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<p>CHAOS</p><p>MakingaNewScience</p><p>JamesGleick</p><p>ToCynthia</p><p>humanwasthemusic,</p><p>naturalwasthestatic…</p><p>—JOHNUPDIKE</p><p>Contents</p><p>Prologue</p><p>TheButterflyEffect</p><p>EdwardLorenzandhistoyweather.Thecomputermisbehaves.Long-rangeforecastingisdoomed.Order</p><p>masqueradingasrandomness.Aworldofnonlinearity.“Wecompletelymissedthepoint.”</p><p>Revolution</p><p>Arevolutioninseeing.Pendulumclocks,spaceballs,andplaygroundswings.Theinventionofthe</p><p>horseshoe.Amysterysolved:Jupiter’sGreatRedSpot.</p><p>Life’sUpsandDowns</p><p>Modelingwildlifepopulations.Nonlinearscience,“thestudyofnon-elephantanimals.”Pitchfork</p><p>bifurcationsandarideontheSpree.Amovieofchaosandamessianicappeal.</p><p>AGeometryofNature</p><p>Adiscoveryaboutcottonprices.ArefugeefromBourbaki.Transmissionerrorsandjaggedshores.New</p><p>dimensions.Themonstersoffractalgeometry.Quakesintheschizosphere.Fromcloudstobloodvessels.</p><p>Thetrashcansofscience.“Toseetheworldinagrainofsand.”</p><p>StrangeAttractors</p><p>AproblemforGod.Transitionsinthelaboratory.Rotatingcylindersandaturningpoint.DavidRuelle’s</p><p>ideaforturbulence.Loopsinphasespace.Mille-feuillesandsausage.Anastronomer’smapping.</p><p>“Fireworksorgalaxies.”</p><p>Universality</p><p>AnewstartatLosAlamos.Therenormalizationgroup.Decodingcolor.Theriseofnumerical</p><p>experimentation.MitchellFeigenbaum’sbreakthrough.Auniversaltheory.Therejectionletters.Meetingin</p><p>Como.Cloudsandpaintings.</p><p>TheExperimenter</p><p>HeliuminaSmallBox.“Insolidbillowingofthesolid.”Flowandforminnature.AlbertLibchaber’s</p><p>delicatetriumph.Experimentjoinstheory.Fromonedimensiontomany.</p><p>ImagesofChaos</p><p>Thecomplexplane.SurpriseinNewton’smethod.TheMandelbrotset:sproutsandtendrils.Artand</p><p>commercemeetscience.Fractalbasinboundaries.Thechaosgame.</p><p>TheDynamicalSystemsCollective</p><p>SantaCruzandthesixties.Theanalogcomputer.Wasthisscience?“Along-rangevision.”Measuring</p><p>unpredictability.Informationtheory.Frommicroscaletomacroscale.Thedrippingfaucet.Audiovisualaids.</p><p>Aneraends.</p><p>InnerRhythms</p><p>Amisunderstandingaboutmodels.Thecomplexbody.Thedynamicalheart.Resettingthebiologicalclock.</p><p>Fatalarrhythmia.Chickembryosandabnormalbeats.Chaosashealth.</p><p>ChaosandBeyond</p><p>Newbeliefs,newdefinitions.TheSecondLaw,thesnowflakepuzzle,andloadeddice.Opportunityand</p><p>necessity.</p><p>Afterword</p><p>NotesonSourcesandFurtherReading</p><p>Acknowledgments</p><p>Index</p><p>CHAOS</p><p>Prologue</p><p>THE POLICE IN THE SMALL TOWN of Los Alamos, New Mexico, worried</p><p>brieflyin1974aboutamanseenprowlinginthedark,nightafternight,thered</p><p>glowofhiscigarette floatingalong thebackstreets.Hewouldpace forhours,</p><p>headingnowhereinthestarlightthathammersdownthroughthethinairofthe</p><p>mesas.Thepolicewerenottheonlyonestowonder.Atthenationallaboratory</p><p>somephysicistshadlearnedthattheirnewestcolleaguewasexperimentingwith</p><p>twenty-six–hourdays,whichmeantthathiswakingschedulewouldslowlyroll</p><p>in and out of phase with theirs. This bordered on strange, even for the</p><p>TheoreticalDivision.</p><p>In the three decades since J. Robert Oppenheimer chose this unworldly</p><p>New Mexico landscape for the atomic bomb project, Los Alamos National</p><p>Laboratoryhad spread across an expanseofdesolateplateau, bringingparticle</p><p>accelerators and gas lasers and chemical plants, thousands of scientists and</p><p>administrators and technicians, as well as one of the world’s greatest</p><p>concentrationsofsupercomputers.Someoftheolderscientistsrememberedthe</p><p>woodenbuildingsrisinghastilyoutoftherimrockinthe1940s,buttomostof</p><p>the Los Alamos staff, young men and women in college-style corduroys and</p><p>work shirts, the first bombmakers were just ghosts. The laboratory’s locus of</p><p>purest thought was the Theoretical Division, known as T division, just as</p><p>computingwasCdivision andweaponswasXdivision.More than a hundred</p><p>physicists and mathematicians worked in T division, well paid and free of</p><p>academic pressures to teach and publish. These scientists had experiencewith</p><p>brillianceandwitheccentricity.Theywerehardtosurprise.</p><p>But Mitchell Feigenbaum was an unusual case. He had exactly one</p><p>published article to his name, and hewasworking on nothing that seemed to</p><p>haveanyparticularpromise.Hishairwasa raggedmane,sweepingbackfrom</p><p>hiswidebrowinthestyleofbustsofGermancomposers.Hiseyesweresudden</p><p>andpassionate.Whenhespoke,always rapidly,he tended todroparticlesand</p><p>pronouns in a vaguelymiddleEuropeanway, even though hewas a native of</p><p>Brooklyn.Whenheworked,heworkedobsessively.Whenhecouldnotwork,he</p><p>walkedand thought,dayornight, andnightwasbestof all.The twenty-four–</p><p>hour day seemed too constraining. Nevertheless, his experiment in personal</p><p>quasiperiodicity came to an end when he decided he could no longer bear</p><p>wakingtothesettingsun,ashadtohappeneveryfewdays.</p><p>At the age of twenty-nine he had already become a savant among the</p><p>savants, an ad hoc consultant whom scientists would go to see about any</p><p>especially intractable problem, when they could find him. One evening he</p><p>arrived at work just as the director of the laboratory, Harold Agnew, was</p><p>leaving. Agnew was a powerful figure, one of the original Oppenheimer</p><p>apprentices. He had flown over Hiroshima on an instrument plane that</p><p>accompaniedtheEnolaGay,photographingthedeliveryofthelaboratory’sfirst</p><p>product.</p><p>“Iunderstandyou’rerealsmart,”AgnewsaidtoFeigenbaum.“Ifyou’reso</p><p>smart,whydon’tyoujustsolvelaserfusion?”</p><p>EvenFeigenbaum’sfriendswerewonderingwhetherhewasevergoingto</p><p>produceanyworkofhisown.Aswillingashewastodoimpromptumagicwith</p><p>theirquestions,hedidnotseeminterested indevotinghisownresearch toany</p><p>problemthatmightpayoff.Hethoughtaboutturbulenceinliquidsandgases.He</p><p>thought about time—did it glide smoothly forward or hop discretely like a</p><p>sequenceofcosmicmotion-pictureframes?Hethoughtabouttheeye’sabilityto</p><p>seeconsistentcolorsandformsinauniversethatphysicistsknewtobeashifting</p><p>quantumkaleidoscope.He thought about clouds,watching them fromairplane</p><p>windows(until,in1975,hisscientifictravelprivilegeswereofficiallysuspended</p><p>ongroundsofoveruse)orfromthehikingtrailsabovethelaboratory.</p><p>In the mountain towns of the West, clouds barely resemble the sooty</p><p>indeterminatelow-flyinghazesthatfilltheEasternair.AtLosAlamos,inthelee</p><p>ofagreatvolcaniccaldera,thecloudsspillacrossthesky,inrandomformation,</p><p>yes, but also not-random, standing in uniform spikes or rolling in regularly</p><p>furrowed patterns like brain matter. On a stormy afternoon, when the sky</p><p>shimmersand trembleswith theelectricity tocome, theclouds standout from</p><p>thirtymilesaway,filteringthelightandreflectingit,untilthewholeskystartsto</p><p>seem like a spectacle staged as a subtle reproach to physicists. Clouds</p><p>representedasideofnaturethatthemainstreamofphysicshadpassedby,aside</p><p>thatwasatonce,fuzzyanddetailed,structuredandunpredictable.Feigenbaum</p><p>thoughtaboutsuchthings,quietlyandunproductively.</p><p>Toaphysicist,creatinglaserfusionwasalegitimateproblem;puzzlingout</p><p>thespinandcolorandflavorofsmallparticleswasalegitimateproblem;dating</p><p>theoriginoftheuniversewasalegitimateproblem.Understandingcloudswasa</p><p>problem for a meteorologist. Like other physicists, Feigenbaum used an</p><p>understated, tough-guy vocabulary to rate such problems. Such a thing is</p><p>obvious,hemightsay,meaningthataresultcouldbe</p><p>lines of inquiry. Thesis proposals are turned down or articles are</p><p>refused publication.The theorists themselves are not surewhether theywould</p><p>recognize an answer if they sawone.They accept risk to their careers.A few</p><p>freethinkers working alone, unable to explain where they are heading, afraid</p><p>eventotelltheircolleagueswhattheyaredoing—thatromanticimageliesatthe</p><p>heartofKuhn’sscheme,andithasoccurredinreallife,timeandtimeagain,in</p><p>theexplorationofchaos.</p><p>Every scientist who turned to chaos early had a story to tell of</p><p>discouragement or open hostility. Graduate students were warned that their</p><p>careers could be jeopardized if theywrote theses in an untested discipline, in</p><p>which their advisors had no expertise.A particle physicist, hearing about this</p><p>new mathematics, might begin playing with it on his own, thinking it was a</p><p>beautifulthing,bothbeautifulandhard—butwouldfeelthathecouldnevertell</p><p>his colleagues about it. Older professors felt they were suffering a kind of</p><p>midlifecrisis,gamblingonalineofresearchthatmanycolleagueswerelikelyto</p><p>misunderstandorresent.Buttheyalsofeltanintellectualexcitementthatcomes</p><p>with the truly new. Even outsiders felt it, those who were attuned to it. To</p><p>FreemanDyson at the Institute for Advanced Study, the news of chaos came</p><p>“likeanelectric shock” in the1970s.Others felt that for the first time in their</p><p>professionallivestheywerewitnessingatrueparadigmshift,atransformationin</p><p>awayofthinking.</p><p>Thosewhorecognizedchaosintheearlydaysagonizedoverhowtoshape</p><p>theirthoughtsandfindingsintopublishableform.Workfellbetweendisciplines</p><p>—for example, too abstract for physicists yet too experimental for</p><p>mathematicians.Tosomethedifficultyofcommunicatingthenewideasandthe</p><p>ferociousresistancefromtraditionalquartersshowedhowrevolutionarythenew</p><p>science was. Shallow ideas can be assimilated; ideas that require people to</p><p>reorganizetheirpictureoftheworldprovokehostility.AphysicistattheGeorgia</p><p>InstituteofTechnology,JosephFord,startedquotingTolstoy:“Iknowthatmost</p><p>men, including those at ease with problems of the greatest complexity, can</p><p>seldomaccepteven thesimplestandmostobvious truth if itbesuchaswould</p><p>oblige them to admit the falsity of conclusions which they have delighted in</p><p>explaining tocolleagues,which theyhaveproudly taught toothers, andwhich</p><p>theyhavewoven,threadbythread,intothefabricoftheirlives.”</p><p>Manymainstream scientists remained only dimly aware of the emerging</p><p>science.Some,particularlytraditionalfluiddynamicists,activelyresentedit.At</p><p>first, the claimsmade on behalf of chaos soundedwild and unscientific. And</p><p>chaosreliedonmathematicsthatseemedunconventionalanddifficult.</p><p>As the chaos specialists spread, some departments frowned on these</p><p>somewhat deviant scholars; others advertised for more. Some journals</p><p>established unwritten rules against submissions on chaos; other journals came</p><p>forthtohandlechaosexclusively.Thechaoticistsorchaologists(suchcoinages</p><p>couldbeheard)turnedupwithdisproportionatefrequencyontheyearlylistsof</p><p>important fellowships and prizes. By the middle of the eighties a process of</p><p>academicdiffusionhadbroughtchaosspecialistsintoinfluentialpositionswithin</p><p>university bureaucracies. Centers and institutes were founded to specialize in</p><p>“nonlineardynamics”and“complexsystems.”</p><p>Chaos has become not just theory but also method, not just a canon of</p><p>beliefsbutalsoawayofdoingscience.Chaoshascreateditsowntechniqueof</p><p>usingcomputers,a techniquethatdoesnotrequire thevastspeedofCraysand</p><p>Cybers but instead favorsmodest terminals that allow flexible interaction. To</p><p>chaos researchers,mathematics has become an experimental science, with the</p><p>computer replacing laboratories full of test tubes and microscopes. Graphic</p><p>imagesarethekey.“It’smasochismforamathematiciantodowithoutpictures,”</p><p>onechaosspecialistwouldsay.“Howcantheyseetherelationshipbetweenthat</p><p>motionand this?Howcan theydevelop intuition?”Somecarryout theirwork</p><p>explicitlydenyingthatitisarevolution;othersdeliberatelyuseKuhn’slanguage</p><p>ofparadigmshiftstodescribethechangestheywitness.</p><p>Stylistically, early chaospapers recalled theBenjaminFranklin era in the</p><p>waytheywentbacktofirstprinciples.AsKuhnnotes,establishedsciencestake</p><p>for granted a body of knowledge that serves as a communal starting point for</p><p>investigation. To avoid boring their colleagues, scientists routinely begin and</p><p>endtheirpaperswithesoterica.Bycontrast,articlesonchaosfromthelate1970s</p><p>onward sounded evangelical, from their preambles to their perorations. They</p><p>declarednewcredos,and theyoftenendedwithpleas foraction.Theseresults</p><p>appeartoustobebothexcitingandhighlyprovocative.Atheoreticalpictureof</p><p>the transition to turbulence is just beginning to emerge.Theheart of chaos is</p><p>mathematicallyaccessible.Chaosnowpresagesthefutureasnonewillgainsay.