside." He led medown 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 overflowrepository."Theoretically, the best way to coola supercomputer is to submerge it inwater," Gregory said, from his bed inthe junk yard."Then we could add goldfish," Davidsaid."That would solve all our problems.""We are not good plumbers, Greg-ory. As long as I am alive, we will notcool a machine with water.""What is the temperature in there?"Gregory asked, nodding toward m zero'sroom."It grows to thirty-four degrees Cel-sius. Above ninety Fahrenheit. This isnot good. Things begin to fry."David took Gregory under the arm,and we passed through the Frenchdoor into gloom and pestilential heat.The shades were drawn, the lightswere off, and an air-conditioner in awindow ran in vain. Sweat immedi-ately began to pour down my body. "Idon't like to go into this room," Greg-ory said. The steel frame in the centerof the room&emdash;the heart ofm zero&emdash;had acquired morelogic boards, and more red lightsblinked inside the machine. Icould hear disk drives murmur-ing. The drives were copyingand recopying segments of tran-scendental numbers, to check the digitsfor perfect accuracy. Gregory knelt onthe floor, facing the steel frame.David opened a cardboard box andremoved an electronic board. He be-gan to fit it into m zero. I noticed thathis hands were marked with smallcuts, which he had got from reachinginside the machine."David, could you give me theflashlight?" Gregory said.David pulled the Mini Mag-Litefrom his shirt pocket and handed it toGregory. The brothers knelt besideeach other, Gregory shining the flash-light into the supercomputer. Davidreached inside with his fingers andpalpated a logic board."Don't!" Gregory said. "O.K., look.No! No!" They muttered to each otherin Russian. "It's too small," Gregorysaid.David adjusted an electric fan. "Webought it at a hardware store down thestreet," he said to me. "We buy ourfans in the winter. It saves money."He pointed to a gauge that had adial on it. "Here we have a meatthermometer."The brothers had thrust the ther-mometer between two circuit boards inorder 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 machinebelow 'Pork,' " Gregory remarked. Helifted a keyboard out of the steel frameand typed something on it, staring ata display screen. Numbers filled thescreen. "The machine is checking itsmemory," he said. A buzzer sounded."It shut down!" he said. "It's a disk-drive controller. The stupid thingobviously has problems.""It's mentally deficient," Davidcommented. He went over to a book-shelf and picked up a hunting knife.I thought he was going to plunge itinto the supercomputer, but he used itto rip open a cardboard box. "We'regoing to ship the part back to themanufacturer," he said to me. "Youhad better send it in the original boxor you may not get your money back.Now you know the reason this apart-ment is full of empty boxes. We haveto save them. Gregory, I wonder if youare tired.""If I stand up now, I will falldown," Gregory said, from the floor."Therefore, I will sit in my center ofgravity. I will maintain my center ofgravity. Let me see, meanwhile, whatis happening with this machine." Hetyped something on his keyboard. "Youwon't believe it, Dave, but the control-ler now seems to work.""We need to buy a new one," Davidsaid."Try Nevada."David dialled a mail-order house inNevada that here will be called Search-light Computers. He said loudly, in athick Russian accent, "Hi, Searchlight.I need a fifteen-forty controller....No! No! No! I don't need any-thing else! Just the controller! Just anaked unit! Naked! How much youcharge? . . . Two hundred and fifty-seven dollars?"Gregory glanced at his brother andshrugged. "Eh.""Look, Searchlight, can you ship itto me Federal Express? For tomorrowmorning. How much?. . . Thirty-ninedollars for Fed Ex? Come on! Whatabout afternoon delivery? . . . Twenty-nine dollars before 3 P.M.? Relax. Whatis your name? . . . Bob. Fine. O.K. Soit'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 thesteel frame and tapped the keys. An-other display screen came alive andfilled with numbers."Tell me this," David said to Bobin Nevada. "Do you have thirty-daymoney-back guarantee? . . . No? Comeon! Look, any device might not work.""Of course, a part might work,"Gregory muttered to his brother. "Butit usually doesn't.""Question Number Two: The FedEx should not cost twenty-nine bucks,"David said to Bob. "No, nothing! I'mjust 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 needanother controller, like the one yousent. Can you send it today FedEx? . . . How much you charge? . . .Naked! I want a naked unit! Not ina shoebox, nothing!"A rhythmic clicking sound camefrom one of the disk drives. Gregoryremarked to me, "We are calculatingpi right now.""Do you want my MasterCard?Look, it's really imperative that Iget my unit tomorrow. A.K., please,I really need my unit bad." Davidhung up the telephone and sighed."This is what has happened to a puremathematician."GREGORY and David are both ex-tremely childlike, but I don'tmean childish at all," Gregory's wife,Christine Pardo Chudnovsky, said onemuggy summer day, at the dining-room table. "There is a certain amountof play in everything they do, a certainamount of fooling around between two52brothers." She is six years youngerthan Gregory; she was an undergradu-ate at Barnard College when she firstmet him. "I fell in love with Gregoryimmediately. His illness came with thepackage." She is still in love with him,even if at times they fight over hisheaps of paper. ("I don't have roomto put my things down," she says tohim.) As we talked, though, pyramidsof boxes and stacks of paper leanedagainst the dining-room windows,pressing against the glass and blockingdaylight, and a smell of hot electricalgear crept through the room. "Thishouse is an example of mathematics infamily life," she said. At night, shedreams that she is dancing from roomto room through an empty apartmentthat has parquet floors.David brought his mother out of herbedroom, settled her at the table, andkissed her on the cheek. Malka Ben-jaminovna seemed frail but alert. Sheis a small, white-haired woman witha fresh face and clear blue eyes, whospeaks limited English. A mathemati-cian once described Malka Ben-jaminovna as the glue that holds theChudnovsky family together. She wasan engineer during the Second WorldWar, when she designed buildings,laboratories, and proving grounds inthe Urals for testing the Katyusharocket; later, she taught engineering atschools around Kiev. She handed meplates of roast chicken, kasha, pickles,cream cheese, brown bread, and littlewedges of The Laughing Cow cheesein foil. "Mother thinks you aren'tgetting enough to eat," Christine said.Malka Benjaminovna slid a jug ofGatorade across the table at me.After