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MATH LABComputer experiments are transforming mathematics

BY ERICA KLARREICH

Many people regard mathematics as the crownjewel of the sciences. Yet math has histori-cally lacked one of the defining trappingsof science: laboratory equipment. Physicistshave their particle accelerators; biologists,

their electron microscopes; and astronomers, their tel-escopes. Mathematics, by contrast, concerns not the physical landscape but an idealized, abstract world. For explor-ing that world, mathematicians have traditionally had only theirintuition.

Now, computers are starting to give mathematicians the labinstrument that they have beenmissing. Sophisticated software isenabling researchers to travel fur-ther and deeper into the mathe-matical universe. They’re calcu-lating the number pi withmind-boggling precision, forinstance, or discovering patternsin the contours of beautiful, infi-nite chains of spheres that ariseout of the geometry of knots.

Experiments in the computer labare leading mathematicians to dis-coveries and insights that they mightnever have reached by traditionalmeans. “Pretty much every [math-ematical] field has been transformedby it,” says Richard Crandall, a math-ematician at Reed College in Port-land, Ore. “Instead of just being anumber-crunching tool, the com-puter is becoming more like a gar-den shovel that turns over rocks, andyou find things underneath.”

At the same time, the new workis raising unsettling questions abouthow to regard experimental resultsin a discipline for which rigorous proof is the gold standard.

At a workshop late in March in Oakland, Calif., mathematiciansgathered to discuss current efforts in computer-assisted researchand to consider the approach’s promise for the future. The work-shop’s organizers, David Bailey of Lawrence Berkeley (Calif.)National Laboratory and Jonathan Borwein of Dalhousie Uni-versity in Halifax, Nova Scotia, argue that computer experimen-tation is launching a new epoch in mathematics.

Computer power, Borwein says, is enabling mathematicians tomake a quantum leap akin to the one that took place whenLeonardo of Pisa introduced Arabic numerals—1, 2, 3, . . .—toEuropean mathematicians in the 12th century.

“I have some of the excitement that Leonardo of Pisa must havefelt when he encountered Arabic arithmetic. It suddenly made cer-tain calculations flabbergastingly easy,” Borwein says. “That’s whatI think is happening with computer experimentation today.”

EXPERIMENTERS OF OLD In one sense, math experimentsare nothing new. Despite their field’s reputation as a purely deduc-tive science, the great mathematicians over the centuries havenever limited themselves to formal reasoning and proof.

For instance, in 1666, sheer curiosity and love of numbers led IsaacNewton to calculate directly the first 16 digits of the number pi,later writing, “I am ashamed to tell you to how many figures I car-ried these computations, having no other business at the time.”

Carl Friedrich Gauss, one of the towering figures of 19th-cen-tury mathematics, habitually dis-covered new mathematical resultsby experimenting with numbers andlooking for patterns. When Gausswas a teenager, for instance, hisexperiments led him to one of themost important conjectures in thehistory of number theory: that thenumber of prime numbers less thana number x is roughly equal to xdivided by the logarithm of x.

Gauss often discovered resultsexperimentally long before he couldprove them formally. Once, he com-plained, “I have the result, but I donot yet know how to get it.”

In the case of the prime numbertheorem, Gauss later refined hisconjecture but never did figure outhow to prove it. It took more than acentury for mathematicians to comeup with a proof.

Like today’s mathematicians,math experimenters in the late 19thcentury used computers—but inthose days, the word referred to peo-ple with a special facility for calcu-

lation. These specialists would often spend days or months mak-ing enormous tables of computations. Mathematicians of the timealso built expensive three-dimensional geometric models to try tobolster their insight about solid geometry.

Today, electronic computers take only seconds to carry outcalculations and to create beautiful graphics of three-dimen-sional shapes. Whereas Newton labored to calculate 16 digitsof pi, for instance, the current computer-assisted record is morethan 1 trillion digits (MathTrek, Science News Online: http://www.sciencenews.org/articles/20021214/mathtrek.asp).

David Mumford, a mathematician at Brown University in Prov-idence, R.I., says that he tells his students that merely “having Excel

UNSOLVED MYSTERIES — A computer experiment produced this plot of all the solutions to a collection of simple equations in 2001. Mathematicians are still trying toaccount for its many features.

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software on their desktops gives them power that mathematiciansin the past would have drooled over.” That power is now enablingresearchers to discover hidden corners of the mathematical universe,many of which earlier mathematicians never dreamed existed.