</p><p>Buttoacceptthefuture,onemustrenouncemuchofthepast.</p><p>New hopes, new styles, and, most important, a new way of seeing.</p><p>Revolutionsdonotcomepiecemeal.Oneaccountofnaturereplacesanother.Old</p><p>problemsareseeninanewlightandotherproblemsarerecognizedforthefirst</p><p>time.Something takesplace that resemblesawhole industry retooling fornew</p><p>production.InKuhn’swords,“Itisratherasiftheprofessionalcommunityhad</p><p>beensuddenlytransportedtoanotherplanetwherefamiliarobjectsareseenina</p><p>differentlightandarejoinedbyunfamiliaronesaswell.”</p><p>THELABORATORYMOUSEofthenewsciencewasthependulum:emblemof</p><p>classical mechanics, exemplar of constrained action, epitome of clockwork</p><p>regularity.Abobswingsfreeattheendofarod.Whatcouldbefurtherremoved</p><p>fromthewildnessofturbulence?</p><p>WhereArchimedeshadhisbathtubandNewtonhisapple,so,accordingto</p><p>the usual suspect legend,Galileo had a church lamp, swaying back and forth,</p><p>time and again, on and on, sending its message monotonously into his</p><p>consciousness.ChristianHuygensturnedthepredictabilityofthependuluminto</p><p>ameansoftimekeeping,sendingWesterncivilizationdownaroadfromwhich</p><p>therewasnoreturn.Foucault,inthePanthéonofParis,usedatwenty-story–high</p><p>pendulumtodemonstratetheearth’srotation.Everyclockandeverywristwatch</p><p>(until theeraofvibratingquartz) reliedonapendulumof somesizeor shape.</p><p>(For thatmatter, theoscillationof quartz is not sodifferent.) In space, freeof</p><p>friction,periodicmotioncomesfromtheorbitsofheavenlybodies,butonearth</p><p>virtuallyanyregularoscillationcomesfromsomecousinofthependulum.Basic</p><p>electronic circuits are described by equations exactly the same as those</p><p>describing a swinging bob. The electronic oscillations are millions of times</p><p>faster, but the physics is the same.By the twentieth century, though, classical</p><p>mechanics was strictly a business for classrooms and routine engineering</p><p>projects.Pendulumsdecoratedsciencemuseumsandenlivenedairportgiftshops</p><p>intheformofrotatingplastic“spaceballs.”Noresearchphysicistbotheredwith</p><p>pendulums.</p><p>Yetthependulumstillhadsurprisesinstore.Itbecameatouchstone,asit</p><p>had for Galileo’s revolution.WhenAristotle looked at a pendulum, he saw a</p><p>weighttryingtoheadearthwardbutswingingviolentlybackandforthbecauseit</p><p>wasconstrainedbyitsrope.Tothemodern</p><p>earthissoundsfoolish.Forsomeone</p><p>bound by classical concepts of motion, inertia, and gravity, it is hard to</p><p>appreciate the self-consistent world view that went with Aristotle’s</p><p>understandingofapendulum.Physicalmotion,forAristotle,wasnotaquantity</p><p>or a force but rather a kind of change, just as a person’s growth is a kind of</p><p>change.Afallingweightissimplyseekingitsmostnaturalstate,thestateitwill</p><p>reach if left to itself. In context, Aristotle’s view made sense. When Galileo</p><p>looked at a pendulum, on the other hand, he saw a regularity that could be</p><p>measured.Toexplainitrequiredarevolutionarywayofunderstandingobjectsin</p><p>motion.Galileo’sadvantageovertheancientGreekswasnotthathehadbetter</p><p>data.Onthecontrary,hisideaoftimingapendulumpreciselywastogetsome</p><p>friends together to count the oscillations over a twenty-four–hour period—a</p><p>labor-intensiveexperiment.Galileosawtheregularitybecausehealreadyhada</p><p>theorythatpredictedit.HeunderstoodwhatAristotlecouldnot: thatamoving</p><p>object tends tokeepmoving, thatachangeinspeedordirectioncouldonlybe</p><p>explainedbysomeexternalforce,likefriction.</p><p>In fact, so powerful was his theory that he saw a regularity that did not</p><p>exist.He contended that a pendulumof a given length not only keeps precise</p><p>time but keeps the same time nomatter howwide or narrow the angle of its</p><p>swing.Awide-swingingpendulumhasfarthertotravel,butithappenstotravel</p><p>just that much faster. In other words, its period remains independent of its</p><p>amplitude. “If two friends shall set themselves to count the oscillations, one</p><p>counting thewide ones and the other the narrow, theywill see that theymay</p><p>countnotjusttens,butevenhundreds,withoutdisagreeingbyevenone,orpart</p><p>of one.”Galileophrasedhis claim in termsof experimentation, but the theory</p><p>made it convincing—somuch so that it is still taught as gospel inmost high</p><p>school physics courses.But it iswrong.The regularityGalileo saw is only an</p><p>approximation. The changing angle of the bob’s motion creates a slight</p><p>nonlinearityintheequations.Atlowamplitudes,theerrorisalmostnonexistent.</p><p>But it is there,and it ismeasurableeven inanexperimentascrudeas theone</p><p>Galileodescribes.</p><p>Small nonlinearities were easy to disregard. People who conduct</p><p>experiments learnquicklythat theylive inanimperfectworld.In thecenturies</p><p>since Galileo and Newton, the search for regularity in experiment has been</p><p>fundamental.Anyexperimentalist looks forquantities that remain thesame,or</p><p>quantities that are zero. But that means disregarding bits of messiness that</p><p>interfere with a neat picture. If a chemist finds two substances in a constant</p><p>proportionof2.001oneday,and2.003thenextday,and1.998thedayafter,he</p><p>wouldbeafoolnottolookforatheorythatwouldexplainaperfecttwo-to–one</p><p>ratio.</p><p>Togethisneatresults,Galileoalsohadtodisregardnonlinearities thathe</p><p>knewof: friction and air resistance.Air resistance is a notorious experimental</p><p>nuisance,acomplicationthathadtobestrippedawaytoreachtheessenceofthe</p><p>new science of mechanics. Does a feather fall as rapidly as a stone? All</p><p>experiencewithfallingobjectssaysno.ThestoryofGalileodroppingballsoff</p><p>the tower of Pisa, as a piece ofmyth, is a story about changing intuitions by</p><p>inventinganidealscientificworldwhereregularitiescanbeseparatedfromthe</p><p>disorderofexperience.</p><p>To separate the effectsofgravityon agivenmass from the effectsof air</p><p>resistancewasabrilliantintellectualachievement.ItallowedGalileotoclosein</p><p>on the essence of inertia and momentum. Still, in the real world, pendulums</p><p>eventuallydoexactlywhatAristotle’squaintparadigmpredicted.Theystop.</p><p>In laying thegroundwork for thenextparadigmshift, physicistsbegan to</p><p>faceuptowhatmanybelievedwasadeficiencyintheireducationaboutsimple</p><p>systems like the pendulum. By our century, dissipative processes like friction</p><p>were recognized, and students learned to include them in equations. Students</p><p>alsolearnedthatnonlinearsystemswereusuallyunsolvable,whichwastrue,and</p><p>that they tended to be exceptions—which was not true. Classical mechanics</p><p>described the behavior of whole classes of moving objects, pendulums and</p><p>double pendulums, coiled springs and bent rods, plucked strings and bowed</p><p>strings.Themathematicsappliedtofluidsystemsandtoelectricalsystems.But</p><p>almost no one in the classical era suspected the chaos that could lurk in</p><p>dynamicalsystemsifnonlinearitywasgivenitsdue.</p><p>Aphysicistcouldnot trulyunderstand turbulenceorcomplexityunlesshe</p><p>understoodpendulums—andunderstood them in away thatwas impossible in</p><p>the first half of the twentieth century. As chaos began to unite the study of</p><p>different systems, pendulum dynamics broadened to cover high technologies</p><p>from lasers to superconducting Josephson junctions. Some chemical reactions</p><p>displayed pendulum-like behavior, as did the beating heart. The unexpected</p><p>possibilities extended, one physicist wrote, to “physiological and psychiatric</p><p>medicine,economicforecasting,andperhapstheevolutionofsociety.”</p><p>Consider a playground swing. The swing accelerates on its way down,</p><p>deceleratesonitswayup,allthewhilelosingabitofspeedtofriction.Itgetsa</p><p>regularpush—say,fromsomeclockworkmachine.Allourintuitiontellsusthat,</p><p>nomatterwheretheswingmightstart,themotionwilleventuallysettledownto</p><p>aregularbackandforthpattern,withtheswingcomingtothesameheighteach</p><p>time.Thatcanhappen.Yet,oddasitseems,themotioncanalsoturnerratic,first</p><p>high,thenlow,neversettlingdowntoasteadystateandneverexactlyrepeating</p><p>apatternofswingsthatcamebefore.</p><p>Thesurprising,erraticbehaviorcomesfromanonlineartwistintheflowof</p><p>energyinandoutofthissimpleoscillator.Theswingisdampedanditisdriven:</p><p>dampedbecausefrictionistryingtobringittoahalt,drivenbecauseitisgetting</p><p>aperiodicpush.Evenwhenadamped,drivensystemisatequilibrium,itisnot</p><p>atequilibrium,andtheworldisfullofsuchsystems,beginningwiththeweather,</p><p>dampedbythefrictionofmovingairandwaterandbythedissipationofheatto</p><p>outerspace,anddrivenbytheconstantpushofthesun’senergy.</p><p>But unpredictability was not the reason physicists and mathematicians</p><p>began taking pendulums seriously again in the sixties and seventies.</p><p>Unpredictability was only the attention-grabber. Those studying chaotic</p><p>dynamicsdiscovered that thedisorderlybehaviorof simple systemsactedas a</p><p>creativeprocess. Itgeneratedcomplexity: richlyorganizedpatterns, sometimes</p><p>stable and sometimes unstable, sometimes finite and sometimes infinite, but</p><p>alwayswiththefascinationoflivingthings.Thatwaswhyscientistsplayedwith</p><p>toys.</p><p>Onetoy,soldunderthename“SpaceBalls”or“SpaceTrapeze,”isapairof</p><p>ballsatoppositeendsofarod,sittinglikethecrossbarofaTatopapendulum</p><p>withathird,heavierballatitsfoot.Thelowerballswingsbackandforthwhile</p><p>theupperrodrotatesfreely.Allthreeballshavelittlemagnetsinside,andonce</p><p>set in motion the device keeps going because it has a battery-powered</p><p>electromagnet embedded in the base. The device senses the approach of the</p><p>lowestballandgivesitasmallmagnetickickeachtimeitpasses.Sometimesthe</p><p>apparatus settles into a steady, rhythmic swinging.But other</p><p>times, itsmotion</p><p>seemstoremainchaotic,alwayschangingandendlesslysurprising.</p><p>Another common pendulum toy is no more than a so-called spherical</p><p>pendulum—a pendulum free to swing not just back and forth but in any</p><p>direction.Afewsmallmagnetsareplacedarounditsbase.Themagnetsattract</p><p>themetalbob,andwhenthependulumstops,itwillhavebeencapturedbyone</p><p>ofthem.Theideaistosetthependulumswingingandguesswhichmagnetwill</p><p>win.Evenwith just threemagnetsplaced ina triangle, thependulum’smotion</p><p>cannotbepredicted.ItwillswingbackandforthbetweenAandBforawhile,</p><p>thenswitchtoBandC,andthen,justasitseemstobesettlingonC,jumpback</p><p>to A. Suppose a scientist systematically explores the behavior of this toy by</p><p>makingamap,as follows:Pickastartingpoint;hold thebob thereand letgo;</p><p>colorthepointred,blue,orgreen,dependingonwhichmagnetendsupwiththe</p><p>bob.What will the map look like? It will have regions of solid red, blue, or</p><p>green, as one might expect—regions where the bob will swing reliably to a</p><p>particular magnet. But it can also have regions where the colors are woven</p><p>togetherwithinfinitecomplexity.Adjacent toaredpoint,nomatterhowclose</p><p>onechoosestolook,nomatterhowmuchonemagnifiesthemap,therewillbe</p><p>greenpointsandbluepoints.Forallpracticalpurposes, then, thebob’sdestiny</p><p>willbeimpossibletoguess.</p><p>Traditionally, a dynamicist would believe that to write down a system’s</p><p>equations is to understand the system. How better to capture the essential</p><p>features? For a playground swing or a toy, the equations tie together the</p><p>pendulum’sangle,itsvelocity,itsfriction,andtheforcedrivingit.Butbecause</p><p>of the little bits of nonlinearity in these equations, a dynamicist would find</p><p>himselfhelplesstoanswertheeasiestpracticalquestionsaboutthefutureofthe</p><p>system.Acomputercanaddresstheproblembysimulatingit,rapidlycalculating</p><p>each cycle. But simulation brings its own problem: the tiny imprecision built</p><p>intoeachcalculationrapidlytakesover,becausethisisasystemwithsensitive</p><p>dependenceoninitialconditions.Beforelong,thesignaldisappearsandallthat</p><p>remainsisnoise.</p><p>Orisit?Lorenzhadfoundunpredictability,buthehadalsofoundpattern.</p><p>Others, too, discovered suggestions of structure amid seemingly random</p><p>behavior. The example of the pendulumwas simple enough to disregard, but</p><p>thosewhochosenottodisregarditfoundaprovocativemessage.Insomesense,</p><p>they realized, physics understood perfectly the fundamental mechanisms of</p><p>pendulummotionbutcouldnotextendthatunderstandingtothelongterm.The</p><p>microscopic pieceswere perfectly clear; themacroscopic behavior remained a</p><p>mystery.Thetraditionoflookingatsystemslocally—isolatingthemechanisms</p><p>andthenaddingthemtogether—wasbeginningtobreakdown.Forpendulums,</p><p>for fluids, for electronic circuits, for lasers, knowledge of the fundamental</p><p>equationsnolongerseemedtobetherightkindofknowledgeatall.