lunch, and fortified withGatorade, the brothers and I went intothe chamber of m zero, into a pool ofthick heat. The room enveloped us likenoon on the Amazon, and it teemedwith hidden activity. The disk drivesclicked, the red lights flashed, the air-conditioner hummed, and you couldhear dozens of whispering fans. Greg-ory leaned on his cane and contem-plated the machine. "It's doing manyjobs at the moment," he said. "Frankly,I don't know what it's doing. It's doingsome algebra, and I think it's alsobacking up some pieces of pi.""Sit down, Gregory, or you willfall," 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'sgoing 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 tuggedat his beard. "What was I doing beforethe system crashed? With God's help,I will remember." He jotted a fewnotes in a laboratory notebook. Davidslashed open a cardboard box with hishunting knife and lifted out a boardstudded with chips, for making colorimages on a display screen, and pluggedit into m zero. Gregory crawled undera table. "Oh, shit," he said, frombeneath the table."Gregory, you killed the systemagain!""Dave, Dave, can you get me aflashlight?"David handed his Mini Mag-Liteunder the table. Gregory joined somecables 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, "Thesystem is ready.""Ah," David said.The drives began to click, and theparallel processors silently multipliedand conjoined huge numbers. Gregoryheaded for bed, David holding him bythe arm.In the junk yard, his nest, his paper-lined oubliette, Gregory kicked off hisgentleman's slippers, lay down on thebed, and predicted the future. He said,"The gigaflop supercomputers of todayare almost useless. What is needed isa teraflop machine. That's a machinethat can run at a trillion flops, a trillionfloating-point operations per second, orroughly a thousand times as fast as aCray Y-MP8. One such design for ateraflop 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 leastMARCH 2, 1992sixty-four thousand processors in themachine, each of which has the powerof a Cray. And the processors will bejoined by a network that has the totalswitching capacity of the entire tele-phone network in the United States.I think a teraflop machine will existby 1993. Now, a better machine is apetaflop machine. A petaflop is a qua-drillion flops, a quadrillion floating-point operations per second, so a petaflopmachine is a thousand times as fast asa teraflop machine, or a million timesas fast as a Cray Y-MP8. The petaflopmachine will exist by the year 2000,or soon afterward. It will fit into asphere less than a hundred feet indiameter. It will use light and mir-rors&emdash;the machine's network will con-sist of optical cables rather than copperwires. By that time, a gigaflop 'super-computer' will be a single chip. I thinkthat the petaflop machine will be usedmainly to simulate machines like itself,so that we can begin to design somereal machines."I N the nineteenth century, math-ematicians aKacked pi with the helpof human computers. The most pow-erful of these was Johann MartinZacharias Dase, a prodigy from Ham-burg. Dase could multiply large num-bers in his head, and he made a livingexhibiting himself to crowds in Ger-many, Denmark, and England, andhiring himself out to mathematicians.A mathematician once asked Dase tomultiply 79,532,853 by 93,758,479, andDase gave the right answer in fifty-four seconds. Dase extracted the squareroot of a hundred-digit number infifty-two minutes, and he was able tomultiply a couple of hundred-digit num-bers in his head during a period ofeight and three-quarters hours. Dasecould do this kind of thing for weekson end, running as an unattendedsupercomputer. He would break off acalculation at bedtime, store every-thing in his memory for the night, andresume calculation in the morning.Occasionally, Dase had a system crash.In 1845, he bombed while trying todemonstrate his powers to a mathema-tician and astronomer named Hein-rich Christian Schumacher, reckoningwrongly every multiplication that heattempted. He explained to Schumacherthat he had a headache. Schumacheralso noted that Dase did not in the leastunderstand theoretical mathematics. Amathematician named Julius Petersenonce tried in vain for six weeks toteach Dase the rudiments of Euclideangeometry, but they absolutely baffledDase. Large numbers Dase couldhandle, and in 1844 L. K. Schulzvon Strassnitsky hired him to computepi. Dase ran the job for almost twomonths iri his brain, and at the end ofthe time he wrote down pi correctly tothe first two hundred decimal places&emdash;then a world record.To many mathematicians, math-ematical objects such as the number piseem to exist in an external,objective reality. Numbers seemto exist apart from time or theworld; numbers seem to tran-scend the universe; numbersmight exist even if the uni-verse did not. I suspect that intheir hearts most workingmathematicians are Platonists,in that they take it as a matter ofunassailable if unprovable fact thatmathematical reality stands apart fromthe world, and is at least as real as theworld, and possibly gives shape to theworld, as Plato suggested. Most math-ematicians would probably agree thatthe ratio of the circle to its diameterexists brilliantly in the nature beyondnature, and would exist even if thehuman mind was not aware of it, andmight exist even if God had not both-ered to create it. One could imaginethat pi existed before the universe cameinto being and will exist after theuniverse is gone. Pi may even existapart from God, in the opinion of somemathematicians, for while there isreason to doubt the existence of Gd,by their way of thinking there is nogood reason to doubt the existence ofthe circle."To an extent, pi is more real thanthe machine that is computing it,"Gregory remarked to me one day."Plato was right. I am a Platonist. Ofcourse pi is a natural object. Since piis there, it exists. What we are doingis really close to experimental phys-ics&emdash;we are 'observing pi.' Since wecan observe pi, I prefer to think of pias a natural object. Observing pi iseasier than studying physical phenom-ena, because you can prove things inmathematics, whereas you can't proveanything in physics. And, unfortu-nately, the laws of physics change onceevery generation.""Is mathematics a form of art?" Iasked."Mathematics is partially an art,even though it is a natural science," hesaid. "Everything in mathematics doesexist now. It's a matter of naming it.The thing doesn't arrive from God ina fixed form; it's a matter of represent-ing it with symbols. You put it throughyour mind in order to make senseof it."Mathematicians have sorted num-bers into classes in order to make senseof them. One class of numbers is thatof the rational numbers. A rationalnumber is a fraction composed ofinteÛers (whole numbers):l/l, T/3, 3/5, 1¡/7l, and so on.Every rational number, whenit is expressed in decimal form,repeats periodically: I/3, forexample, becomes .333....Next, we come to the irratio-nal numbers. An irrationalnumber can't be expressed asa fraction composed of whole numbers,and, furthermore, its digits go to in-finity without repeating periodically.The square root of two (Û12) is anirrational number. There is simply noway to represent any irrational numberas the ratio of two whole numbers; itcan't be done. Hippasus of Metapontumsupposedly made this discovery in thefifth century B.C., while travelling ina boat with some mathematicians whowere followers of Pythagoras. ThePythagoreans believed that everythingin nature could be reduced to a ratioof two whole numbers, and they threwHippasus overboard for his discovery,since he had wrecked their universe.Expanded as a decimal, the square rootof 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. Thesquare root of two is not a transcen-dental number, because it can be foundwith an equation. It is the solution(root) of an equation. The equation isx2 = 2, and a solution is the squareroot of two. Such numbers are calledalgebraic.While pi is indeed an irrationalnumber&emdash;it can't be expressed as afraction made of whole numbers&emdash;more important, it can't be expressedwith finite algebra. Pi is therefore saidto be a transcendental number, becauseit transcends algebra. Simply and gen-erally speaking, a transcendental num-ber can't be pinpointed through anequation built from a finite number ofintegers. There is no finite algebraic54equation built from whole numbersthat will give an exact value for pi.The statement can be turned aroundthis way: pi is not the solution to anyequation built from a less than infiniteseries of whole numbers. If equationsare trains threading the landscape ofnumbers, then no train stops at pi.Pi is elusive, and can be approachedonly through rational approximations.The approximations hover around thenumber, closing in on it, but do nottouch it. Any formula that heads to-ward pi will consist of a chain ofoperations that never ends. It is aninfinite series. In 1674, GottfriedWilhelm Leibniz (the co-inventor ofthe calculus, along with Isaac New-ton) noticed an extraordinary patternof numbers buried in the circle. TheLeibniz series for pi has been calledone of the most beautiful mathematicaldiscoveries of the seventeenth century:In English: pi over four equals oneminus a third plus a fifth minus aseventh plus a ninth&emdash;and so on. Youfollow the odd numbers out to infinity,and when you arrive there and sumthe terms, you get pi. But since younever arrive at infinity you never getpi. Mathematicians find it deeply mys-terious that a chain of discrete rationalnumbers can connect so easily to ge-ometry, to the smooth and continuouscircle.As an experiment in "observing pi,"as Gregory Chudnovsky puts it, Icomputed the Leibniz series on a pocketcalculator. It was easy, and I gotresults that did seem to wander slowlytoward pi. As the series progresses, theanswers touch on 2.66, 3.46, 2.89, and3.34, in that order. The answers landhigher than pi and lower than pi,skipping back and forth across pi, andgradually closing in on pi. A math-ematician would say that the series"converges on pi." It converges on piforever, playing hopscotch over pi butnever landing on pi.You can take the Leibniz series outa long distance&emdash;you can even dra-matically speed up its movement to-ward pi by adding a few corrections toit&emdash;but no matter how far you takethe Leibniz series, and no matterhow many corrections you hammer intoit, when you stop the operation andsum the terms, you will get a rationalnumber that is somewhere around pibut is not pi, and you will be damnedif you can put your hands on pi.Transcendental numbers continueforever, as an endless non-repeatingstring, in whatever rational form youchoose to display them, whether asdigits or as an equation. The Leibnizseries is a beautiful way to representpi, and it is finally mysterious, becauseit doesn't tell us much about pi. Look-ing at the Leibniz series, you feel theindependence of mathematics fromhuman culture. Surely, on any worldthat knows pi the Leibniz series willalso be known. Leibniz wasn't the firstmathematician to discover the Leibnizseries. Nilakantha, an astronomer,grammarian, and mathematician wholived on the Kerala coast of India,described the formula in Sanskrit po-etry around the year 1500.It is worth thinking about what adecimal place means. Each decimalplace of pi is a range that shows theapproximaÛe location of pi to an accu-racy ten times as great as the previousrange. But as you compute the nextdecimal place you have no idea wherepi will appear in the range. It couldpop up in 3, or just as easily in 9, orin 2. The apparent movement of pi asyou narrow the range is known as therandom walk of pi.Pi does not move; pi is a fixed point.The algebra wobbles around pi. Thereis no such thing as a formula that issteady enough or sharp enough to sticka pin into pi. Mathematicians havediscovered formulas that converge onpi very fast (that is, they skip aroundpi with rapidly increasing accuracy),but they do not and cannot hit pi. TheChudnovsky brothers discovered theirown formula in 1984, and it attacks piwith great ferocity and elegance. TheChudnovsky formula is the fastest seriesfor pi ever found which uses rationalnumbers. Various other series for pi,which use irrational numbers, havealso been found, and they converge onpi faster than the Chudnovsky for-mula, but in practice they run moreslowly on a computer, because irratio-nal numbers are harder to compute.The Chudnovsky formula for pi isthought to be "extremely beautiful" bypersons who have a good feel fornumbers, and it is based on a torus (adoughnut), rather than on a circle. Ituses large assemblages of whole num-bers to hunt for pi, and it owes muchto an earlier formula for pi worked outin 1914 by Srinivasa Ramanujan, amathematician from Madras, who wasa number theorist of unsurpassed ge-nius. Gregory says that the Chudnovskyformula "is in the style of Ramanujan,"and that it "is really very simple, andcan be programmed into a computerwith a few lines of code."In 1873, Georg Cantor, a Russian-born mathematician who was one ofthe towering intellectual figures of thenineteenth century, proved that the setof transcendental numbers is infinitelymore extensive than the set of alge-braic numbers. That is, finite algebracan't find or describe most numbers. Toput it another way, most numbers areinfinitely long and non-repeating inany rational form of representation. Inthis respect, most numbers are like pi.Cantor's proof was a disturbing pieceof news, for at that time very fewtranscendental numbers were