SQUARING theCide is no Pieceofn

BY BRUCE WATSON

MATHEMATICIANS HAVE SLICED,

AND NOW SUPERCOMPUTERS HAVE

CRUNCHED, BUT THE MYSTERY OF

GOES ON AND ON AND.. .

ON AN OTHERWISE RATIONAL FALL DAY, A CLASS OF SIXTH-GRADERS IS ABOUT TO TAKE ON AN irrational task. Students at the Smith College Campus School in Northampton, Massachusetts, spread coffee can lids, compact disks, coins and cups on their desks. Ready or not, these innocent kids are about to have a rendezvous with the most intriguing of numbers. “Today,” teacher Tom Weiner tells them, “we’re going to learn how to measure circles.” Little do they suspect that they are on the verge of a mystery that has vexed mathematicians for ages. Fol- lowing Weiner’s instructions, students begin cutting strings to measure the distance around each cir- cular object. They wrap strings around lids and cups. They crimp them around CDs. They strug- gle for accuracy, but their strings slacken or slip. “I’m having a hard time getting an exact measurement,” Sarah reports. Her teacher wonders: “Could there be an easier way?” Finally, Wein- er suggests a different plan, one based on a little number called pi. Many of us recall our first encounter with pi. It began with dry formulas: C = 7cD. A = m2. (Remember? C = circumference; D = diameter; r = radius; A = area.) Inevitably, somebody told the old, old joke: “Pi are squared? [m2?] No way! Pie are round!” Compared with the exact figures that drive our digital age-stock prices, interest rates, computer bytes-pi is as mystical as a zen koan. Stated simply, it is the number you get when you divide the distance around a circle (circumference) by the distance through the middle (diameter). The distance around every circle is about three times the distance across it. Easy, eh? But in that “about” lies the puzzle of pi. Mathematicians call pi an irrational number. That is, when you divide circumference by diameter, the answer comes out in decimals that go on forever without any pattern. Pi begins pp5g265 and it never ends. Ever. This curiosity makes pi part cult and part of our culture, the subject of several books, such as TheJoy ofx and A History of Pi, and many Websites (one of the best is at www.cecm.sfu.cafpi).

On the face of it, pi seems as easy as pie. But Archimedes refused to settle for a half-baked solution.