</p><p>Asthe1960swenton,individualscientistsmadediscoveriesthatparalleled</p><p>Lorenz’s: a French astronomer studying galactic orbits, for example, and a</p><p>Japaneseelectricalengineermodelingelectroniccircuits.Butthefirstdeliberate,</p><p>coordinatedattempt tounderstandhowglobalbehaviormightdiffer fromlocal</p><p>behavior came frommathematicians. Among themwas Stephen Smale of the</p><p>University of California at Berkeley, already famous for unraveling the most</p><p>esoteric problems of many-dimensional topology. A young physicist, making</p><p>small talk, asked what Smale was working on. The answer stunned him:</p><p>“Oscillators.” It was absurd. Oscillators—pendulums, springs, or electrical</p><p>circuits—were the sort of problem that a physicist finished off early in his</p><p>training. They were easy. Why would a great mathematician be studying</p><p>elementaryphysics?NotuntilyearslaterdidtheyoungmanrealizethatSmale</p><p>was looking at nonlinear oscillators, chaotic oscillators, and seeing things that</p><p>physicistshadlearnednottosee.</p><p>SMALEMADEABADCONJECTURE. In themost rigorousmathematical terms,</p><p>heproposedthatpracticallyalldynamicalsystemstendedtosettle,mostofthe</p><p>time,intobehaviorthatwasnottoostrange.Ashesoonlearned,thingswerenot</p><p>sosimple.</p><p>Smalewasamathematicianwhodidnotjustsolveproblemsbutalsobuilt</p><p>programs of problems for others to solve. He parlayed his understanding of</p><p>historyandhisintuitionaboutnatureintoanabilitytoannounce,quietly,thata</p><p>wholeuntried areaof researchwasnowworth amathematician’s time.Like a</p><p>successfulbusinessman,heevaluatedrisksandcoollyplannedhisstrategy,and</p><p>he had a Pied Piper quality.Where Smale led,many followed.His reputation</p><p>wasnotconfinedtomathematics,though.EarlyintheVietnamwar,heandJerry</p><p>Rubin organized “InternationalDays of Protest” and sponsored efforts to stop</p><p>the trains carrying troops through California. In 1966, while the House Un-</p><p>AmericanActivitiesCommitteewastryingtosubpoenahim,hewasheadingfor</p><p>Moscow to attend the International Congress of Mathematicians. There he</p><p>receivedtheFieldsMedal,thehighesthonorofhisprofession.</p><p>Thescene inMoscowthatsummerbecamean indeliblepartof theSmale</p><p>legend. Five thousand agitated and agitating mathematicians had gathered.</p><p>Politicaltensionswerehigh.Petitionswerecirculating.Astheconferencedrew</p><p>towarditsclose,SmalerespondedtoarequestfromaNorthVietnamesereporter</p><p>bygivingapressconferenceonthebroadstepsofMoscowUniversity.Hebegan</p><p>bycondemningtheAmericaninterventioninVietnam,andthen,justashishosts</p><p>begantosmile,addedacondemnationoftheSovietinvasionofHungaryandthe</p><p>absence of political freedom in the SovietUnion.When hewas done, hewas</p><p>quickly hustled away in a car for questioning by Soviet officials. When he</p><p>returnedtoCalifornia,theNationalScienceFoundationcanceledhisgrant.</p><p>Smale’s Fields Medal honored a famous piece of work in topology, a</p><p>branch of mathematics that flourished in the twentieth century and had a</p><p>particular heyday in the fifties. Topology studies the properties that remain</p><p>unchanged when shapes are deformed by twisting or stretching or squeezing.</p><p>Whether a shape is square or round, large or small, is irrelevant in topology,</p><p>becausestretchingcanchangethoseproperties.Topologistsaskwhetherashape</p><p>isconnected,whetherithasholes,whetheritisknotted.Theyimaginesurfaces</p><p>not just in the one–, two–, and three-dimensional universes of Euclid, but in</p><p>spaces ofmanydimensions, impossible to visualize.Topology is geometry on</p><p>rubbersheets. Itconcerns thequalitativerather than thequantitative. Itasks, if</p><p>you don’t know themeasurements, what can you say about overall structure.</p><p>Smale had solved one of the historic, outstanding problems of topology, the</p><p>Poincaréconjecture, forspacesof fivedimensionsandhigher,and insodoing</p><p>establishedasecurestandingasoneofthegreatmenofthefield.Inthe1960s,</p><p>though, he left topology for untried territory. He began studying dynamical</p><p>systems.</p><p>Both subjects, topology and dynamical systems, went back to Henri</p><p>Poincaré,who saw themas two sidesofone coin.Poincaré, at the turnof the</p><p>century,hadbeenthelastgreatmathematiciantobringageometricimagination</p><p>to bear on the laws of motion in the physical world. He was the first to</p><p>understand the possibility of chaos;</p><p>his writings hinted at a sort of</p><p>unpredictability almost as severe as the sort Lorenz discovered. But after</p><p>Poincaré’sdeath,whiletopologyflourished,dynamicalsystemsatrophied.Even</p><p>the name fell into disuse; the subject to which Smale nominally turned was</p><p>differential equations. Differential equations describe the way systems change</p><p>continuouslyovertime.Thetraditionwastolookatsuchthingslocally,meaning</p><p>thatengineersorphysicistswouldconsideronesetofpossibilitiesatatime.Like</p><p>Poincaré,Smalewantedtounderstandthemglobally,meaningthathewantedto</p><p>understandtheentirerealmofpossibilitiesatonce.</p><p>Any set of equations describing a dynamical system—Lorenz’s, for</p><p>example—allowscertainparameterstobesetatthestart.Inthecaseofthermal</p><p>convection,oneparameterconcernstheviscosityofthefluid.Largechangesin</p><p>parameterscanmakelargedifferencesinasystem—forexample,thedifference</p><p>between arriving at a steady state and oscillating periodically. But physicists</p><p>assumedthatverysmallchangeswouldcauseonlyverysmalldifferencesinthe</p><p>numbers,notqualitativechangesinbehavior.</p><p>Linkingtopologyanddynamicalsystemsisthepossibilityofusingashape</p><p>tohelpvisualizethewholerangeofbehaviorsofasystem.Forasimplesystem,</p><p>the shapemight be some kind of curved surface; for a complicated system, a</p><p>manifoldofmanydimensions.A singlepointon sucha surface represents the</p><p>state of a systemat an instant frozen in time.As a systemprogresses through</p><p>time,thepointmoves,tracinganorbitacrossthissurface.Bendingtheshapea</p><p>little corresponds to changing the system’s parameters, making a fluid more</p><p>viscousordrivingapendulumalittleharder.Shapesthatlookroughlythesame</p><p>giveroughlythesamekindsofbehavior.Ifyoucanvisualizetheshape,youcan</p><p>understandthesystem.</p><p>When Smale turned to dynamical systems, topology, like most pure</p><p>mathematics,wascarriedoutwithanexplicitdisdainforreal-worldapplications.</p><p>Topology’s origins had been close to physics, but for mathematicians the</p><p>physical origins were forgotten and shapes were studied for their own sake.</p><p>Smalefullybelievedinthatethos—hewasthepurestofthepure—yethehadan</p><p>idea that the abstract, esoteric development of topology might now have</p><p>somethingtocontributetophysics, justasPoincaréhadintendedat theturnof</p><p>thecentury.</p><p>One of Smale’s first contributions, as it happened, was his faulty</p><p>conjecture.Inphysical terms,hewasproposingalawofnaturesomethinglike</p><p>this:Asystemcanbehaveerratically,buttheerraticbehaviorcannotbestable.</p><p>Stability—“stabilityinthesenseofSmale,”asmathematicianswouldsometimes</p><p>say—was a crucial property. Stable behavior in a system was behavior that</p><p>would not disappear just because some number was changed a tiny bit. Any</p><p>systemcouldhaveboth stable andunstablebehaviorswithin it.The equations</p><p>governingapencilstandingonitspointhaveagoodmathematicalsolutionwith</p><p>thecenterofgravitydirectlyabovethepoint—butyoucannotstandapencilon</p><p>its point because the solution is unstable. The slightest perturbation draws the</p><p>systemawayfromthatsolution.Ontheotherhand,amarblelyingatthebottom</p><p>of a bowl stays there, because if themarble is perturbed slightly it rolls back.</p><p>Physicists assumed that any behavior they could actually observe regularly</p><p>wouldhavetobestable,sinceinrealsystemstinydisturbancesanduncertainties</p><p>areunavoidable.Youneverknowtheparametersexactly. Ifyouwantamodel</p><p>thatwillbebothphysicallyrealisticandrobustinthefaceofsmallperturbations,</p><p>physicistsreasonedthatyoumustsurelywantastablemodel.</p><p>Thebadnewsarrived in themailsoonafterChristmas1959,whenSmale</p><p>was living temporarily in an apartment in Rio de Janeiro with his wife, two</p><p>infant children, and a mass of diapers. His conjecture had defined a class of</p><p>differential equations, all structurally stable. Any chaotic system, he claimed,</p><p>couldbeapproximatedascloselyasyoulikedbyasysteminhisclass.Itwasnot</p><p>so.Aletterfromacolleagueinformedhimthatmanysystemswerenotsowell-</p><p>behavedashehadimagined,anditdescribedacounterexample,asystemwith</p><p>chaosandstability,together.Thissystemwasrobust.Ifyouperturbeditslightly,</p><p>asanynaturalsystemisconstantlyperturbedbynoise,thestrangenesswouldnot</p><p>go away. Robust and strange—Smale studied the letter with a disbelief that</p><p>meltedawayslowly.</p><p>Chaos and instability, concepts only beginning to acquire formal</p><p>definitions, were not the same at all. A chaotic system could be stable if its</p><p>particular brand of irregularity persisted in the face of small disturbances.</p><p>Lorenz’ssystemwasanexample,althoughyearswouldpassbeforeSmaleheard</p><p>aboutLorenz.ThechaosLorenzdiscovered,withallitsunpredictability,wasas</p><p>stableasamarbleinabowl.Youcouldaddnoisetothissystem,jiggleit,stirit</p><p>up, interfere with its motion, and then when everything settled down, the</p><p>transientsdying away like echoes in a canyon, the systemwould return to the</p><p>same peculiar pattern of irregularity as before. It was locally unpredictable,</p><p>globally stable. Real dynamical systems played by amore complicated set of</p><p>rules than anyone had imagined. The example described in the letter from</p><p>Smale’s colleague was another simple system, discovered more than a</p><p>generation earlier and all but forgotten.As it happened, itwas a pendulum in</p><p>disguise:anoscillatingelectroniccircuit.Itwasnonlinearanditwasperiodically</p><p>forced,justlikeachildonaswing.</p><p>Itwas just avacuum tube, really, investigated in the twentiesbyaDutch</p><p>electrical engineer named Balthasar van der Pol. A modern physics student</p><p>wouldexplorethebehaviorofsuchanoscillatorbylookingatthelinetracedon</p><p>the screenofanoscilloscope.VanderPoldidnothaveanoscilloscope, sohe</p><p>hadtomonitorhiscircuitbylisteningtochangingtonesinatelephonehandset.</p><p>Hewaspleasedtodiscoverregularitiesinthebehaviorashechangedthecurrent</p><p>that fed it.The tonewould leap from frequency to frequency as if climbing a</p><p>staircase,leavingonefrequencyandthenlockingsolidlyontothenext.Yetonce</p><p>inawhilevanderPolnotedsomethingstrange.Thebehaviorsoundedirregular,</p><p>inawaythathecouldnotexplain.Underthecircumstanceshewasnotworried.</p><p>“Oftenanirregularnoiseisheardinthetelephonereceiversbeforethefrequency</p><p>jumpstothenextlowervalue,”hewroteinalettertoNature.“However,thisisa</p><p>subsidiaryphenomenon.”Hewasoneofmanyscientistswhogotaglimpseof</p><p>chaosbuthadno language tounderstandit.Forpeople trying tobuildvacuum</p><p>tubes,thefrequency-lockingwasimportant.Butforpeopletryingtounderstand</p><p>thenatureofcomplexity,thetrulyinterestingbehaviorwouldturnouttobethe</p><p>“irregular noise” created by the conflicting pulls of a higher and lower</p><p>frequency.</p><p>Wrongthoughitwas,Smale’sconjectureputhimdirectlyonthetrackofa</p><p>new way of conceiving the full complexity of dynamical systems. Several</p><p>mathematicians had taken another look at the possibilities of the van der Pol</p><p>oscillator, and Smale now took their work into a new realm. His only</p><p>oscilloscope screen was his mind, but it was a mind shaped by his years of</p><p>exploring the topological universe. Smale conceived of the entire range of</p><p>possibilitiesintheoscillator,theentirephasespace,asphysicistscalledit.Any</p><p>state of the</p><p>system at amoment frozen in timewas represented as a point in</p><p>phasespace;all the informationabout itspositionorvelocitywascontainedin</p><p>the coordinates of that point. As the system changed in some way, the point</p><p>would move to a new position in phase space. As the system changed</p><p>continuously,thepointwouldtraceatrajectory.</p><p>For a simple system like a pendulum, the phase space might just be a</p><p>rectangle:thependulum’sangleatagiveninstantwoulddeterminetheeast-west</p><p>positionofapointand thependulum’s speedwoulddetermine thenorth-south</p><p>position. For a pendulum swinging regularly back and forth, the trajectory</p><p>through phase spacewould be a loop, around and around as the system lived</p><p>throughthesamesequenceofpositionsoverandoveragain.