actuallyknown. ( Not until nearly a decadelater did Ferdinand Lindemann finallyprove the transcendence of pi; beforethat, mathematicians had only conjec-tured that pi was transcendental.) Per-haps even more disturbing, Cantoroffered no clue, in his proof, to whata transcendental number might looklike, or how to construct such a beast.Cantor's celebrated proof of the exis-tence of uncountable multitudes of tran-scendental numbers resembled a proofthat the world is packed with micro-scopic angels&emdash;a proof, however, thatdoes not tell us what the angels looklike or where they can be found; itmerely proves that they exist in un-countable multitudes. While Cantor'sproof lacked any specific description ofa transcendental number, it showedthat algebraic numbers (such as thesquare root of two) are few and farbetween: they poke up like markerbuoys 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 yetdiscovered most numbers, which weretranscendentals, and in the secondplace that they lacked any tools ormethods that would determine whethera given number was transcendentalor not. Leopold Kronecker, an influ-ential older mathematician, rejectedCantor's proof, and resisted the wholenotion of "discovering" a number. (Heonce said, in a famous remark, "Godmade the integers, all else is the workof 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 hassubsided, mathematicians having de-cided to work with transcendentalnumbers no matter who made them.The Chudnovsky brothers claim thatthe digits of pi form the most nearlyperfect random sequence of digits thathas ever been discovered. They saythat nothing known to humanity ap-pears to be more deeply unpredictablethan the succession of digits in pi,except, perhaps, the haphazard clicksof a Geiger counter as it detects thedecay of radioactive nuclei. But pi isnot random. The fact that pi can beproduced by a relatively simple for-mula means that pi is orderly. Pi looksrandom only because the pattern in thedigits is fantastically complex. TheLudolphian number is fixed in eter-nity&emdash;not a digit out of place, all char-acters in their proper order, an endlesssentence written to the end of theworld by the division of the circle'sdiameter into its circumference. Vari-ous simple methods of approximationwill always yield the same successionof digits in the same order. If a singledigit in pi were to be changed any-where between here and infinity, theresulting number would no longer bepi; it would be "garbage," in David'sword, because to change a single digitin pi is to throw all the following digitsout of whack and miles from pi."Pi is a damned good fake of arandom number," Gregory said. "Ijust wish it were not as good a fake.It would make our lives a lot easier."Around the three-hundred-millionthdecimal place of pi, the digits go88888888&emdash;eight eights pop up in arow. Does this mean anything? Itappears to be random noise. Later,ten sixes erupt: 6666666666. Whatdoes this mean? Apparently nothing,only more noise. Somewhere past thehalf-billion mark appears the string123456789. It's an accident, as it were."We do not have a good, clear, crys-tallized idea of randomness," Gregorysaid. "It cannot be that pi is trulyrandom. Actually, a truly random se-quence of numbers has not yet beendiscovered."No one knows what happens to thedigits of pi in the deeper regions, as thenumber is resolved toward infinity. Dothe digits turn into nothing but eightsand fives, say? Do they show a pre-dominance of sevens? Similarly, noone knows if a digit stops appearing inpi. This conjecture says that after acertain point in the sequence a digitdrops out completely. For example, nomore fives appear in pi&emdash;somethinglike that. Almost certainly, pi does notdo such things, Gregory Chudnovskythinks, because it would be stupid, andnature isn't stupid. Nevertheless, noone has ever been able to prove ordisprove a certain basic conjecture aboutpi: that every digit has an equal chanceof appearing in pi. This is known asthe normality conjecture for pi. Thenormality conjecture says that, onaverage, there is no more or less of anydigit in pi: for example, there is noexcess of sevens in pi. If all digits doappear with the same average fre-quency in pi, then pi is a "normal"number&emdash;"normal" by the narrowmathematical definition of the word."This is the simplest possible conjec-ture about pi," Gregory said. "Thereis absolutely no doubt that pi is a'normal' number. Yet we can't proveit. We don't even know how to try toprove it. We know very little abouttranscendental numbers, and, what isworse, the number of conjectures aboutthem isn't growing." No one knowseven how to tell the difference betweenthe square root of two and pi merelyby looking at long strings of theirdigits, though the two numbers havecompletely distinct mathematical prop-erties, one being algebraic and theother transcendental.Even if the brothers couldn't proveanything about the digits of pi, they feltthat by looking at them through thewindow of their machine they mighÛat least see something that could leadto an important conjecture about pi orabout transcendental numbers as a class.You can learn a lot about all cats bylooking closely at one of them. So ifyou wanted to look closely at pi howmuch of it could you see with a verylarge supercomputer? What if youturned the universe into a supercom-puter? What then? How much pi couldyou see? Naturally, the brothers hadconsidered this project. They hadimagined a computer built from theuniverse. Here's how they estimatedthe machine's size. It has been calcu-lated that there are about 1079 electronsand protons in the observable universe;this is the so-called Eddington numberof the universe. (Sir Arthur StanleyEddingtonÛ the astrophysicist, first cameup with the number.) The Edding-ton number is the digit 1 followed byseventy-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 declaresthe power of the Eddington machine.The Eddington machine would bethe universal supercomputer. It wouldbe made of all the atoms in the uni-verse. The Eddington machine wouldcontain ten vigintsextillion parts, andif the Chudnovsky brothers could figureout how to program it with FORTRANthey might make it churn toward pi."In order to study the sequence of pi,you have to store it in the Eddingtonmachine's memory," Gregory said. Tobe realistic, the brothers thought thata practical Eddington machine wouldn'tbe able to store pi much beyond 1077digits&emdash;a number that is only a hun-dredth of the Eddington number. Now,what if the digits of pi only begin toshow regularity beyond 1077 digits?Suppose, for example, that pi manifestsa regularity starting at 101¡¡ decimalplaces? That number is known as agoogol. If the design in pi appears onlyafter a googol of digits, then not eventhe Eddington machine