RELATING TO PI In 1995, Bailey used computer experimenta-tion to discover something very much in the spirit of earlier exper-imenters. It was a new formula for pi (MathTrek, Science NewsOnline: http://www.sciencenews.org/pages/sn_arc98/2_28_98/mathland.htm). Over the centuries, mathematicians have foundmany amazingly simple ways to express pi as an infinite sum, forinstance, 1 – 1/3 + 1/5 – 1/7 + 1/9 . . . .

Bailey’s collaborators Peter Borwein of Simon Fraser Universityin Burnaby, British Columbia, and Simon Plouffe, now at the Uni-versity of Québec in Montreal, had recently noticed that the loga-rithm of 2 has a simple infinite-sum formula with an unusual prop-erty. The formula can reveal, say, themillionth binary digit of log 2 with noneed to calculate the 999,999 digitsbefore it. Peter Borwein and Ploufferealized that about 20 other mathe-matical constants have similar short-cut formulas.

Together with Bailey, the pair won-dered whether pi also has a shortcutformula. A thorough search of themathematical literature failed to turnup any such formula.

“At the time, it seemed extremelyunlikely to us that such a formulaexisted,” Bailey says. “We presumedthat if it did, it would have been dis-covered 200 or 300 years ago.”

Nevertheless, the researchersdecided to use a computer programdesigned by Bailey and Helaman Fer-guson, a sculptor and mathematicianbased in Laurel, Md., to seek numerical relationships between piand the constants the math team already knew to have shortcut for-mulas (MathTrek, Science News Online: http://www.sciencenews.org/articles/20000212/mathtrek.asp). If they could find a relationshipof the right kind—one that represented pi as a sum of the other con-stants multiplied by whole numbers—they knew they could use theknown shortcut formulas to write a shortcut formula for pi.

The researchers calculated pi and the other constants to anaccuracy of several hundred digits and set the computer search-ing for a relationship among these long strings of digits. Even-tually, the computer found an equation that related pi to log 5and two other constants.

Bailey recalls, “After months of runs with different constants, inthe middle of the night, the computer found the relation, and itsent Peter and Simon an e-mail. The next morning, they wrote itout, and, sure enough, it gave a formula for pi.”

Once the computer had produced the formula, proving that itwas correct was embarrassingly easy, Bailey says. “The proof is lit-erally a six-line exercise in freshman calculus,” he explains.

This is frequently the case with experimental results, Jonathan Bor-wein says. “Often, knowing what is true is 99 percent of the battle,”he notes.

KNOTTY SHAPES It might seem that computers’ calculatingpower makes them particularly well suited for tackling numericalquestions rather than geometric ones. However, computer exper-imentation has also become a valuable tool for geometry. Sophis-ticated software packages can perform complicated geometric cal-culations and produce shapes and patterns that mathematicianshad never before visualized.

One program called Snappea—created by Jeffrey Weeks, a free-

lance mathematician in Canton, N.Y.—has revolutionized the studyof three-dimensional shapes with hyperbolic geometry, an alter-native geometry to the one that Euclid compiled and most school-children still study. Last spring, for example, Colin Adams, a math-ematician at Williams College in Williamstown, Mass., usedSnappea to discover an unexpected property of certain knots.

Adams was interested in what mathematicians call knot com-plements: the three-dimensional shapes left behind when a knot-ted loop is drilled out of three-dimensional space, like a wormholethrough an apple (SN: 12/8/01, p. 360). Mathematicians haveknown for decades that many knot complements have hyperbolicgeometry.

One of the ways that mathematicians study such a shape is toexamine a pattern of balls, called horoballs, that encodes the sym-metries of the shape. While using Snappea to draw the horoballpatterns corresponding to certain knots, Adams and his student

Eric Schoenfeld stumbled upon some-thing they had never seen before. Itwas a knot whose horoballs lined upinto perfectly straight chains. Nor-mally, the pattern of horoballs is ran-dom— “a mess,” Adams says.

“We thought, ‘That’s weird,’” Adamsrecalls. “It was really exciting.”

He and Schoenfeld eventually real-ized that the straight lines of horoballsindicated that the knot complementcontains a special surface that is com-pletely flat from the perspective ofhyperbolic geometry. Previously,Adams says, mathematicians had noreason to expect such a surface to exist.After looking at more examples,Adams and several of his studentsproved that a whole family of knotshas such surfaces.

“There’s no way we would have beenled to these results without the computer,” Adams says. Snappea hasbecome an indispensable tool for studying shapes with hyperbolicgeometry, he adds.

“I’m incredibly dependent on it and use it all the time as my lab-oratory,” he says. “These patterns just pop out at you that you haveno explanation for, and then slowly you explain what you see.”