</p><p>Smale, instead of looking at any one trajectory, concentrated on the</p><p>behavioroftheentirespaceasthesystemchanged—asmoredrivingenergywas</p><p>added,forexample.Hisintuitionleaptfromthephysicalessenceofthesystem</p><p>toanewkindofgeometricalessence.Histoolsweretopologicaltransformations</p><p>of shapes in phase space—transformations like stretching and squeezing.</p><p>Sometimes these transformations had clear physicalmeaning.Dissipation in a</p><p>system, the loss of energy to friction,meant that the system’s shape in phase</p><p>spacewouldcontractlikeaballoonlosingair—finallyshrinkingtoapointatthe</p><p>momentthesystemcomestoacompletehalt.Torepresentthefullcomplexityof</p><p>thevanderPoloscillator,herealizedthatthephasespacewouldhavetosuffera</p><p>complexnewkindofcombinationoftransformations.Hequicklyturnedhisidea</p><p>aboutvisualizingglobalbehaviorintoanewkindofmodel.Hisinnovation—an</p><p>enduringimageofchaosintheyearsthatfollowed—wasastructurethatbecame</p><p>knownasthehorseshoe.</p><p>MAKINGPORTRAITSINPHASESPACE.Traditionaltimeseries(above)andtrajectoriesinphasespace</p><p>(below)are twowaysofdisplayingthesamedataandgainingapictureofasystem’slongtermbehavior.</p><p>The first system (left) converges on a steady state—a point in phase space. The second repeats itself</p><p>periodically,formingacyclicalorbit.Thethirdrepeatsitselfinamorecomplexwaltzrhythm,acyclewith</p><p>“periodthree.”Thefourthischaotic.</p><p>TomakeasimpleversionofSmale’shorseshoe,youtakearectangleand</p><p>squeezeittopandbottomintoahorizontalbar.Takeoneendofthebarandfold</p><p>it and stretch it around the other, making a C-shape, like a horseshoe. Then</p><p>imagine the horseshoe embedded in a new rectangle and repeat the same</p><p>transformation,shrinkingandfoldingandstretching.</p><p>The processmimics thework of amechanical taffy-maker, with rotating</p><p>arms that stretch the taffy, double it up, stretch it again, and so on until the</p><p>taffy’ssurfacehasbecomevery long,very thin,and intricatelyself-embedded.</p><p>Smale put his horseshoe through an assortment of topological paces, and, the</p><p>mathematics aside, the horseshoe provided a neat visual analogue of the</p><p>sensitive dependence on initial conditions that Lorenz would discover in the</p><p>atmosphereafewyearslater.Picktwonearbypointsintheoriginalspace,and</p><p>you cannot guess where they will end up. They will be driven arbitrarily far</p><p>apartbyallthefoldingandstretching.Afterward,twopointsthathappentolie</p><p>nearbywillhavebegunarbitrarilyfarapart.</p><p>SMALE’SHORSESHOE.This topological transformationprovidedabasis forunderstanding thechaotic</p><p>propertiesofdynamicalsystems.Thebasicsaresimple:Aspaceisstretchedinonedirection,squeezedin</p><p>another,andthenfolded.Whentheprocessisrepeated,itproducesakindofstructuredmixingfamiliarto</p><p>anyonewhohas rolledmany-layeredpastrydough.Apairofpoints thatendupclose togethermayhave</p><p>begunfarapart.</p><p>Originally,Smalehadhoped toexplainalldynamicalsystems in termsof</p><p>stretching and squeezing—with no folding, at least no folding that would</p><p>drastically undermine a system’s stability. But folding turned out to be</p><p>necessary, and folding allowed sharp changes in dynamical behavior. Smale’s</p><p>horseshoe stood as the first of many new geometrical shapes that gave</p><p>mathematiciansandphysicistsanewintuitionaboutthepossibilitiesofmotion.</p><p>In some ways it was too artificial to be useful, still too much a creature of</p><p>mathematicaltopologytoappealtophysicists.Butitservedasastartingpoint.</p><p>As the sixties went on, Smale assembled around him at Berkeley a group of</p><p>young mathematicians who shared his excitement about this new work in</p><p>dynamicalsystems.Anotherdecadewouldpassbeforetheirworkfullyengaged</p><p>theattentionoflesspuresciences,butwhenitdid,physicistswouldrealizethat</p><p>Smalehadturnedawholebranchofmathematicsbacktowardtherealworld.It</p><p>wasagoldenage,theysaid.</p><p>“It’stheparadigmshiftofparadigmshifts,”saidRalphAbraham,aSmale</p><p>colleaguewhobecameaprofessorofmathematicsattheUniversityofCalifornia</p><p>atSantaCruz.</p><p>“WhenIstartedmyprofessionalworkinmathematicsin1960,whichisnot</p><p>solongago,modernmathematicsinitsentirety—initsentirety—wasrejectedby</p><p>physicists, including the most avant-garde mathematical physicists. So</p><p>differentiable dynamics, global analysis, manifolds of mappings, differential</p><p>geometry—everything justayearor twobeyondwhatEinsteinhadused—was</p><p>all rejected.Theromancebetweenmathematiciansandphysicistshadended in</p><p>divorce in the 1930s. These people were no longer speaking. They simply</p><p>despised each other. Mathematical physicists refused their graduate students</p><p>permission to takemathcourses frommathematicians:Takemathematics from</p><p>us.Wewillteachyouwhatyouneedtoknow.Themathematiciansareonsome</p><p>kind of terrible ego trip and they will destroy yourmind. Thatwas 1960. By</p><p>1968 this had completely turned around.” Eventually physicists, astronomers,</p><p>andbiologistsallknewtheyhadtohavethenews.</p><p>AMODESTCOSMICMYSTERY: theGreatRedSpotofJupiter,avast,swirling</p><p>oval,likeagiantstormthatnevermovesandneverrunsdown.Anyonewhosaw</p><p>the pictures beamed across space from Voyager 2 in 1978 recognized the</p><p>familiarlookofturbulenceonahugelyunfamiliarscale.Itwasoneofthesolar</p><p>system’s most venerable landmarks—“the red spot roaring like an anguished</p><p>eye/amida turbulenceofboilingeyebrows,”asJohnUpdikedescribed it.But</p><p>whatwasit?TwentyyearsafterLorenz,Smale,andotherscientistssetinmotion</p><p>anewwayofunderstandingnature’sflows,theother-worldlyweatherofJupiter</p><p>proved to be one of themany problems awaiting the altered sense of nature’s</p><p>possibilitiesthatcamewiththescienceofchaos.</p><p>For threecenturies ithadbeenacaseof themoreyouknow, the lessyou</p><p>know.AstronomersnoticedablemishonthegreatplanetnotlongafterGalileo</p><p>firstpointedhistelescopesatJupiter.RobertHookesawitinthe1600s.Donati</p><p>CretipainteditintheVatican’spicturegallery.Asapieceofcoloration,thespot</p><p>called for little explaining. But telescopes got better, and knowledge bred</p><p>ignorance. The last century produced a steady march of theories, one on the</p><p>heelsofanother.Forexample:</p><p>TheLavaFlowTheory,Scientistsinthelatenineteenthcenturyimagineda</p><p>hugeovallakeofmoltenlavaflowingoutofavolcano.Orperhapsthelavahad</p><p>flowedoutofaholecreatedbyaplanetoidstrikingathinsolidcrust.</p><p>TheNewMoonTheory.AGermanscientistsuggested,bycontrast,thatthe</p><p>spotwasanewmoononthepointofemergingfromtheplanet’ssurface.</p><p>TheEggTheory.An awkward</p><p>new fact: the spotwas seen to be drifting</p><p>slightlyagainsttheplanet’sbackground.Soanotionputforwardin1939viewed</p><p>thespotasamoreorlesssolidbodyfloatingintheatmospherethewayanegg</p><p>floats in water. Variations of this theory—including the notion of a drifting</p><p>bubbleofhydrogenorhelium—remainedcurrentfordecades.</p><p>TheColumn-of-GasTheory.Anothernewfact:eventhoughthespotdrifted,</p><p>somehowitneverdriftedfar.Soscientistsproposed in thesixties that thespot</p><p>wasthetopofarisingcolumnofgas,possiblycomingthroughacrater.</p><p>Then came Voyager. Most astronomers thought the mystery would give</p><p>wayassoonastheycouldlookcloselyenough,andindeed,theVoyagerfly-by</p><p>providedasplendidalbumofnewdata,butthedata,intheend,wasnotenough.</p><p>Thespacecraftpicturesin1978revealedpowerfulwindsandcolorfuleddies.In</p><p>spectaculardetail,astronomerssawthespotitselfasahurricane-likesystemof</p><p>swirling flow, shoving aside the clouds, embedded in zonesof east-westwind</p><p>that made horizontal stripes around the planet. Hurricane was the best</p><p>description anyone could think of, but for several reasons it was inadequate.</p><p>Earthlyhurricanesarepoweredbytheheatreleasedwhenmoisturecondensesto</p><p>rain; no moist processes drive the Red Spot. Hurricanes rotate in a cyclonic</p><p>direction, counterclockwise above the Equator and clockwise below, like all</p><p>earthly storms; the Red Spot’s rotation is anticyclonic. And most important,</p><p>hurricanesdieoutwithindays.</p><p>Also, as astronomers studied theVoyager pictures, they realized that the</p><p>planetwasvirtuallyallfluidinmotion.Theyhadbeenconditionedtolookfora</p><p>solid planet surrounded by a paper-thin atmosphere like earth’s, but if Jupiter</p><p>had a solid core anywhere, it was far from the surface. The planet suddenly</p><p>looked like one big fluid dynamics experiment, and there sat the Red Spot,</p><p>turningsteadilyaroundandaround,thoroughlyunperturbedbythechaosaround</p><p>it.</p><p>Thespotbecameagestalttest.Scientistssawwhattheirintuitionsallowed</p><p>themtosee.Afluiddynamicistwhothoughtofturbulenceasrandomandnoisy</p><p>hadnocontextforunderstandinganislandofstabilityinitsmidst.Voyagerhad</p><p>made the mystery doubly maddening by showing small-scale features of the</p><p>flow, too small to be seen by the most powerful earthbound telescopes. The</p><p>smallscalesdisplayedrapiddisorganization,eddiesappearinganddisappearing</p><p>withinadayorless.Yetthespotwasimmune.Whatkeptitgoing?Whatkeptit</p><p>inplace?</p><p>TheNationalAeronautics and SpaceAdministration keeps its pictures in</p><p>archives, a half-dozen or so around the country. One archive is at Cornell</p><p>University.Nearby,intheearly1980s,PhilipMarcus,ayoungastronomerand</p><p>appliedmathematician,hadanoffice.AfterVoyager,Marcuswasoneofahalf-</p><p>dozenscientistsintheUnitedStatesandBritainwholookedforwaystomodel</p><p>the Red Spot. Freed from the ersatz hurricane theory, they found more</p><p>appropriate analogues elsewhere. The Gulf Stream, for example, winding</p><p>through thewesternAtlanticOcean, twists and branches in subtly reminiscent</p><p>ways.Itdevelopslittlewaves,whichturnintokinks,whichturnintoringsand</p><p>spin off from the main current—forming slow, long-lasting, anticyclonic</p><p>vortices. Another parallel came from a peculiar phenomenon in meteorology</p><p>knownasblocking.Sometimesa systemofhighpressure sitsoffshore, slowly</p><p>turning,forweeksormonths,indefianceoftheusualeast-westflow.Blocking</p><p>disrupted the global forecastingmodels, but it also gave the forecasters some</p><p>hope,sinceitproducedorderlyfeatureswithunusuallongevity.</p><p>Marcus studied those NASA pictures for hours, the gorgeous Hasselblad</p><p>pictures of men on the moon and the pictures of Jupiter’s turbulence. Since</p><p>Newton’s laws apply everywhere, Marcus programmed a computer with a</p><p>systemof fluidequations.TocaptureJovianweathermeantwritingrules fora</p><p>massofdensehydrogenandhelium,resemblinganunlitstar.Theplanetspins</p><p>fast,eachdayflashingbyintenearthhours.ThespinproducesastrongCoriolis</p><p>force, the sidelong force that shoves against a personwalking across amerry-</p><p>go–round,andtheCoriolisforcedrivesthespot.</p><p>Where Lorenz used his tiny model of the earth’s weather to print crude</p><p>lines on rolled paper, Marcus used far greater computer power to assemble</p><p>strikingcolorimages.Firsthemadecontourplots.Hecouldbarelyseewhatwas</p><p>going on. Then he made slides, and then he assembled the images into an</p><p>animated movie. It was a revelation. In brilliant blues, reds, and yellows, a</p><p>checkerboardpatternofrotatingvorticescoalescesintoanovalwithanuncanny</p><p>resemblanceto theGreatRedSpot inNASA’sanimatedfilmof thereal thing.</p><p>“You see this large-scale spot, happy as a clam amid the small-scale chaotic</p><p>flow,andthechaoticflowissoakingupenergylikeasponge,”hesaid.“Yousee</p><p>theselittletinyfilamentarystructuresinabackgroundseaofchaos.”</p><p>The spot is a self-organizing system, created and regulated by the same</p><p>nonlineartwiststhatcreatetheunpredictableturmoilaroundit.Itisstablechaos.</p><p>Asagraduatestudent,Marcushadlearnedstandardphysics,solvinglinear</p><p>equations, performing experiments designed tomatch linear analysis. Itwas a</p><p>shelteredexistence,butafterall,nonlinearequationsdefysolution,sowhywaste</p><p>a graduate student’s time?Gratificationwas programmed into his training.As</p><p>longashekepttheexperimentswithincertainbounds,thelinearapproximations</p><p>would suffice andhewouldbe rewardedwith the expectedanswer.Once in a</p><p>while, inevitably, therealworldwouldintrude,andMarcuswouldseewhathe</p><p>realizedyears laterhadbeenthesignsofchaos.Hewouldstopandsay,“Gee,</p><p>whatabout this little fluffhere.”Andhewouldbe told,“Oh, it’sexperimental</p><p>error,don’tworryaboutit.”