will see anysystem in pi; pi will look totally dis-ordered to the universe, even if picontains a slow, vast, delicate struc-ture. A mere googol of pi might be onlythe first knot at the corner of a kindof limitless Persian rug, which is woveninto increasingly elaborate diamonds,cross-stars, gardens, and cosmogonies.It may never be possible, in principle,to see the order in the digits of pi. Noteven nature itself may know the natureof pi."If pi doesn't show systematic be-havior until more than ten to theseventy-seven decimal places, it wouldMARCH 2, 1992really be a disaster," Gregory said. "Itwould be actually horrifying.""I wouldn't give up," David said."There might be some other way ofleaping over the barrier&emdash;""And of attacking the son of a bitch,"Gregory said.THE brothers first came in contactwith the membrane that dividesthe dreamlike earth from mathemati-cal reality when they were boys, grow-ing up in Kiev, and their father gaveDavid a book entitled "What Is Math-ematics?," by two mathematicians namedRichard Courant and Herbert Rob-bins. The book is a classic&emdash;millionsof copies of it have been printed inunauthorized Russian and Chinese edi-tions alone&emdash;and after the brothersfinished reading "Robbins," as the bookis called in Russia, David decided tobecome a mathematician, and Gregorysoon followed his brother's footstepsinto the nature beyond nature. Gregory'sfirst publication, in the journal SovietMathematics&emdash;Doklady, came when hewas sixteen years old: "Some Resultsin the Theory of Infinitely Long Ex-pressions." Already you can see wherehe was headed. David, sensing hisyounger brother's power, encouragedhim to grapple with central problemsin mathematics. Gregory made his firstmajor discovery at the age of seven-teen, when he solved Hilbert's TenthProblem. (It was one of twenty-threegreat problems posed by David Hilbertin 1900.) To solve a Hilbert problemwould be an achievement for a life-time; Gregory was a high-school stu-dent who had read a few books onmathematics. Strangely, a young Rus-sian mathematician named Yuri Matya-sevich had just solved Hilbert's TenthProblem, and the brothers hadn't heardthe news. Matyasevich has recentlysaid that the Chudnovsky method isthe preferred way to solve Hilbert'sTenth Problem.The brothers enrolled at Kiev StateUniversity, and both graduated summacum laude. They took their Ph.D.s atthe Institute of Mathematics at theUkrainian Academy of Sciences. Atfirst, they published their papers sepa-rately, but by the mid-nineteen-seventiesthey were collaborating on much oftheir work. They lived with their parentsin Kiev until the family decided to tryto take Gregory abroad for treatment,and in 1976 Volf and Malka Chud-novsky applied to the government toemigrate. Volf was immediately firedfrom his job.The K.G.B. began tailing the broth-ers. "Gregory would not believe meuntil it became totally obvious," Davidsaid. "I had twelve K.G.B. agents onmy tail. No, look, I'm not kidding!They shadowed me around the clockin two cars, six agents in each car.Three in the front seat and three inthe back seat. That was how the K.G. B.operated." One day, in 1976, Davidwas walking down the street whenK.G.B. officers attacked him, breakinghis skull. He went home and nearlydied, but didn't go to the hospital. "IfI had gone to the hospital, I wouldhave died for sure," he told me. "Thehospital is run by the state. I wouldforget to breathe."On July 22, 1977, plainclothesmenfrom the K.G.B. accosted Volf andMalka on a street in Kiev and beatthem up. They broke Malka's arm andfractured her skull. David took hismother to the hospital. "The doctor inthe emergency room said there was nofracture," David said.Gregory, at home in bed, was notso vulnerable. Also, he was conspicu-ous in the West. Edwin Hewitt, amathematician at the University ofWashington, in Seattle, had visitedKiev in 1976 and collaborated withGregory on a paper, and later, whenHewitt learned that the Chudnovskyfamily was in trouble, he persuadedSenator Henry M. Jackson, the pow-erful member of the Senate ArmedServices Committee, to take up theChudnovskys' case. Jackson put pres-sure on the Soviets to let the familyleave the country. Just before the K.G.B.attacked the parents, two members ofa French parliamentary delegation thatwas in Kiev made an unofficial visit tothe Chudnovskys to see what was goingon. One of the visitors, a staff memberof the delegation, was Nicole Lanne-grace, who married David in 1983.Andrei Sakharov also helped to drawattention to the Chudnovskys' increas-ingly desperate situation. Two monthsafter the parents were attacked, theSoviet government unexpectedly let thefamily go. "That summer when I wasgeKing killed by the K.G.B., I couldnever have imagined that the next yearI would be in Paris or that I wouldwind up in New York, married to abeautiful Frenchwoman," David said.The Chudnovsky family settled in NewYork, near Columbia University.I F pi is truly random, then attimes pi will appear to be ordered.Therefore, if pi is random it containsaccidental order. For example, some-where in pi a sequence may run07070707070707 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 ofzeros and sevens may appear, only thistime interrupted by a single occurrenceof the digit 3. Another accident. Thoseand all other "accidental" arrange-ments of digits almost certainly eruplin pi, but their presence has never beenproved. "Even if pi is not truly ran-dom, you can still assume that youget every string of digits in pi," Greg-ory said.If you were to assign letters of thealphabet to combinations of digits, andwere to do this for all human alpha-bets, syllabaries, and ideograms, thenyou could fit any written character inany language to a combination of digitsin pi. According to this system, pi couldbe turned into literature. Then, if youcould look far enough into pi, youwould probably find the expression"See the U.S.A. in a Chevrolet!" abillion times in a row. Elsewhere, youwould find Christ's Sermon on theMount in His native Aramaic tongue,and you would find versions of theSermon on the Mount that are pureblasphemy. Also, you would find adictionary of Yanomamo curses. Aguide to the pawnshops of Lubbock.The book about the sea which JamesJoyce supposedly declared he wouldwrite after he finished "FinnegansWake." The collected transcripts of"The Tonight Show" rendered intoEtruscan. "Knowledge of All ExistingThings," by Ahmes the Egyptian scribe.Each occurrence of an apparently- or-dered string in pi, such as the wordst "Ruin hath taught me thus to rumi-nate/That Time will come and takemy love away," is followed by unimag-inable deserts of babble. No book andnone but the shortest poems will everbe seen in pi, since it is infinitesimallyunlikely that even as brief a text asan English