ILLUSTRATIONS BY JEFF SEAVER a stone, the double helix of DNA, and one of our most important inven- tions-the wheel. Billions of digits do not make very interesting reading, yet pi has a surprising story to tell. “God ever geometrizes,” Plato wrote, but he might as well have been referring to the great calculator of pi-Archimedes. By the time Archimedes began playing with num- bers-in the third century B.C.-math- ematicians already knew of the fired ratio in every circle. But exactly how much was that ratio? Pi shows up in history for the first time in Egypt. In 1858, a Scottish anti- quary, Henry Rhind, came upon a wonder, written on papyrus. From square one, the Rhind papyrus boast- ed of its own importance: “Accurate reckoning. The entrance into the knowledge of all existing things and obscure secrets.” One of the mathe- matical solutions instructed: “Cut off ‘/9 of a diameter [of a circle] and con- struct a square upon the remainder; There are many irrational numbers this has the same area as the circle.” No mutter how you whose decimals go on forever. But has The Babylonians calculated pi as 3 ‘1s. cut it, pi will never anyone ever written poems about the The Bible, in 1 Kings 7:23, measures a square root of 23 How many other circular vessel built by Solomon using irrational numbers appear in college a ratio of 3. But the Greeks wanted an comeout even,hence football cheers, in Star Trek or as the exact figure. To pin down pi, they title of a recent film? In “x1’-the drew straight-sided shapes inside cir- d circle cun never be movie-a mathematical prodigy goes cles, like a stop sign fit neatly inside a mad seeking number patterns in the wheel. The more sides the polygon squared exuctly. stock market and the Torah. Though had, the closer its area matched that the film is fictional, such a fixation on of the circle. The Greeks called this numbers is not. For millennia, pi has the exhaustion method, and it did plagued exacting minds. Last year, a exhaust many a mathematician. Japanese computer scientist calculated Archimedes, however, could “geo- it to some 206 billion digits. A pi that metrize” longer than most. Working big has no practical use beyond test- from g6-sided figures, he estimated pi ing supercomputers or theorizing in at between 3 iol7i and 3 L/7. For the arcane realms of advanced mathemat- next 700 years, his was the most accu- ics. Yet for , calcu- rate pi. Only in the fifth century A.D., lating pi is “like Mount Everest. I do it when a Chinese astronomer and his because it’s there.” son crammed 24,576 sides inside a cir- Although pi is just a string of num- cle, did the world have a better pi. bers, it can also be exquisite. As the Exhausting a circle may seem blueprint of every circle, pi is in the tedious, but you can do the math in ripples of a pond when you throw in your own home! First bake a pie- banana cream, say. Cut your pie into close enough.) Now turn the can on its four equal slices. Now divide each of side and roll it one full revolution. The those in half. Do it again (16). And can traces a rectangle that is equal to again (32). And just for fun, one more the area of its lid. But Leonardo’s pro- time. Now take a knife and cut cedure was too crude to be of any use around the 64 pieces, slicing off the in refining pi. The search continued. crust in straight lines as close as pos- In 1610, an unlikely monument to sible to the pie dish. Eureka! as pi was built in Holland. A tombstone Archimedes might have said. To cal- in the graveyard of Peter’s Church in culate pi, merely measure around the Leiden was supposedly engraved crust with a ruler and divide by the with the numbers 2-8-8, representing pie’s diameter. Watch out for those the 33i-d through 35th digits of pi, cal- crumbs! Yes, these pieces are pretty culated by the mathematician Ludolph sloppy. You should have tried a stur- van Ceulen. To arrive at pi, van To imm0rt&e his lubors, dier pie-pecan, perhaps. Or maybe Ceulen formulated a y-billion-sided you shouldn’t try to calculate your pi polygon. Computing its area using Ludolph vdn Ceulen was and eat it, too. the state-of-the-art invention called Pi has posed a messy problem for “decimals,” van Ceulen took pi to 20 said to hdve had decimals minds far greater than yours and mine. digits. “Whoever wants to can come The Greeks referred to it as squaring closer,” he concluded, but he was the ofpi carvedinto his the circle. The problem: using straight- only one who wanted to. He spent edge and compass, can you draw a cir- his last 14 years expanding pi to 35 tombstone. cle and a square with equal areas? Pi digits. The tombstone said to have is at the heart of the problem because immortalized his work was lost, and the equals pi times its van Ceulen’s labor was soon rendered radius squared. And once squaring the moot by a new invention-calculus. circle became a passion, the search for As the plague ravaged London in a perfect pi became an obsession. 1665, Isaac Newton retired to his cot- Greek philosophe r Anax- tage in Woolsthorpe. There he devel- .as, imprisoned for teaching oped the calculus, the means of cal- the sun was not : a god, culating the shapes of curves, the arc iled away his time : behind of artillery, the trajectory of a rocket rs trying to square a circle. to the moon. While in Woolsthorpe, Centuries later, the philoso- Newton also calculated pi. Later he pher Thomas Hobbes admitted that the little number had claimed to have solved preoccupied him. “I am ashamed to the problem, and when tell you to how many figures I have proved wrong he lashed carried these computations, having no out, rationalizing that other business at the time,” Newton long-standing principles wrote. Yet with calculus, Newton had of geometry must surely created a new-and much more ele- be wrong. gant-means of finding pi. Leonardo da Vinci Soon mathematicians expanded the came close to a solution little number even further. It reached with a “hands-on” ap- loo digits in 1706, the same year a proach. Get a cylinder British mathematician gave pi its whose height is half its Greek name. (As if the math weren’t radius, he said. (Transla- hard enough, pi had been called “the tion: Go to the kitchen quantity which, when the diameter is and get a short, squat can; multiplied by it, yields the circumfer- an ordinary can of tuna is ence.“) By the late 17oos, “squaring the