</p><p>Butunlikemostphysicists,MarcuseventuallylearnedLorenz’slesson,that</p><p>adeterministicsystemcanproducemuchmore than justperiodicbehavior.He</p><p>knew to look for wild disorder, and he knew that islands of structure could</p><p>appearwithinthedisorder.SohebroughttotheproblemoftheGreatRedSpot</p><p>an understanding that a complex system can give rise to turbulence and</p><p>coherenceat the same time.Hecouldworkwithinanemergingdiscipline that</p><p>was creating its own tradition of using the computer as an experimental tool.</p><p>Andhewaswillingtothinkofhimselfasanewkindofscientist:notprimarily</p><p>an astronomer, not a fluid dynamicist, not an applied mathematician, but a</p><p>specialistinchaos.</p><p>Life’sUps</p><p>andDowns</p><p>The result of a mathematical development should be continuously checked</p><p>against one’s own intuition about what constitutes reasonable biological</p><p>behavior. When such a check reveals disagreement, then the following</p><p>possibilitiesmustbeconsidered:</p><p>1. Amistakehasbeenmadeintheformalmathematicaldevelopment;</p><p>2. The starting assumptions are incorrect and/or constitute a too drastic</p><p>oversimplification;</p><p>3. One’sownintuitionaboutthebiologicalfieldisinadequatelydeveloped;</p><p>4. Apenetratingnewprinciplehasbeendiscovered.</p><p>—HARVEYJ.GOLD,</p><p>MathematicalModeling</p><p>ofBiologicalSystems</p><p>RAVENOUS FISH AND TASTY plankton. Rain forests drippingwith nameless</p><p>reptiles,birdsglidingundercanopiesofleaves,insectsbuzzinglikeelectronsin</p><p>anaccelerator.Frostbeltswherevolesandlemmingsflourishanddiminishwith</p><p>tidy four-year periodicity in the face</p><p>of nature’s bloody combat. The world</p><p>makesamessy laboratory forecologists, a cauldronof fivemillion interacting</p><p>species.Orisitfiftymillion?Ecologistsdonotactuallyknow.</p><p>Mathematically inclined biologists of the twentieth century built a</p><p>discipline,ecology,thatstrippedawaythenoiseandcolorofreallifeandtreated</p><p>populations as dynamical systems. Ecologists used the elementary tools of</p><p>mathematical physics to describe life’s ebbs and flows. Single species</p><p>multiplying in a place where food is limited, several species competing for</p><p>existence,epidemicsspreadingthroughhostpopulations—allcouldbeisolated,</p><p>ifnotinlaboratoriesthencertainlyinthemindsofbiologicaltheorists.</p><p>In theemergenceofchaosasanewscience in the1970s,ecologistswere</p><p>destinedtoplayaspecialrole.Theyusedmathematicalmodels,buttheyalways</p><p>knewthatthemodelswerethinapproximationsoftheseethingrealworld.Ina</p><p>perverse way, their awareness of the limitations allowed them to see the</p><p>importance of some ideas that mathematicians had considered interesting</p><p>oddities.Ifregularequationscouldproduceirregularbehavior—toanecologist,</p><p>that rang certain bells. The equations applied to population biology were</p><p>elementarycounterpartsofthemodelsusedbyphysicistsfortheirpiecesofthe</p><p>universe.Yetthecomplexityoftherealphenomenastudiedinthelifesciences</p><p>outstripped anything to be found in a physicist’s laboratory. Biologists’</p><p>mathematicalmodels tended to be caricatures of reality, as did themodels of</p><p>economists, demographers, psychologists, andurbanplanners,when those soft</p><p>sciences tried tobring rigor to their studyof systemschangingover time.The</p><p>standardsweredifferent.Toaphysicist,asystemofequationslikeLorenz’swas</p><p>so simple it seemed virtually transparent. To a biologist, even Lorenz’s</p><p>equations seemed forbiddingly complex—three-dimensional, continuously</p><p>variable,andanalyticallyintractable.</p><p>Necessitycreatedadifferentstyleofworkingforbiologists.Thematching</p><p>of mathematical descriptions to real systems had to proceed in a different</p><p>direction. A physicist, looking at a particular system (say, two pendulums</p><p>coupledbyaspring),beginsbychoosingtheappropriateequations.Preferably,</p><p>helooksthemupinahandbook;failingthat,hefindstherightequationsfrom</p><p>first principles.Heknowshowpendulumswork, andhe knows about springs.</p><p>Then he solves the equations, if he can.A biologist, by contrast, could never</p><p>simply deduce the proper equations by just thinking about a particular animal</p><p>population.Hewouldhavetogatherdataandtrytofindequationsthatproduced</p><p>similar output. What happens if you put one thousand fish in a pond with a</p><p>limited foodsupply?Whathappens ifyouadd fifty sharks that like toeat two</p><p>fishperday?Whathappenstoavirusthatkillsatacertainrateandspreadsata</p><p>certain rate depending on population density? Scientists idealized these</p><p>questionssothattheycouldapplycrispformulas.</p><p>Oftenitworked.Populationbiologylearnedquiteabitaboutthehistoryof</p><p>life, how predators interact with their prey, how a change in a country’s</p><p>populationdensityaffectsthespreadofdisease.Ifacertainmathematicalmodel</p><p>surged ahead, or reached equilibrium, or died out, ecologists could guess</p><p>somethingaboutthecircumstancesinwhicharealpopulationorepidemicwould</p><p>dothesame.</p><p>Onehelpfulsimplificationwastomodeltheworldintermsofdiscretetime</p><p>intervals, like a watch hand that jerks forward second by second instead of</p><p>gliding continuously. Differential equations describe processes that change</p><p>smoothly over time, but differential equations are hard to compute. Simpler</p><p>equations—“difference equations”—can be used for processes that jump from</p><p>statetostate.Fortunately,manyanimalpopulationsdowhattheydoinneatone-</p><p>yearintervals.Changesyeartoyearareoftenmoreimportantthanchangesona</p><p>continuum.Unlikepeople,manyinsects,forexample,sticktoasinglebreeding</p><p>season,sotheirgenerationsdonotoverlap.Toguessnextspring’sgypsymoth</p><p>populationornextwinter’smeaslesepidemic, anecologistmightonlyneed to</p><p>knowthecorrespondingfigureforthisyear.Ayear-by–yearfacsimileproduces</p><p>nomore thanashadowofasystem’s intricacies,but inmanyrealapplications</p><p>theshadowgivesalltheinformationascientistneeds.</p><p>ThemathematicsofecologyistothemathematicsofSteveSmalewhatthe</p><p>TenCommandmentsaretotheTalmud:agoodsetofworkingrules,butnothing</p><p>toocomplicated.Todescribeapopulationchangingeachyear,abiologistusesa</p><p>formalism that a high school student can follow easily. Suppose next year’s</p><p>populationofgypsymothswilldependentirelyon thisyear’spopulation.You</p><p>could imaginea table listingall thespecificpossibilities—31,000gypsymoths</p><p>this year means 35,000 next year, and so forth. Or you could capture the</p><p>relationshipbetweenall thenumbersfor thisyearandall thenumbersfornext</p><p>yearasarule—afunction.Thepopulation(x)nextyearisafunction(F)ofthe</p><p>population this year: xnext = F(x). Any particular function can be drawn on a</p><p>graph,instantlygivingasenseofitsoverallshape.</p><p>In a simplemodel like thisone, followingapopulation through time is a</p><p>matter of taking a starting figure and applying the same function again and</p><p>again.Toget thepopulationforathirdyear,youjustapplythefunctiontothe</p><p>result for the second year, and so on. The whole history of the population</p><p>becomesavailablethroughthisprocessoffunctionaliteration—afeedbackloop,</p><p>each year’s output serving as the next year’s input. Feedback can get out of</p><p>hand, as it does when sound from a loudspeaker feeds back through a</p><p>microphone and is rapidly amplified to anunbearable shriek.Or feedback can</p><p>producestability,asathermostatdoesinregulatingthetemperatureofahouse:</p><p>anytemperatureaboveafixedpointleadstocooling,andanytemperaturebelow</p><p>itleadstoheating.</p><p>Many different types of functions are possible. A naive approach to</p><p>populationbiologymight suggesta function that increases thepopulationbya</p><p>certainpercentageeachyear.Thatwouldbealinearfunction—xnext=rx—andit</p><p>would be the classic Malthusian scheme for population growth, unlimited by</p><p>foodsupplyormoralrestraint.Theparameterrrepresentstherateofpopulation</p><p>growth.Sayitis1.1;thenifthisyear’spopulationis10,nextyear’sis11.Ifthe</p><p>inputis20,000,theoutputis22,000.Thepopulationriseshigherandhigher,like</p><p>moneyleftforeverinacompound-interestsavingsaccount.</p><p>Ecologists realizedgenerationsago that theywouldhave todobetter.An</p><p>ecologistimaginingrealfishinarealpondhadtofindafunctionthatmatched</p><p>the crude realities of life—for example, the reality of hunger, or competition.</p><p>Whenthefishproliferate,theystarttorunoutoffood.Asmallfishpopulation</p><p>will grow rapidly. An overly large fish population will dwindle. Or take</p><p>Japanese beetles. Every August 1 you go out to your garden and count the</p><p>beetles. For simplicity’s sake, you ignore birds, ignore beetle diseases, and</p><p>consideronlythefixedfoodsupply.Afewbeetleswillmultiply;manywilleat</p><p>thewholegardenandstarvethemselves.</p><p>In the Malthusian scenario of unrestrained growth, the linear growth</p><p>functionrisesforeverupward.Foramorerealisticscenario,anecologistneeds</p><p>an equation with some extra term that restrains growth when the population</p><p>becomes</p><p>large.Themostnaturalfunctiontochoosewouldrisesteeplywhenthe</p><p>populationissmall,reducegrowthtonearzeroatintermediatevalues,andcrash</p><p>downward when the population is very large. By repeating the process, an</p><p>ecologistcanwatchapopulationsettleintoitslongtermbehavior—presumably</p><p>reachingsomesteadystate.Asuccessfulforayintomathematicsforanecologist</p><p>would let him say something like this: Here’s an equation; here’s a variable</p><p>representing reproductive rate; here’s a variable representing the natural death</p><p>rate; here’s avariable representing the additional death rate fromstarvationor</p><p>predation; and look—thepopulationwill rise at this speeduntil it reaches that</p><p>levelofequilibrium.</p><p>Howdoyou find such a function?Manydifferent equationsmightwork,</p><p>and possibly the simplest is a modification of the linear,Malthusian version:</p><p>xnext=rx(1–x).Again,theparameterrrepresentsarateofgrowththatcanbe</p><p>sethigherorlower.Thenewterm,1–x,keepsthegrowthwithinbounds,since</p><p>asxrises,1–xfalls.*Anyonewithacalculatorcouldpicksomestartingvalue,</p><p>pick some growth rate, and carry out the arithmetic to derive next year’s</p><p>population.</p><p>Bythe1950sseveralecologistswerelookingatvariationsofthatparticular</p><p>equation, known as the logistic difference equation. InAustralia, for example,</p><p>W.E.Rickerappliedittorealfisheries.Ecologistsunderstoodthatthegrowth-</p><p>rateparameter r representedan important featureof themodel. In thephysical</p><p>systems from which these equations were borrowed, that parameter</p><p>correspondedtotheamountofheating,ortheamountoffriction,ortheamount</p><p>ofsomeothermessyquantity.Inshort,theamountofnonlinearity.Inapond,it</p><p>might correspond to the fecundityof the fish, thepropensityof thepopulation</p><p>notjusttoboombutalsotobust(“bioticpotential”wasthedignifiedterm).The</p><p>questionwas,howdidthesedifferentparametersaffecttheultimatedestinyofa</p><p>changingpopulation?Theobviousanswer is thata lowerparameterwillcause</p><p>thisidealizedpopulationtoendupatalowerlevel.Ahigherparameterwilllead</p><p>toahighersteadystate.This turnsout tobecorrect formanyparameters—but</p><p>not all.Occasionally, researchers likeRicker surely triedparameters thatwere</p><p>evenhigher,andwhentheydid,theymusthaveseenchaos.</p><p>Apopulationreachesequilibriumafterrising,overshooting,andfallingback.</p><p>Oddly, the flow of numbers begins to misbehave, quite a nuisance for</p><p>anyone calculatingwith a hand crank. The numbers still do not growwithout</p><p>limit,ofcourse,but theydonotconverge toasteadylevel,either.Apparently,</p><p>though,noneoftheseearlyecologistshadtheinclinationorthestrengthtokeep</p><p>churning out numbers that refused to settle down. Anyway, if the population</p><p>keptbouncingbackandforth,ecologistsassumedthatitwasoscillatingaround</p><p>someunderlyingequilibrium.Theequilibriumwastheimportantthing.Itdidnot</p><p>occurtotheecologiststhattheremightbenoequilibrium.</p><p>Referencebooksandtextbooksthatdealtwiththelogisticequationandits</p><p>more complicated cousins generally did not even acknowledge that chaotic</p><p>behaviorcouldbeexpected.J.MaynardSmith,intheclassic1968Mathematical</p><p>Ideas in Biology, gave a standard sense of the possibilities: populations often</p><p>remain approximately constant or else fluctuate “with a rather regular</p><p>periodicity”aroundapresumedequilibriumpoint.Itwasn’tthathewassonaive</p><p>as to imagine that real populations could never behave erratically. He simply</p><p>assumed that erratic behavior hadnothing to dowith the sort ofmathematical</p><p>modelshewasdescribing. In anycase,biologistshad tokeep thesemodels at</p><p>arm’slength.Ifthemodelsstartedtobetraytheirmakers’knowledgeofthereal</p><p>population’s behavior, some missing feature could always explain the</p><p>discrepancy: the distribution of ages in the population, some consideration of</p><p>territoryorgeography,orthecomplicationofhavingtocounttwosexes.