sonnet will appear in thefirst 1077 digits of pi, which is thelongest piece of pi that can be calcu-lated in this universe.Anything that can be produced by asimple method is by definition orderly.Pi can be produced by various simplemethods of rational approximation, andthose methods yield the same digits ina fixed order forever. Therefore, pi is- orderly in the extreme. Pi may also bea powerful random-number generator,spinning out any and all possible com-binations of digits. We see that thedistinction between chance and fixitydissolves in pi. The deep connectionbetween disorder and order, betweencacophony and harmony, in the mostfamous ratio in mathematics fascinatedGregory and David Chudnovsky. Theywondered if the digits of pi had apersonality."We are looking for the appearanceof some rules that will distinguish thedigits of pi from other numbers,"Gregory explained. "It's like studyingwriters by studying their use of words,their grammar. If you see a Russiansentence that extends for a whole page,with hardly a comma, it is definitelyTolstoy. If someone were to give youa million digits from somewhere in pi,could you tell it was from pi? We don'treally look for patterns; we look forrules. Think of games for children. IfI give you the sequence one, two,three, four, five, can you tell me whatthe next digit is? Even a child can doit; the next digit is six. How about thisgame? Three, one, four, one, five,nine. Just by looking at that sequence,can you tell me the next digit? Whatif I gave you a sequence of a milliondigits from pi? Could you tell me thenext digit just by looking at the se-quence? Why does pi look like a totallyunpredictable sequence with the high-est complexity? We need to find out therules that govern this game. For all weknow, we may never find a rule in pi."HERBERT ROBBINS, the co-author of"What Is Mathematics?," is anemeritus professor of mathematicalstatistics at Columbia University. Forthe past six years, he has been teachingat Rutgers. The Chudnovskys call himonce in a while to get his advice onhow to use statistical tools to search forsigns of order in pi. Robbins lives ina rectilinear house that has a lot ofglass 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 andprobability theory happened inside hishead. Robbins is a tall, restless manin his seventies, with a loud voice,furrowed cheeks, and penetrating eyes.One recent day, he stretched himselfout on a daybed in a garden roomin his house and played with a rub-ber band, making a harp across hisfingertips."It is a very difficult philosophicalquestion, the question of what 'ran-dom' is," he said. He plucked therubber band with his thumb, boink,boink. "Everyone knows the famousremark of Albert Einstein, that Goddoes not throw dice. Einstein just wouldnot believe that there is an element ofrandomness in the construction of theworld. The question of whether theuniverse is a random process or isdetermined in some way is a basicphilosophical question that has nothingto do with mathematics. The questionis important. People consider it whenthey decide what to do with their lives.It concerns religion. It is the questionof whether our fate will be revealed orwhether we live by blind chance. MyGod, how many people have beenmurdered over an answer to that ques-tion! Mathematics is a lesser activitythan religion in the sense that we'veagreed not to kill each other but todiscuss things."Robbins got up from the daybed andsat in an armchair. Then he stood upand paced the room, and sat at a tablein the room, and sat on a couch, andwent back to the table, and finallyreturned to the daybed. The man wasin constant motion. It looked randomto me, but it may have been systematic.It was the random walk of HerbertRobbins."Mathematics is broken into tinyspecialties today, but Gregory Chud-novsky is a generalist who knows thewhole 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 asGregory Chudnovsky. He's like Mozart:he's the last of his breed. I happen tothink the brothers' pi project is a will-o'-the-wisp, and is one of the leastinteresting things they've ever done.But what do I know? Gregory seemsto be asking questions that can't beanswered. To ask for the system in piis like asking 'Is there life after death?'When you die, you'll find out. Mostmathematicians are not interested inthe digits of pi, because the question isof no practical importance. In order fora mathematician to become interestedin a problem, there has to be a possi-bility of solving it. If you are anathlete, you ask yourself if you canjump thirty feet. Gregory likes to askif he can jump around the world. Helikes to do things that are impossible."At some point after the brotherssettled in New York, it became obvi-ous that Columbia University was notgoing to be able to invite them tobecome full-fledged members of thefaculty. Since then, the brothers havealways enjoyed cordial personal rela-tionships with various members of thefaculty, but as an institution the Math-ematics Department has been unableto create permanent faculty positionsfor them. Robbins and a couple offellow-mathematicians&emdash;Lipman Bersand the late Mark Kac&emdash;once tried toraise money from private sources foran endowed chair at Columbia to beshared by the brothers, but the effortfailed. Then the John D. and Cath-erine T. MacArthur Foundation award-ed Gregory Chudnovsky a "genius"fellowship; that happened in 1981, thefirst year the awards were given, as ifto suggest that Gregory is a personfor whom the MacArthur prize wasinvented. The brothers can exhibitother fashionable paper&emdash;a Prix Peccot-Vimont, a couple of Guggenheims, aDoctor of Science honorÛs causa fromBard College, the Moscow Mathemati-cal Society Prize&emdash;but there is onedefect in their rŽsumŽ, which is the factthat Gregory has to lie in bed most ofthe day. The ugly truth is that GregoryChudnovsky can't get a permanent jobat any American institution of higherlearning because he is physically dis-abled. But there are other, more per-plexing reasons that have led the Chud-novsky brothers to pursue their workin solitude, outside the normal aca-demic hierarchy, since the day theyarrived in the United States.Columbia University has awardedeach brother the title of senior researchscientist in the Department of Math-ematics. Their position at Columbia isambiguous. The university officiallyconsiders them to be members of thefaculty, but they don't have tenure, andColumbia doesn't spend its own fundsto pay their salaries or to support theirresearch. However, Columbia does givethem heakh-insurance benefits and ahousing subsidy.The brothers have been living onmodest grants from the National Sci-ence Foundation and various otherresearch agencies, which are funnelledthrough Columbia and have to beapplied for regularly. Nicole