s M I r H s 0 N I A N m circle” had become the pastime of hands, you’d expect 200 to be all five cranks, and the French Academy of of the same digit-l, for example. Sciences, inundated with “solutions” When pi is the dealer, you do actual to what it suspected was an impossi- ly get 200. ble problem, refused to accept any Throughout the 19th century, more circle-squaring “proofs” or, for human calculators tried, but could that matter, blueprints for perpetual not finish off pi. One such brain, motion machines. Johann Dase of Hamburg, was able to So what does a big piece of pi look multiply two eight-digit numbers in like? I estimate that the 206 billion dig- his head. Dase could calculate for its currently known would fill more hours at a time, go to sleep, then than a million issues of SMITHSONIAN. wake up and continue where he left In pi’sfirst twomillion Here’s a snapshot of pi’s first 100 dig- off. In 1844, he put his mental calcu- fiuc-digit “hands,” says its: 3.~159265358979~3846264~8~795028 lator to work on pi, and in two 8419~69399~~o582og7474944592307816406 months computed it to a new record probability, you shouldget 286208998628034825342117067 and of 205 places. Another true believer, counting. Using just 3.14, you can pass William Shanks, spent 20 years with your geometry final. Using the first ten pencil and paper calculating pi to 707 about onemillion pairs. decimals, you can measure the earths digits. Shanks’ mark stood into the circumference to a fraction of an inch. 20th century, though it was later dis- So why take pi any further? Some covered that he made a mistake in the mathematicians, like the mad genius in 527th digit. Twenty years on the job the movie “x,” hate to see numbers and his pi was incorrect. Yet Shanks lead to chaos. Every pi seeker hopes wasn’t the only one to get pi wrong. that in mining its depths, some won- A circle is a circle is a circle derful order might emerge. Yet when throughout the universe. Whether it’s taken to extremes, pi’s sequence re- in a wedding ring or the rings of Sat- mains amazingly random. urn, pi is the same number. Except in Suppose pi were a poker hand. Indiana. Or so the state’s House of Imagine that its decimals were cards Representatives proclaimed in 1897.It dealt in a digital game of five-card all began with a country doctor draw. According to the laws of prob- named Edwin Goodwin, of Solitude, ability, based on formulas way, way Indiana, who claimed to have been beyond sixth-grade math, cards and “supernaturally taught the exact mea- random numbers are distributed in a sure of the circle in due confir- way that defies prediction. Never- mation of scriptural promises.” The Bible makes no such promises, of I matical pattern. These odds say course. Goodwin was just another that in pi’s first two million “circle squarer” who refused to let five-digit “hands,” YOU mathematical truth stand in the way should get about one mil- of fame. Never mind that in 1882, a lion pairs-hands such as German mathematician had proved 34459. And in fact you that pi was “transcendental,” which do. The odds also say leads to the conclusion that no mat- you should get 216,000 ter how you cut it, pi will never hands with two pairs, come out even, hence a circle can such as 88663. You get never be squared exactly. Undeterred, 216,520. Pi also obeys the Goodwin set out to do the impossi- laws of probability for three ble. He squared his circle all right, though he had to use a pi valued at of a kind, __a full house and__ four of a kind. Finally, in two million poker 9.2376, almost three times the actual artillery trajectories. Slower than the laptop on which this article is being written, ENIAC nonetheless weighed 30 tons and ran up an electricity bill of $650 an hour. In 1949, ENIAC spewed out its calculation of pi in 70 hours. Mathematicians scratched their heads and wondered. After 2,oy dig- its, there was still no rhyme or reason to the progression of pi. Computers proved the perfect match for extending pi. First in min- utes, later in fractions of a second, they crunched the numbers it had taken William Shanks decades to compute, sending pi into the strato- sphere. It reached a half-million digits in 1967, a million six years later. By 1983, it was up