</p><p>Most important, in the back of ecologists’ minds was always the</p><p>assumptionthatanerraticstringofnumbersprobablymeant that thecalculator</p><p>wasactingup,orjustlackedaccuracy.Thestablesolutionsweretheinteresting</p><p>ones.Orderwasitsownreward.Thisbusinessoffindingappropriateequations</p><p>andworking out the computationwas hard, after all.No onewanted towaste</p><p>timeonalineofworkthatwasgoingawry,producingnostability.Andnogood</p><p>ecologist ever forgot that his equationswere vastly oversimplified versions of</p><p>the real phenomena. The whole point of oversimplifying was to model</p><p>regularity.Whygotoallthattroublejusttoseechaos?</p><p>LATER, PEOPLE WOULD SAY that James Yorke had discovered Lorenz and</p><p>giventhescienceofchaositsname.Thesecondpartwasactuallytrue.</p><p>Yorkewasamathematicianwholikedtothinkofhimselfasaphilosopher,</p><p>though this was professionally dangerous to admit. Hewas brilliant and soft-</p><p>spoken,amildlydisheveledadmirerofthemildlydisheveledSteveSmale.Like</p><p>everyone else, he found Smale hard to fathom. But unlike most people, he</p><p>understoodwhySmalewashardtofathom.Whenhewasjusttwenty-twoyears</p><p>old, Yorke joined an interdisciplinary institute at the University of Maryland</p><p>calledtheInstituteforPhysicalScienceandTechnology,whichhelaterheaded.</p><p>Hewasthekindofmathematicianwhofeltcompelledtoputhisideasofreality</p><p>tosomeuse.Heproducedareportonhowgonorrheaspreadsthatpersuadedthe</p><p>federalgovernmenttoalteritsnationalstrategiesforcontrollingthedisease.He</p><p>gaveofficialtestimonytotheStateofMarylandduringthe1970sgasolinecrisis,</p><p>arguing correctly (but unpersuasively) that the even-odd system of limiting</p><p>gasoline sales would only make lines longer. In the era of antiwar</p><p>demonstrations, when the government released a spy-plane photograph</p><p>purporting to show sparse crowds around the Washington Monument at the</p><p>height of a rally, he analyzed the monument’s shadow to prove that the</p><p>photograph had actually been taken a half-hour later, when the rally was</p><p>breakingup.</p><p>At the institute,Yorke enjoyed an unusual freedom towork on problems</p><p>outside traditional domains, and he enjoyed frequent contactwith experts in a</p><p>wide range of disciplines.One of these experts, a fluid dynamicist, had come</p><p>acrossLorenz’s1963paper“DeterministicNonperiodicFlow”in1972andhad</p><p>fallen in lovewith it, handing out copies to anyonewhowould take one. He</p><p>handedonetoYorke.</p><p>Lorenz’s paper was a piece of magic that Yorke had been looking for</p><p>withoutevenknowingit.Itwasamathematicalshock,tobeginwith—achaotic</p><p>systemthatviolatedSmale’soriginaloptimisticclassificationscheme.Butitwas</p><p>not just mathematics; it was a vivid physical model, a picture of a fluid in</p><p>motion,andYorkeknewinstantlythatitwasathinghewantedphysiciststosee.</p><p>Smalehadsteeredmathematicsinthedirectionofsuchphysicalproblems,but,</p><p>as Yorke well understood, the language of mathematics remained a serious</p><p>barrier to communication. If only the academic world had room for hybrid</p><p>mathematician/physicists—but it did not. Even though Smale’s work on</p><p>dynamical systems had begun to close the gap, mathematicians continued to</p><p>speakonelanguage,physicistsanother.AsthephysicistMurrayGell-Mannonce</p><p>remarked: “Faculty members are familiar with a certain kind of person who</p><p>lookstothemathematicianslikeagoodphysicistandlooksto</p><p>thephysicistslike</p><p>a good mathematician. Very properly, they do not want that kind of person</p><p>around.” The standards of the two professionswere different.Mathematicians</p><p>proved theorems by ratiocination; physicists’ proofs used heavier equipment.</p><p>The objects that made up their worlds were different. Their examples were</p><p>different.</p><p>Smalecouldbehappywithanexamplelikethis:takeanumber,afraction</p><p>betweenzeroandone,anddoubleit.Thendroptheintegerpart,theparttothe</p><p>left of the decimal point. Then repeat the process. Since most numbers are</p><p>irrationalandunpredictableintheirfinedetail,theprocesswilljustproducean</p><p>unpredictable sequenceofnumbers.Aphysicistwould seenothing therebut a</p><p>tritemathematicaloddity,utterlymeaningless,toosimpleandtooabstracttobe</p><p>of use. Smale, though, knew intuitively that this mathematical trick would</p><p>appearintheessenceofmanyphysicalsystems.</p><p>Toaphysicist,a legitimateexamplewasadifferentialequationthatcould</p><p>bewrittendowninsimpleform.WhenYorkesawLorenz’spaper,eventhough</p><p>it was buried in a meteorology journal, he knew it was an example that</p><p>physicistswould understand.He gave a copy toSmale,with his address label</p><p>pasted on so that Smale would return it. Smale was amazed to see that this</p><p>meteorologist—ten years earlier—had discovered a kind of chaos that Smale</p><p>himself had once considered mathematically impossible. He made many</p><p>photocopiesof“DeterministicNonperiodicFlow,”andthusarosethelegendthat</p><p>Yorke had discovered Lorenz. Every copy of the paper that ever appeared in</p><p>BerkeleyhadYorke’saddresslabelonit.</p><p>Yorke felt that physicists had learned not to see chaos. In daily life, the</p><p>Lorenzian quality of sensitive dependence on initial conditions lurks</p><p>everywhere. A man leaves the house in the morning thirty seconds late, a</p><p>flowerpotmisses his head by a fewmillimeters, and then he is run over by a</p><p>truck.Or, less dramatically, hemisses a bus that runs every tenminutes—his</p><p>connection to a train that runs every hour. Small perturbations in one’s daily</p><p>trajectorycanhavelargeconsequences.Abatterfacingapitchedballknowsthat</p><p>approximately the same swing will not give approximately the same result,</p><p>baseballbeingagameofinches.Science,though—sciencewasdifferent.</p><p>Pedagogically speaking, a good share of physics andmathematicswas—</p><p>andis—writingdifferentialequationsonablackboardandshowingstudentshow</p><p>tosolvethem.Differentialequationsrepresentrealityasacontinuum,changing</p><p>smoothlyfromplacetoplaceandfromtimetotime,notbrokenindiscretegrid</p><p>points or time steps. As every science student knows, solving differential</p><p>equations is hard. But in two and a half centuries, scientists have built up a</p><p>tremendous body of knowledge about them: handbooks and catalogues of</p><p>differentialequations,alongwithvariousmethodsforsolvingthem,or“finding</p><p>aclosed-formintegral,”asascientistwillsay.It isnoexaggerationtosaythat</p><p>the vast business of calculusmade possiblemost of the practical triumphs of</p><p>post-medieval science; nor to say that it stands as one of the most ingenious</p><p>creationsofhumans trying tomodel thechangeableworldaroundthem.Soby</p><p>the time a scientist masters this way of thinking about nature, becoming</p><p>comfortablewiththetheoryandthehard,hardpractice,heislikelytohavelost</p><p>sightofonefact.Mostdifferentialequationscannotbesolvedatall.</p><p>“If you could write down the solution to a differential equation,” Yorke</p><p>said,“thennecessarilyit’snotchaotic,becausetowriteitdown,youmustfind</p><p>regularinvariants, thingsthatareconserved,likeangularmomentum.Youfind</p><p>enough of these things, and that lets you write down a solution. But this is</p><p>exactlythewaytoeliminatethepossibilityofchaos.”</p><p>The solvable systems are the ones shown in textbooks. They behave.</p><p>Confrontedwith a nonlinear system, scientistswould have to substitute linear</p><p>approximations or find some other uncertain backdoor approach. Textbooks</p><p>showed students only the rare nonlinear systems thatwould giveway to such</p><p>techniques. They did not display sensitive dependence on initial conditions.</p><p>Nonlinearsystemswithrealchaoswererarelytaughtandrarelylearned.When</p><p>people stumbled across such things—andpeopledid—all their training argued</p><p>fordismissingthemasaberrations.Onlyafewwereable torememberthat the</p><p>solvable, orderly, linear systems were the aberrations. Only a few, that is,</p><p>understoodhownonlinearnatureisinitssoul.EnricoFermionceexclaimed,“It</p><p>doesnot say in theBible that all lawsofnature are expressible linearly!”The</p><p>mathematician Stanislaw Ulam remarked that to call the study of chaos</p><p>“nonlinear science” was like calling zoology “the study of non elephant</p><p>animals.”</p><p>Yorke understood. “The firstmessage is that there is disorder. Physicists</p><p>and mathematicians want to discover regularities. People say, what use is</p><p>disorder.Butpeoplehavetoknowaboutdisorderiftheyaregoingtodealwith</p><p>it.Theautomechanicwhodoesn’tknowabout sludge invalves isnot agood</p><p>mechanic.”Scientistsandnonscientistsalike,Yorkebelieved,caneasilymislead</p><p>themselves about complexity if they are not properly attuned to it. Why do</p><p>investors insist on the existence of cycles in gold and silver prices? Because</p><p>periodicity is themost complicated orderly behavior they can imagine.When</p><p>theyseeacomplicatedpatternofprices,theylookforsomeperiodicitywrapped</p><p>inalittlerandomnoise.Andscientificexperimenters,inphysicsorchemistryor</p><p>biology, are no different. “In the past, people have seen chaotic behavior in</p><p>innumerable circumstances,” Yorke said. “They’re running a physical</p><p>experiment,andtheexperimentbehavesinanerraticmanner.Theytrytofixit</p><p>ortheygiveup.Theyexplaintheerraticbehaviorbysayingthere’snoise,orjust</p><p>thattheexperimentisbad.”</p><p>YorkedecidedtherewasamessageintheworkofLorenzandSmalethat</p><p>physicistswerenothearing.Sohewroteapaperforthemostbroadlydistributed</p><p>journal he thought he could publish in, the AmericanMathematicalMonthly.</p><p>(As amathematician, he foundhimself helpless to phrase ideas in a form that</p><p>physicsjournalswouldfindacceptable; itwasonlyyearslater thathehitupon</p><p>the trick of collaboratingwith physicists.)Yorke’s paperwas important on its</p><p>merits, but in the end its most influential feature was its mysterious and</p><p>mischievoustitle:“PeriodThreeImpliesChaos.”Hiscolleaguesadvisedhimto</p><p>choosesomethingmoresober,butYorkestuckwithawordthatcametostand</p><p>for thewholegrowingbusinessofdeterministicdisorder.Healso talked tohis</p><p>friendRobertMay,abiologist.</p><p>MAYCAMETOBIOLOGYthroughthebackdoor,asithappened.Hestartedas</p><p>a theoretical physicist in his native Sydney, Australia, the son of a brilliant</p><p>barrister, and he did postdoctoralwork in appliedmathematics atHarvard. In</p><p>1971, he went for a year to the Institute for Advanced Study in Princeton;</p><p>insteadofdoingtheworkhewassupposedtobedoing,hefoundhimselfdrifting</p><p>overtoPrincetonUniversitytotalktothebiologiststhere.</p><p>Evennow,biologiststendnottohavemuchmathematicsbeyondcalculus.</p><p>People who like mathematics and have an aptitude for it tend more toward</p><p>mathematics or physics than the life sciences. May was an exception. His</p><p>interestsatfirst</p><p>tendedtowardtheabstractproblemsofstabilityandcomplexity,</p><p>mathematicalexplanationsofwhatenablescompetitors tocoexist.Buthesoon</p><p>began to focuson the simplest ecological questionsof how single populations</p><p>behaveover time.The inevitablysimplemodelsseemedlessofacompromise.</p><p>By the time he joined the Princeton faculty for good—eventually he would</p><p>become the university’s dean for research—he had already spent many hours</p><p>studying a version of the logistic difference equation, using mathematical</p><p>analysisandalsoaprimitivehandcalculator.</p><p>Once, in fact, on a corridor blackboard back in Sydney, he wrote the</p><p>equationoutasaproblemforthegraduatestudents.Itwasstartingtoannoyhim.</p><p>“What the Christ happens when lambda gets bigger than the point of</p><p>accumulation?”Whathappened,thatis,whenapopulation’srateofgrowth,its</p><p>tendency toward boom and bust, passed a critical point. By trying different</p><p>valuesofthisnonlinearparameter,Mayfoundthathecoulddramaticallychange</p><p>the system’s character. Raising the parameter meant raising the degree of</p><p>nonlinearity,andthatchangednotjustthequantityoftheoutcome,butalsoits</p><p>quality.Itaffectednotjustthefinalpopulationatequilibrium,butalsowhether</p><p>thepopulationwouldreachequilibriumatall.</p><p>Whentheparameterwaslow,May’ssimplemodelsettledonasteadystate.