Lanne-grace and Christine Chudnovsky fi-nanced m zero out of their paychecks.Christine's father, Gonzalo Pardo, whois a professor of dentistry at the StateUniversity of New York at Stony Brook,built the steel frame for m zero in hisbasement during a few weekends, usinga wrench and a hacksaw.The brothers' mode of existence hascome to be known among mathema-ticians as the Chudnovsky Problem.Herbert Robbins eventually decidedthat it was time to ask the entireAmerican mathematics profession whyit could not solve the ChudnovskyProblem. Robbins is a member of theNational Academy of Sciences, and in1986 he sent a letter to all of themathematicians in the academy:I fear that unless a decent and honorableposition in the American educational and re-search system is found for the brothers soon,a personal and scientific tragedy will takeplace for which all American mathematicianswill share responsibility....I have asked many of my colleagues whythis situation exists, and what can be done toput an end to what I regard as a nationaldisgrace. I have never received an answerthat 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 toacquaint yourselves with their present cir-cumstances, and if you are convinced of themerits of their case, to find a suitable positionsomewhere in the country for them as a pair.There wasn't much of a response.Robbins says that he received threewritten replies to his letter. One, froma faculty member at a well-knownEast Coast university, complained aboutDavid Chudnovsky's personality. Heremarked that "when David learns tobe less overbearing" the brothers mighthave better luck. He also did not fullyunderstand the tone of Rob- .bins' letter: while he agreedthat some resolution to theChudnovsky Problem mustbe found, he thought thatHerb Robbins ought to ap-proach the subject realistically and with more candor.("More candor? How could I havebeen more candid?" Robbins asked.)Academic administrator. I'm sorry I havenothing more effective to propose."An emotional reaction to Robbins'campaign on behalf of the Chudnovskyscame a bit later from Edwin HewiK,the mathematician who had helped getthe family out of the Soviet Union, andone of the few Americans who hasever worked with Gregory Chudnovsky.Hewitt wrote to colleagues, "I havecollaborated with many excelIent math-ematicians . . . but with no one elsehave I witnessed an outpouring ofmathematics like that from Gregory.He simply KNOWS what is true andwhat is not." In another letter, Hewittwrote:The Chudnovsky situation is a nationaldisgrace. Everyone says, "Oh, what a cryingshame" & then suggests that they be placed atsomebody else's institution. No one seems towant the admittedly burdensome task of car-ing for the Chudnovsky family. I imagine itwould be a full-time, if not an impossible,job. We may remember that both Mozart andBeethoven were disagreeable people, to saynothing of Gauss.The brothers would have to be hiredas a pair. Gregory won't take any jobunless David gets one, and vice versa.Physically and intellectually com-mingled, like two trees that have growntogether at the root and bole, the broth-ers claim that they can't be separatedwithout becoming deadfalls and crash-ing to the ground. To hire the Chud-novsky pair, a department would haveto create a joint opening for them.Gregory can't teach classes in the normalway, because he is more or less confined to bed.It would require a small degree offlexibility in a department toallow Gregory to concentrate on research, while Davidhandled the teaching. TheA problem is that Gregory mightAnother letter came from a facultymember at Princeton University, whooffered to put in a good word with theNational Science Foundation to helpthe brothers get their grants, but didnot mention a job at Princeton oranywhere else. The most thoughtfulresponse came from a faculty memberat M.I.T., who remarked, "It doesseem odd that they have not been moresought after." He wondered if in somepart this might be a consequence oftheir breadth. "A specialist appears asa safer investment to a cautious aca-working with a few brilliant graduatestudents&emdash;a privilege that might not godown well in an American academicdepartment."They are prototypical Russians,"Robbins said. "They combine a rathergrandiose vision of themselves with anability to live on scraps rather thancompromise their principles. These arepeople the world is not able to copewith, and they are not making it anyeasier for the world. I don't see thatthe world is particularly trying to keepGregory Chudnovsky alive. The trag-edy&emdash;the disgrace, so to speak&emdash;is thatthe American scientific and educationalestablishment is not benefitting fromthe Chudnovskys' assistance. Thirteenyears have gone by since the Chud-novskys arrived here, and where areall the graduate students who wouldhave worked with the brothers? Howmany truly great mathematicians haveyou ever heard of who couldn't get ajob? I think the Chudnovskys are theonly example in history. This vasteducational system of ours has pouredthe Chudnovskys out on the sand, towaste. Yet Gregory is one of the re-markable personalities of our time.When I go up to that apartment andsit by his bed, I think, My God, whenI was a student at Harvard I was incontact with people far less interestingthan this. What happens to me inGregory's room is like that line in theGerard Manley Hopkins poem: 'Mar-garet, are you grieving/Over Golden-grove unleaving?' I'm grieving, and Iguess it's me I'm grieving for."T" WO billion digits of pi? Wheredo they keep them?" SamuelEilenberg said to me. Eilenberg is agifted and distinguished topologist, andan emeritus professor of mathematicsat Columbia University. He was thechairman of the department when thequestion of hiring the brothers firstbecame troublesome to Columbia."There is an element of fatigue in theChudnovsky Problem," he said. "Inthe academic world, we have to becareful who our colleagues are. Davidis a pain in the neck. He interruptspeople, and he is not interested inanything except what concerns himand his brother. He is a nudnick!Gregory is certainly unusual, but he isnot great. You can spend all your lifecomputing digits. What for? You knowin advance that you can't see anyregularity in pi. It's about as interest-ing as going to the beach and countingsand. I wouldn't be caught dead doingthat kind of work! Most mathemati-cians probably feel this way. An im-portant ingredient in mathematics istaste. Mathematics is mostly about givingpleasure. The ultimate criterion ofmathematics is aesthetic, and to calcu-late the two-billionth digit of pi is tome abhorrent.""Abhorrent&emdash;yes, most mathemati-cians would probably agree with that,"said Dale Brownawell, a respectednumber theorist at Penn State. "Tasteschange, though. If something were tobegin