to 16 million digits. But computers are only as good as the formulas programmed into them. Such calculations were made possible by one of the 20th century’s most dazzling minds. was a poor clerk in southern India. value. Goodwin had his “solution” Yet, captivated by , he Johunn Ddse couza published in the Americun Mathemat- studied on his own. In 1913, he sent ical Monthly, then set about getting solutions to several problems to G. H. calculate pifor hours, government approval for his own pri- Hardy at Cambridge University. Rec- vate pi. He convinced his local legis- ognizing Ramanujan’s genius, Hardy go to sleep, then lator to introduce a bill before Indi- brought him to Cambridge. In 194, ana’s House offering state schools free one of the former clerk’s papers wake up and continue use of his “new mathematical truth.” revealed the fastest equation yet for The bill, chock full of math jargon, expanding pi. where he left ofi fooled the House and passed by a In 1984, two Russian brothers using 67-o vote. But not even politicians can their own formula, “in the style of do a number on everybody, and the Ramanujan,” began taking pi toward fraud was soon exposed by a math a billion decimals. They did the math professor. Mercifully bogged down in on a supercomputer they built in bureaucracy, Goodwin’s new truth their apartment. never passed the state Senate. Pi When I visited him at Brooklyn’s remained 3.4159. Polytechnic University, Gregory Following centuries of baflling the Chudnovsky warned me about pi. greatest mathematicians, in the 20th “Look,” he said, “in considering pi century pi came up against a bunch you should not apply your pedestri- of wires and tubes. The world’s first an, Euclidian way of thinking.” I large computer was not designed to agreed that I wouldn’t. Then I settled run circles around a single number. back as Chudnovsky, wrapped in a ENIAC (Electronic Numerical Inte- brown coat and huddled on his office grator and Computer) was built dur- couch, proceeded to open an entire ing World War II to compute universe. More than two decades had passed since Chudnovsky and his a mess.’ Look at pi and it seems brother David fled the Soviet Union equally chaotic.” following severe harassment by the So why bother? I asked. Was it the K.G.B., which wanted to keep them pursuit of a world record? Soon after in the Soviet fold. Four years later, the Chudnovskys computed nearly a Gregory won one of the first Mac- half-billion digits, Yasumasa Kanada Arthur Foundation “genius” grants. topped them by 50 million. The The grant helped keep the brothers Chudnovskys were the first to calcu- Chudnovsky working in spite of ill- late a billion digits. Kanada did a few ness. Gregory, a prodigy who pub- million more. The Chudnovskys took lished a major math treatise while still pi to two billion. Kanada churned out in high school, has myasthenia gravis, six billion. In 1996, the Chudnovskys a muscular disorder that confines him reached eight billion. to a wheelchair. Unable to teach, he But they didn’t do it for the record, began to tinker with supercomputers, they assured me. They were hot on which perform billions of calculations the trail of “hypergeometric function per second. Pi is the acid test for such identities from the point of view of beasts. Like Sisyphus endlessly push- applications to diophantine approxi- During a three-hour ing his rock uphill, a supercomputer mations to numbers. ” Right. In can calculate pi forever without fin- other words, the brothers wanted to memory marathon, Simon ishing. So the Chudnovskys, experts know whether the formulas for pi in number theory, began putting pi could be grouped into categories Ploufe spewedout a record through its paces. At first, the broth- according to theories almost no one ers tapped into a Cray supercom- else in the world understands. At 4,096digits of pi. puter over a phone line, but this eight billion places, and finding no proved too costly. profound pattern in pi, the Chud- In 1989, they built a home comput- novskys decided they had gone far er-but not your garden variety desk- enough. “Pi was interesting up to a top. Ordering parts from tech stores point,” Gregory concluded, “but you around the country, they assembled don’t want to