</p><p>When the parameter was high, the steady state would break apart, and the</p><p>populationwouldoscillatebetweentwoalternatingvalues.Whentheparameter</p><p>was very high, the system—the very same system—seemed to behave</p><p>unpredictably. Why? What exactly happened at the boundaries between the</p><p>differentkindsofbehavior?Maycouldn’tfigureitout.(Norcouldthegraduate</p><p>students.)</p><p>May carried out a program of intense numerical exploration into the</p><p>behaviorof thissimplestofequations.Hisprogramwasanalogous toSmale’s:</p><p>hewastryingtounderstandthisonesimpleequationallatonce,notlocallybut</p><p>globally. The equation was far simpler than anything Smale had studied. It</p><p>seemed incredible that its possibilities for creating order and disorder had not</p><p>beenexhaustedlongsince.Buttheyhadnot.Indeed,May’sprogramwasjusta</p><p>beginning.Heinvestigatedhundredsofdifferentvaluesoftheparameter,setting</p><p>the feedback loop in motion and watching to see where—and whether—the</p><p>stringofnumberswouldsettledowntoafixedpoint.Hefocusedmoreandmore</p><p>closelyonthecriticalboundarybetweensteadinessandoscillation.Itwasasif</p><p>hehadhisownfishpond,wherehecouldwieldfinemasteryover the“boom-</p><p>and–bustiness”ofthefish.Stillusingthelogisticequation,xnext=rx(1–x),May</p><p>increasedtheparameterasslowlyashecould.Iftheparameterwas2.7,thenthe</p><p>population would be .6292. As the parameter rose, the final population rose</p><p>slightly,too,makingalinethatroseslightlyasitmovedfromlefttorightonthe</p><p>graph.</p><p>Suddenly,though,astheparameterpassed3,thelinebrokeintwo.May’s</p><p>imaginaryfishpopulationrefusedtosettledowntoasinglevalue,butoscillated</p><p>betweentwopointsinalternatingyears.Startingatalownumber,thepopulation</p><p>would rise and then fluctuate until it was steadily flipping back and forth.</p><p>Turninguptheknobabitmore—raisingtheparameterabitmore—wouldsplit</p><p>the oscillation again, producing a string of numbers that settled down to four</p><p>differentvalues,eachreturningeveryfourthyear.*Nowthepopulationroseand</p><p>fell on a regular four-year schedule. The cycle had doubled again—first from</p><p>yearly to every twoyears, and now to four.Once again, the resulting cyclical</p><p>behaviorwasstable;differentstartingvaluesforthepopulationwouldconverge</p><p>onthesamefour-yearcycle.</p><p>PERIOD-DOUBLINGS AND CHAOS. Instead of using individual diagrams to show the behavior of</p><p>populationswithdifferentdegreesoffertility,RobertMayandotherscientistsuseda“bifurcationdiagram”</p><p>toassemblealltheinformationintoasinglepicture.</p><p>Thediagramshowshowchanges inoneparameter—in thiscase,awildlifepopulation’s“boom-and-</p><p>bustiness”—would change the ultimate behavior of this simple system. Values of the parameter are</p><p>representedfromlefttoright;thefinalpopulationisplottedontheverticalaxis.Inasense,turningupthe</p><p>parametervaluemeansdrivingasystemharder,increasingitsnonlinearity.</p><p>Wheretheparameterislow(left), thepopulationbecomesextinct.Astheparameterrises(center),so</p><p>doestheequilibriumlevelofthepopulation.Then,astheparameterrisesfurther,theequilibriumsplitsin</p><p>two,justasturninguptheheatinaconvectingfluidcausesaninstabilitytosetin;thepopulationbeginsto</p><p>alternate between two different levels. The splittings, or bifurcations, come faster and faster. Then the</p><p>systemturnschaotic(right),andthepopulationvisitsinfinitelymanydifferentvalues.</p><p>AsLorenzhaddiscoveredadecadebefore,theonlywaytomakesenseof</p><p>such numbers and preserve one’s eyesight is to create a graph. May drew a</p><p>sketchyoutlinemeanttosumupalltheknowledgeaboutthebehaviorofsucha</p><p>system at different parameters. The level of the parameter was plotted</p><p>horizontally, increasing from left to right. The population was represented</p><p>vertically. For each parameter, May plotted a point representing the final</p><p>outcome,afterthesystemreachedequilibrium.Attheleft,wheretheparameter</p><p>waslow,thisoutcomewouldjustbeapoint,sodifferentparametersproduceda</p><p>linerisingslightlyfromlefttoright.Whentheparameterpassedthefirstcritical</p><p>point, May would have to plot two populations: the line would split in two,</p><p>making a sideways Y or a pitchfork. This split corresponded to a population</p><p>goingfromaone-yearcycletoatwo-yearcycle.</p><p>As the parameter rose further, the number of points doubled again, then</p><p>again, then again. It was dumbfounding—such complex behavior, and yet so</p><p>tantalizinglyregular.“Thesnakeinthemathematicalgrass”washowMayputit.</p><p>Thedoublingsthemselveswerebifurcations,andeachbifurcationmeantthatthe</p><p>pattern of repetitionwas breaking down a step further. A population that had</p><p>been stable would alternate between different levels every other year. A</p><p>populationthathadbeenalternatingonatwo-yearcyclewouldnowvaryonthe</p><p>thirdandfourthyears,thusswitchingtoperiodfour.</p><p>These bifurcations would come faster and faster—4, 8, 16, 32…—and</p><p>suddenly break off. Beyond a certain point, the “point of accumulation,”</p><p>periodicitygiveswaytochaos,fluctuationsthatneversettledownatall.Whole</p><p>regionsofthegrapharecompletelyblackedin.Ifyouwerefollowingananimal</p><p>populationgovernedbythissimplestofnonlinearequations,youwouldthinkthe</p><p>changes fromyear toyearwere absolutely random, as thoughblownaboutby</p><p>environmental noise. Yet in the middle of this complexity, stable cycles</p><p>suddenly return. Even though the parameter is rising, meaning that the</p><p>nonlinearity is driving the system harder and harder, a windowwill suddenly</p><p>appearwitharegularperiod:anoddperiod,like3or7.Thepatternofchanging</p><p>population repeats itself on a three-year or seven-year cycle.Then the period-</p><p>doubling bifurcations begin all over at a faster rate, rapidly passing through</p><p>cyclesof3,6,12…or7,14,28…,andthenbreakingoffonceagaintorenewed</p><p>chaos.</p><p>Atfirst,Maycouldnotseethiswholepicture.Butthefragmentshecould</p><p>calculatewereunsettlingenough.Inareal-worldsystem,anobserverwouldsee</p><p>understoodbyanyskilled</p><p>physicistafterappropriatecontemplationandcalculation.Notobviousdescribed</p><p>work that commanded respect andNobelprizes.For thehardestproblems, the</p><p>problemsthatwouldnotgivewaywithoutlonglooksintotheuniverse’sbowels,</p><p>physicistsreservedwordslikedeep.In1974,thoughfewofhiscolleaguesknew</p><p>it,Feigenbaumwasworkingonaproblemthatwasdeep:chaos.</p><p>WHERECHAOSBEGINS,classicalsciencestops.Foraslongastheworldhas</p><p>had physicists inquiring into the laws of nature, it has suffered a special</p><p>ignorance about disorder in the atmosphere, in the turbulent sea, in the</p><p>fluctuationsofwildlifepopulations,intheoscillationsoftheheartandthebrain.</p><p>Theirregularsideofnature,thediscontinuousanderraticside—thesehavebeen</p><p>puzzlestoscience,orworse,monstrosities.</p><p>Butinthe1970safewscientistsintheUnitedStatesandEuropebeganto</p><p>findaway throughdisorder.Theyweremathematicians,physicists,biologists,</p><p>chemists, all seeking connections between different kinds of irregularity.</p><p>Physiologists foundasurprisingorder in thechaos thatdevelops in thehuman</p><p>heart,theprimecauseofsudden,unexplaineddeath.Ecologistsexploredtherise</p><p>andfallofgypsymothpopulations.Economistsdugoutoldstockpricedataand</p><p>tried a new kind of analysis. The insights that emerged led directly into the</p><p>natural world—the shapes of clouds, the paths of lightning, the microscopic</p><p>intertwiningofbloodvessels,thegalacticclusteringofstars.</p><p>WhenMitchellFeigenbaumbeganthinkingaboutchaosatLosAlamos,he</p><p>wasoneofahandfulofscatteredscientists,mostlyunknowntooneanother.A</p><p>mathematician inBerkeley,California, had formed a small group dedicated to</p><p>creating a new study of “dynamical systems.” A population biologist at</p><p>PrincetonUniversitywasabouttopublishanimpassionedpleathatallscientists</p><p>shouldlookatthesurprisinglycomplexbehaviorlurkinginsomesimplemodels.</p><p>AgeometerworkingforIBMwaslookingforanewwordtodescribeafamily</p><p>ofshapes—jagged,tangled,splintered,twisted,fractured—thatheconsideredan</p><p>organizingprinciple innature.AFrenchmathematicalphysicisthad justmade</p><p>thedisputatiousclaimthatturbulenceinfluidsmighthavesomethingtodowith</p><p>abizarre,infinitelytangledabstractionthathecalledastrangeattractor.</p><p>A decade later, chaos has become a shorthand name for a fast-growing</p><p>movement that is reshaping the fabric of the scientific establishment. Chaos</p><p>conferences and chaos journals abound. Government program managers in</p><p>chargeofresearchmoneyforthemilitary,theCentralIntelligenceAgency,and</p><p>theDepartmentofEnergyhaveputevergreatersumsintochaosresearchandset</p><p>upspecialbureaucraciestohandlethefinancing.Ateverymajoruniversityand</p><p>everymajorcorporateresearchcenter,sometheoristsallythemselvesfirstwith</p><p>chaosandonlysecondwiththeirnominalspecialties.AtLosAlamos,aCenter</p><p>forNonlinearStudieswasestablished tocoordinateworkonchaosand related</p><p>problems;similar institutionshaveappearedonuniversitycampusesacross the</p><p>country.</p><p>Chaoshascreatedspecialtechniquesofusingcomputersandspecialkinds</p><p>of graphic images, pictures that capture a fantastic and delicate structure</p><p>underlying complexity. The new science has spawned its own language, an</p><p>elegant shop talkof fractals andbifurcations, intermittencies andperiodicities,</p><p>folded-towel diffeomorphisms and smooth noodle maps. These are the new</p><p>elementsofmotion,justas,intraditionalphysics,quarksandgluonsarethenew</p><p>elementsofmatter.Tosomephysicistschaosisascienceofprocessratherthan</p><p>state,ofbecomingratherthanbeing.</p><p>Now that science is looking, chaos seems to be everywhere. A rising</p><p>columnofcigarettesmokebreaksintowildswirls.Aflagsnapsbackandforth</p><p>in the wind. A dripping faucet goes from a steady pattern to a random one.</p><p>Chaos appears in the behavior of the weather, the behavior of an airplane in</p><p>flight, the behavior of cars clustering on an expressway, the behavior of oil</p><p>flowinginundergroundpipes.Nomatterwhatthemedium,thebehaviorobeys</p><p>thesamenewlydiscoveredlaws.Thatrealizationhasbeguntochangetheway</p><p>businessexecutivesmakedecisionsabout insurance, thewayastronomers look</p><p>atthesolarsystem,thewaypoliticaltheoriststalkaboutthestressesleadingto</p><p>armedconflict.</p><p>Chaosbreaksacrossthelinesthatseparatescientificdisciplines.Becauseit</p><p>isascienceoftheglobalnatureofsystems,ithasbroughttogetherthinkersfrom</p><p>fields thathadbeenwidely separated.“Fifteenyearsago, sciencewasheading</p><p>foracrisisof increasingspecialization,”aNavyofficial inchargeofscientific</p><p>financingremarkedtoanaudienceofmathematicians,biologists,physicists,and</p><p>medical doctors. “Dramatically, that specialization has reversed because of</p><p>chaos.”Chaosposesproblemsthatdefyacceptedwaysofworkinginscience.It</p><p>makesstrongclaimsabouttheuniversalbehaviorofcomplexity.Thefirstchaos</p><p>theorists, the scientists who set the discipline in motion, shared certain</p><p>sensibilities. They had an eye for pattern, especially pattern that appeared on</p><p>different scales at the same time. They had a taste for randomness and</p><p>complexity, for jagged edges and sudden leaps. Believers in chaos—and they</p><p>sometimes call themselves believers, or converts, or evangelists—speculate</p><p>aboutdeterminismandfreewill,aboutevolution,aboutthenatureofconscious</p><p>intelligence. They feel that they are turning back a trend in science toward</p><p>reductionism,theanalysisofsystemsintermsoftheirconstituentparts:quarks,</p><p>chromosomes,orneurons.Theybelievethattheyarelookingforthewhole.</p><p>Themostpassionateadvocatesof thenewsciencegosofaras tosaythat</p><p>twentieth-century science will be remembered for just three things: relativity,</p><p>quantummechanics,andchaos.Chaos,theycontend,hasbecomethecentury’s</p><p>third great revolution in the physical sciences. Like the first two revolutions,</p><p>chaos cuts away at the tenets of Newton’s physics. As one physicist put it:</p><p>“Relativity eliminated the Newtonian illusion of absolute space and time;</p><p>quantumtheoryeliminatedtheNewtoniandreamofacontrollablemeasurement</p><p>process; and chaos eliminates the Laplacian fantasy of deterministic</p><p>predictability.”Of the three, therevolution inchaosapplies to theuniversewe</p><p>seeandtouch,toobjectsathumanscale.Everydayexperienceandrealpictures</p><p>oftheworldhavebecomelegitimatetargetsforinquiry.Therehaslongbeena</p><p>feeling, not always expressed openly, that theoretical physics has strayed far</p><p>from human intuition about the world.Whether this will prove to be fruitful</p><p>heresy or just plain heresy, no one knows. But some of those who thought</p><p>physicsmightbeworkingitswayintoacornernowlooktochaosasawayout.