to show up in the digits of pi,it would boggle everyone's mind."Brownawell met the Chudnovskys atthe Vienna airport when they escapedfrom the Soviet Union. "They didn'tbring much with them, just a pile olbags and boxes. David would walkthrough a wall to do what is right forMARCH 2, 1992his brother. In the situation they arein, how else can they survive? To seethe Chudnovskys carrying on scienceat such a high level with such meagresupport is awe-inspiring."Richard Askey, a prominent math-ematician at the University of Wis-consin at Madison, occasionally flies toNew York to sit at the foot of GregoryChudnovsky's bed and learn aboutmathematics. "David Chudnovsky is avery good mathematician," Askey saidto me. "Gregory is one of the few greatmathematicians of our time. Gregoryis so much beKer than I am that it isimpossible for me to say how good hereally 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 smallpart of their work. They are reallytrying to find out what the word 'ran-dom' means. I've heard some peoplesay that the brothers are wasting theirtime with that machine, but GregoryChudnovsky is a very intelligent man,who has his head screwed on straight,and I wouldn't begin to question hispriorities. The tragedy is that Gregoryhas had hardly any students. If he dieswithout having passed on not only hisknowledge 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 theblame lies with all American math-ematicians. Gregory Chudnovsky is anational problem."I" T looks like kvetching," Gregorysaid from his bed. "It looks cheap,and it is cheap. We are here in theUnited States by our own choice. Idon't think we were somehow wronged.I really can't teach. So what does onewant to do about it? Attempts to changethe system are very expensive andtime-consuming and largely a waste oftime. We barely have time to do thethings 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 totalitariansystem.""Yes, you just make a decree,"Gregory said. "Anyway, this sort oftalk moves into philosophical ques-tions. What is life, and where does themoney come from?" He shrugged.F Toward the end of the summer of1991, the brothers halted their probeinto pi. They had surveyed pi to twobillion two hundred and sixty millionthree hundred and twenty-one thou-sand three hundred and thirty-six dig-its. It was a world record, doubling therecord that the Chudnovskys had set in1989. If the digits were printed inordinary type, they would stretch fromNew York to Southern California. Thebrothers had temporarily ditched theirchief competitor, Yasumasa Kanada&emdash;a pleasing development when the broth-ers considered that Kanada had accessto a half-megawatt Hitachi monsterthat was supposed to be faster than aCray. Kanada reacted gracefully to theChudnovskys' achievement, and he toldScience News that he might be able toget at least a billion and a half digitsof pi if he could obtain enough timeon a Japanese supercomputer."You see the advantage to beingtruly poor. We had to build our ma-chine, but now we get to use it, too,"Gregory said.The Chudnovskys' machine hadspent its time both calculating pi andchecking the result. The job had takenabout two hundred and fifty hours onm zero. The machine had spent mostof its time checking the answer, tomake sure each digit was correct, ratherthan doing the fundamental computa-tion of pi."We have done our tests for pat-terns, and there is nothing," Gregorysaid. "It would be rather stupid if therewere something in a few billion digits.There are the usual things. The digitthree is repeated nine times in a row,and we didn't see that before. Unfor-tunately, we still don't have enoughcomputer power to see anything in pi."Such was their scientific conclusion,and yet the brothers felt that they mayhave noticed something in pi. It hov-ered out of reach, but it seemed a littlecloser now. It was a slight sign oforder&emdash;a possible sign&emdash;and it had todo with the running average of thtdigits. You can take an average of anystring of digits in pi. It is like gettinga batting average, an average height,an average weight. The average of thedigits in pi should be 4.5. That's theaverage of the decimal digits zerothrough nine. The brothers noticedthat the average seems to be slightlyskewed. It stays a little high throughmost of the first billion digits, and thenit stays a little low through the nextbillion digits. The running average ofpi looks like a tide that rises andretreats through two billion digits, asif a distant moon were passing over asea of digits, pulling them up anddown. It may or may not be a hint ofa rule in pi. "It's unfortunately notstatistically significant yet," Gregorysaid. "It's close to the edge of signifi-cance." The brothers may have glimpsedonly their desire for order. The tidethat seems to flow through pi may benothing but aimless gabble, but whatif it is a wave rippling through pi?What if the wave begins to show aweird and complicated pulsation as yougo deeper in pi? You could becomeobsessive thinking about things likethis. You might have to build moremachines. "We need a trillion digits,"David said. A trillion digits printed inordinary type would stretch from hereto the moon and back, twice. Thebrothers thought that if they didn't getbored with pi and move on to otherproblems they would easily collect atrillion digits in a few years, with theL help of increasingly powerful super-computing equipment. They would orbitthe moon in digits, and head for ALphaCentauri, and if they lived and theirmachines held, perhaps someday theywould begin to see the true nature of pi.Gregory is lying in bed in the junkyard, a keyboard on his lap. He offersto show me a few digits of pi, and tapsat the keys.On the screen beside his bed, m zeroresponds: "Please, give the beginningof the decimal digit to look."Gregory types a command, andsuddenly the whole screen fills withthe raw Ludolphian number, movinglike Niagara Falls. We observe pi insilence for quite a while, until it endswith:. . . 18820 54573 01261 27678 17413 8777966981 15311 24707 34258 41235 99801 9269352561 92393 53870 24377 10069 16106 2297102523 30027 49528 06378 64067 12852 7785742344 28836 88521 72435 85924 57786 3674132845 66266 96498 68308 59920 06168 6337685976 35341 52906 04621 44710 52106 9907933563 54625 71001 37490 77872 43403 5769001699 82447 20059 93533 82919 46119 8704402125 12329 11964 10087 41341 42633 8824948948 31198 27787 03802 08989 05316 7537543242 20100 43326 74069 33751 86349 4046752687 79749 68922 29914 46047 47109 3167805219 48702 00877 32383 87446 91871 4913690837 88525 51575 35790 83982 20710 5929841193 81740 92975 31."It showed the last digits we'vefound," Gregory says. "The last shallbe first.""Thanks for asking," m zero re-marks, on the screen.&emdash;RICHARD PRESTON--------------5A031751470--