Modern three-dimensional hyperbolic geometry actually owesits origins to computer experiments, Adams says. It was throughexperiments by the late mathematician Robert Riley that WilliamThurston of Cornell University first realized in the 1970s that knotcomplements can have hyperbolic geometry.

Subsequent experiments led Thurston to formulate a famousconjecture about the kinds of geometry three-dimensional shapescan have—work that earned Thurston a Fields Medal, the high-est honor in mathematics. Last year, Russian mathematician Grig-ory Perelman made headlines with a claimed proof of Thurston’sconjecture, which mathematicians are now scrutinizing (SN: 6/14/03, p. 378).

“Without the computer, this field of mathematics wouldn’t exist,”Adams says. Computers have “allowed us to explore areas of mathwe couldn’t explore before,” he says.

PATHS TO ENLIGHTENMENT Although Adams worked outformal proofs to back up his experimental findings in hyperbolicgeometry, computer experiments often lead mathematicians tofindings that they have no idea how to prove.

“One thing that’s happening is you can discover many morethings than you can explain,” Jonathan Borwein says.

If experimental discoveries indeed flood in faster than they canbe proved, could that change the very nature of mathematics? Intheir book Mathematics by Experiment (2003, A.K. Peters Ltd.),

STRAIGHT CIRCLES — When mathematiciansColin Adams and Eric Schoenfeld created this imagewhile playing with the computer program Snappealast year, they were stunned to see perfectly straightchains of spheres. The observation led them to anunexpected discovery about knots.

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Bailey and Jonathan Borwein advance the controversial thesis thatmathematics should move toward a more empirical approach. Init, formal proof would not be the only acceptable way to establishmathematical knowledge.

Mathematicians, Bailey and Borwein argue, should be free towork more like other scientists do, developing hypotheses throughexperimentation and then testingthem in further experiments. Formalproof is still the ideal, they say, but itis not the only path to mathematicaltruth.

“When I started school, I thoughtmathematics was about proofs, butnow I think it’s about having securemathematical knowledge,” Borweinsays. “We claim that’s not the samething.”

Bailey and Borwein point out thatmathematical proofs can run to hun-dreds of pages and require such spe-cialized knowledge that only a fewpeople are capable of reading andjudging them.

“We feel that in many cases, com-putations constitute very strong evidence, evidence that is at leastas compelling as some of the more complex formal proofs in theliterature,” Bailey and Borwein say in Mathematics by Experiment.

Gregory Chaitin, a mathematician at IBM T.J. Watson ResearchCenter in Yorktown Heights, N.Y., argues that if there is enoughexperimental evidence for an important conjecture, mathemati-cians should adopt it as an axiom. He cites the Riemann hypoth-esis. This conjecture postulates that, apart from a few well-under-stood exceptions, all the solutions to a certain famous equationhave a simple relationship to one another.

Mathematicians have calculated billions of solutions to the equa-tion, and they do indeed satisfy the relationship. On the basis ofthis evidence, Chaitin says, a physicist would accept the Riemannhypothesis and its far-reaching ramifications.

“Mathematicians have to start behaving a little more like physi-cists,” he says. “So many useful results are being suggested by exper-imental data that it seems almost criminal to say that we’re goingto ignore the data because we have no proof.”

This view is far from mainstream, however. To Bernd Sturmfels,a mathematician at the University of California, Berkeley who doescomputer experiments in algebra, rigorous proof is precisely whatdistinguishes mathematics from physics.

“I think proof is very much at the heart of mathematics,” he says.“Our understanding can be significantly advanced by experiments,but I think there will always be a clear borderline as to what con-stitutes an acceptable result in pure mathematics.”

Mumford agrees. “I am quite certain that the psychology of puremathematicians is different from that of scientists—that this ideaof proof is central and will not be altered,” he says.

In the case of the Riemann hypothesis, mathematicians observethat although the experimental data may look convincing, otherstatements with similar amounts of supporting evidence haveturned out to be false.

David Eisenbud, the director of the Mathematical SciencesResearch Institute in Berkeley, Calif., points out that it wouldn’tfurther mathematicians’ understanding to accept the truth of amathematical statement such as the Riemann hypothesis on thebasis of computer-generated, experimental evidence. In contrast,the ideas in a proof might provide deep insight into why the state-ment should be true.

“Proof is the path to understanding,” Eisenbud says.Regardless of whether mathematicians start behaving more like

physicists, one thing seems clear. The crown jewel of the sciencesfinally has a lab instrument worthy of it. ■

“One thingthat’shappening is you candiscover manymore thingsthan you canexplain.”— JONATHAN BORWEIN DALHOUSIE UNIVERSITY

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