make a fetish out of it.” one of the world’s most powerful Fetish, no. But a cult? In this age of computers in their Manhattan apart- chaos theory, pi is digital chaos and it ment. With its microprocessors and has a growing number of admirers. cables snaking from room to room, At San Francisco’s Exploratorium, Pi their creation doubled as a heater. Day is celebrated each March 14 at Even when cooled by dozens of fans, qg P.M. (Get it? 3/14 at i:5g?) Pi lovers the computer warmed its surround- circle the science museum’s pi shrine ings to go degrees. Battling frequent 3.14 times, add numbered beads to a breakdowns and unpredictable freight string listing pi’s decimals and hear deliveries of parts, the brothers Chud- music based on the number. They eat novsky were soon in the thick of pi. pie; lemon meringue and apple are Many mathematicians are satisfied favorites. Some perhaps even wear to call pi irrational. The wry and n-the fragrance by Givenchy. Each articulate Chudnovskys call it fall at the Massachusetts Institute of monstrous, subtle, exquisite, Technology, football fans cheer for boring, humbling, annoying, their favorite irrational number: terra incognita and a com- “Cosine, secant, tangent, sine, 3.i415g!” plete mystery. “Look at the Pi turns up like a mantra in Carl sky,” David suggests. “Billions Sagan’s novel Contact, where an alien of stars clumped together, scat- tells an earthling of a message encod- :ered around. You think, ‘What ed within the digits of pi. In one Star more than 5,000. Plouffe, by then, was pursuing higher mathematics. He eventually coauthored an amazing formula for determining any digit of pi, at least in binary form, without calculating all the previous ones. The current memorization record is now well beyond his reach, he admits. The reigning champ is Hiroyuki Goto, who recited 42,195 digits in nine hours. Do not try this math at home. But if you just want to memorize a baker’s dozen digits of pi, recite this: “See, I have a rhyme assisting I my feeble brain, / its tasks oft-times resist- ing.” Count the number of letters in each word, and voila: 3.Mi5g265358g. Such memory aids have been crafted in several languages, but they pale compared with the verse pi has inspired. In her poem Pi, Wislawa Szymborska, winner of the 1996 Nobel Prize in Literature, praised pi for its dogged approach to infinity. Szymborska wove the digits of pi into her poem: ‘The admirable number pi: I three point one four one. I All the fol- Even when cooled by T&k episode, Speck distracts an evil lowing digits are also initial, /five nine computer by instructing it to com- two because it never ends. / It can’t be dozens offans, the pute pi to the last digit. Yet some comprehended six fiue three five at a exacting minds are not content to glance, / eight nine by calculation, / Chudn0dy brothers’ merely chant, eat or smell pi. They seven nine or imagination. ” Awed want to know it by heart. by pi’s infinite reach, Szymborska fol- computer heated As a teenager in Montreal, Simon lows the number as it “goes on across Plouffe became “addicted to num- the table, through the air, / over a its suwclundings bers.” Upon learning of a world wall, a leaf, a bird’s nest, clouds, record for memorizing pi, Plouffe set straight into the sky.” to 90 degrees. out to break it. On his first day, Pi continues to amaze everyone Plouffe memorized 300 digits. Soon who has ever studied a circle. But as he was isolating himself in a dark supercomputers expand, calculating a room, going over the decimals in his trillion or a quadrillion digits, will head. Within six months, he had mathematicians ever find some grand memorized 4,096 digits of pi. “It was order in its chaos? David Chud- a very mystical experience,” Plouffe novsky is optimistic. “It’s very possi- told me. “You develop a matrix in ble that there’s some very simple your head. You can almost see the answer. We just don’t have it yet.” E numbers.” During a three-hour marathon, Plouffe spewed out his An anthology of Bruce Watson’s humorous slice of pi and entered the French edi- essays, Info-Aged, was published in 1999. tion of the Guinness Book of Records. Jeff Seauer has illustrated many articles But not for long. Soon the record was for SMITHSONIAN.