</p><p>Within physics itself, the study of chaos emerged from a backwater. The</p><p>mainstream for most of the twentieth century has been particle physics,</p><p>exploring the building blocks ofmatter at higher and higher energies, smaller</p><p>andsmallerscales,shorterandshortertimes.Outofparticlephysicshavecome</p><p>theories about the fundamental forces of nature and about the origin of the</p><p>universe.Yetsomeyoungphysicistshavegrowndissatisfiedwiththedirection</p><p>ofthemostprestigiousofsciences.Progresshasbeguntoseemslow,thenaming</p><p>ofnewparticlesfutile,thebodyoftheorycluttered.Withthecomingofchaos,</p><p>youngerscientistsbelievedtheywereseeingthebeginnings</p><p>just the vertical slice corresponding to one parameter at a time.Hewould see</p><p>onlyonekindofbehavior—possiblyasteadystate,possiblyaseven-yearcycle,</p><p>possiblyapparentrandomness.Hewouldhavenowayofknowingthatthesame</p><p>system,withsomeslightchangeinsomeparameter,coulddisplaypatternsofa</p><p>completelydifferentkind.</p><p>JamesYorkeanalyzedthisbehaviorwithmathematicalrigorinhis“Period</p><p>ThreeImpliesChaos”paper.Heprovedthatinanyone-dimensionalsystem,ifa</p><p>regularcycleofperiodthreeeverappears,thenthesamesystemwillalsodisplay</p><p>regularcyclesofeveryother length,aswellascompletelychaoticcycles.This</p><p>was the discovery that came as an “electric shock” to physicists likeFreeman</p><p>Dyson.Itwassocontrarytointuition.Youwouldthinkitwouldbetrivialtoset</p><p>up a system thatwould repeat itself in a period-three oscillationwithout ever</p><p>producingchaos.Yorkeshowedthatitwasimpossible.</p><p>Startlingthoughitwas,Yorkebelievedthatthepublicrelationsvalueofhis</p><p>paperoutweighedthemathematicalsubstance.Thatwaspartlytrue.Afewyears</p><p>later,attendinganinternationalconferenceinEastBerlin,hetooksometimeout</p><p>for sightseeing and went for a boat ride on the Spree. Suddenly he was</p><p>approachedby aRussian tryingurgently to communicate something.With the</p><p>helpofaPolishfriend,YorkefinallyunderstoodthattheRussianwasclaiming</p><p>tohaveprovedthesameresult.TheRussianrefusedtogivedetails,sayingonly</p><p>thathewouldsendhispaper.Fourmonthslateritarrived.A.N.Sarkovskiihad</p><p>indeedbeentherefirst,inapapertitled“CoexistenceofCyclesofaContinuous</p><p>Map of a Line into Itself.” But Yorke had offeredmore than amathematical</p><p>result.Hehadsentamessagetophysicists:Chaosisubiquitous;itisstable;itis</p><p>structured.Healsogavereasontobelievethatcomplicatedsystems,traditionally</p><p>modeledbyhardcontinuousdifferentialequations,couldbeunderstoodinterms</p><p>ofeasydiscretemaps.</p><p>WINDOWS OF ORDER INSIDE CHAOS. Even with the simplest equation, the region of chaos in a</p><p>bifurcationdiagramprovestohaveanintricatestructure—farmoreorderlythanRobertMaycouldguessat</p><p>first.First, thebifurcationsproduceperiodsof2,4,8,16….Thenchaosbegins,withnoregularperiods.</p><p>Butthen,asthesystemisdrivenharder,windowsappearwithoddperiods.Astableperiod3appears,and</p><p>then the period-doubling begins again 6, 12, 24…. The structure is infinitely deep.When portions are</p><p>magnified,theyturnouttoresemblethewholediagram.</p><p>The sightseeing encounter between these frustrated, gesticulating</p><p>mathematicianswas a symptom of a continuing communications gap between</p><p>Soviet and Western science. Partly because of language, partly because of</p><p>restricted travel on theSoviet side, sophisticatedWestern scientists haveoften</p><p>repeatedwork that already existed in the Soviet literature. The blossoming of</p><p>chaosintheUnitedStatesandEuropehasinspiredahugebodyofparallelwork</p><p>in the Soviet Union; on the other hand, it also inspired considerable</p><p>bewilderment, becausemuch of the new sciencewas not so new inMoscow.</p><p>Sovietmathematicians and physicists had a strong tradition in chaos research,</p><p>datingback to theworkofA.N.Kolmogorov in the fifties.Furthermore, they</p><p>had a tradition of working together that had survived the divergence of</p><p>mathematicsandphysicselsewhere.</p><p>Thus Soviet scientists were receptive to Smale—his horseshoe created a</p><p>considerablestir in thesixties.Abrilliantmathematicalphysicist,YashaSinai,</p><p>quickly translated similar systems into thermodynamic terms. Similarly, when</p><p>Lorenz’s work finally reached Western physics in the seventies, it</p><p>simultaneously spread in the Soviet Union. And in 1975, as Yorke andMay</p><p>struggled to capture the attention of their colleagues, Sinai and others rapidly</p><p>assembledapowerfulworkinggroupofphysicistscentered inGorki. Inrecent</p><p>years,someWesternchaosexpertshavemadeapointof travelingregularly to</p><p>theSovietUniontostaycurrent;most,however,havehadtocontentthemselves</p><p>withtheWesternversionoftheirscience.</p><p>IntheWest,YorkeandMaywerethefirsttofeelthefullshockofperiod-</p><p>doubling and to pass the shock along to the communityof scientists.The few</p><p>mathematicianswhohadnotedthephenomenontreateditasatechnicalmatter,a</p><p>numerical oddity: almost a kind of game playing. Not that they considered it</p><p>trivial.Buttheyconsidereditathingoftheirspecialuniverse.</p><p>Biologists had overlooked bifurcations on theway to chaos because they</p><p>lackedmathematical sophistication and because they lacked themotivation to</p><p>explore disorderly behavior. Mathematicians had seen bifurcations but had</p><p>moved on.May, a man with one foot in each world, understood that he was</p><p>enteringadomainthatwasastonishingandprofound.</p><p>TO SEE DEEPER INTO this simplest of systems, scientists needed greater</p><p>computing power. Frank Hoppensteadt, at New York University’s Courant</p><p>InstituteofMathematicalSciences,hadsopowerfulacomputerthathedecided</p><p>tomakeamovie.</p><p>Hoppensteadt, a mathematician who later developed a strong interest in</p><p>biologicalproblems,fedthelogisticnonlinearequationthroughhisControlData</p><p>6600 hundreds of millions of times. He took pictures from the computer’s</p><p>displayscreenateachofathousanddifferentvaluesoftheparameter,athousand</p><p>different tunings. The bifurcations appeared, then chaos—and then,within the</p><p>chaos, the little spikes of order, ephemeral in their instability. Fleeting bits of</p><p>periodicbehavior.Staringathisownfilm,Hoppensteadtfeltasifhewereflying</p><p>throughanalienlandscape.Oneinstantitwouldn’tlookchaoticatall.Thenext</p><p>instantitwouldbefilledwithunpredictabletumult.Thefeelingofastonishment</p><p>wassomethingHoppensteadtnevergotover.</p><p>MaysawHoppensteadt’smovie.Healsobegancollectinganaloguesfrom</p><p>otherfields,suchasgenetics,economics,andfluiddynamics.Asatowncrierfor</p><p>chaos,hehad twoadvantagesover thepuremathematicians.Onewas that, for</p><p>him, the simple equations could not represent reality perfectly.He knew they</p><p>were justmetaphors—sohebegan towonderhowwidely themetaphorscould</p><p>apply.Theotherwasthat therevelationsofchaosfeddirectly intoavehement</p><p>controversyinhischosenfield.</p><p>TheoutlineofthebifurcationdiagramasMayfirstsawit,beforemorepowerfulcomputationrevealedits</p><p>richstructure.</p><p>Populationbiologyhadlongbeenamagnetforcontroversyanyway.There</p><p>wastensioninbiologydepartments,forexample,betweenmolecularbiologists</p><p>andecologists.Themolecularbiologiststhoughtthattheydidrealscience,crisp,</p><p>hardproblems,whereas theworkofecologistswasvague.Ecologistsbelieved</p><p>that the technical masterpieces of molecular biology were just clever</p><p>elaborationsofwell-definedproblems.</p><p>Within ecology itself, as May saw it, a central controversy in the early</p><p>1970s dealt with the nature of population change. Ecologists were divided</p><p>almost along lines of personality. Some read the message of the world to be</p><p>orderly:populationsareregulatedandsteady—withexceptions.Othersreadthe</p><p>opposite message: populations fluctuate erratically—with exceptions. By no</p><p>coincidence, these opposing camps also divided over the application of hard</p><p>mathematicstomessybiologicalquestions.Thosewhobelievedthatpopulations</p><p>were steady argued that they must be regulated by some deterministic</p><p>mechanisms.Thosewhobelievedthatpopulationswere</p><p>erraticarguedthatthey</p><p>must be bounced around by unpredictable environmental factors, wiping out</p><p>whatever deterministic signal might exist. Either deterministic mathematics</p><p>producedsteadybehavior,orrandomexternalnoiseproducedrandombehavior.</p><p>Thatwasthechoice.</p><p>Inthecontextofthatdebate,chaosbroughtanastonishingmessage:simple</p><p>deterministic models could produce what looked like random behavior. The</p><p>behavior actually had an exquisite fine structure, yet any piece of it seemed</p><p>indistinguishable from noise. The discovery cut through the heart of the</p><p>controversy.</p><p>AsMaylookedatmoreandmorebiologicalsystemsthroughtheprismof</p><p>simple chaotic models, he continued to see results that violated the standard</p><p>intuitionofpractitioners.Inepidemiology,forexample,itwaswellknownthat</p><p>epidemicstendtocomeincycles,regularorirregular.Measles,polio,rubella—</p><p>all rise and fall in frequency. May realized that the oscillations could be</p><p>reproducedbyanonlinearmodelandhewonderedwhatwouldhappenifsucha</p><p>systemreceivedasuddenkick—aperturbationofthekindthatmightcorrespond</p><p>toaprogramofinoculation.Naïveintuitionsuggeststhatthesystemwillchange</p><p>smoothlyinthedesireddirection.Butactually,Mayfound,hugeoscillationsare</p><p>likely to begin. Even if the longterm trendwas turned solidly downward, the</p><p>path toanewequilibriumwouldbe interruptedbysurprisingpeaks. Infact, in</p><p>data from real programs, such as a campaign to wipe out rubella in Britain,</p><p>doctorshadseenoscillationsjustlikethosepredictedbyMay’smodel.Yetany</p><p>health official, seeing a sharp short-term rise in rubella or gonorrhea, would</p><p>assumethattheinoculationprogramhadfailed.</p><p>Withinafewyears,thestudyofchaosgaveastrongimpetustotheoretical</p><p>biology,bringingbiologistsandphysicistsintoscholarlypartnershipsthatwere</p><p>inconceivable a few years before. Ecologists and epidemiologists dug out old</p><p>datathatearlierscientistshaddiscardedastoounwieldytohandle.Deterministic</p><p>chaoswas found in records ofNewYorkCitymeasles epidemics and in two</p><p>hundredyearsof fluctuationsof theCanadian lynxpopulation, as recordedby</p><p>the trappersof theHudson’sBayCompany.Molecularbiologistsbegan to see</p><p>proteins as systems in motion. Physiologists looked at organs not as static</p><p>structuresbutascomplexesofoscillations,someregularandsomeirregular.</p><p>All throughscience,Mayknew,specialistshadseenandarguedabout the</p><p>complexbehaviorofsystems.Eachdisciplineconsidereditsparticularbrandof</p><p>chaostobespecialuntoitself.Thethoughtinspireddespair.Yetwhatifapparent</p><p>randomness could come from simple models? And what if the same simple</p><p>models applied to complexity in different fields? May realized that the</p><p>astonishingstructureshehadbarelybeguntoexplorehadnointrinsicconnection</p><p>to biology. He wondered how many other sorts of scientists would be as</p><p>astonished as he. He set to work on what he eventually thought of as his</p><p>“messianic”paper,areviewarticlein1976forNature.</p><p>The world would be a better place,May argued, if every young student</p><p>were given a pocket calculator and encouraged to play with the logistic</p><p>differenceequation.Thatsimplecalculation,whichhe laidout infinedetail in</p><p>theNaturearticle,couldcounter thedistortedsenseof theworld’spossibilities</p><p>thatcomesfromastandardscientificeducation.Itwouldchangethewaypeople</p><p>thoughtabouteverythingfromthetheoryofbusinesscyclestothepropagation</p><p>ofrumors.</p><p>Chaos should be taught, he argued. It was time to recognize that the</p><p>standard education of a scientist gave the wrong impression. No matter how</p><p>elaborate linear mathematics could get, with its Fourier transforms, its</p><p>orthogonal functions, its regression techniques, May argued that it inevitably</p><p>misled scientists about their overwhelmingly nonlinear world. “The</p><p>mathematicalintuitionsodevelopedillequipsthestudenttoconfrontthebizarre</p><p>behaviourexhibitedbythesimplestofdiscretenonlinearsystems,”hewrote.</p><p>“Not only in research, but also in the everyday world of politics and</p><p>economics, we would all be better off if more people realized that simple</p><p>nonlinearsystemsdonotnecessarilypossesssimpledynamicalproperties.”</p><p>______________</p><p>*Forconvenience,inthishighlyabstractmodel,“population”isexpressedasafractionbetweenzeroand</p><p>one,zerorepresentingextinction,onerepresentingthegreatestconceivablepopulationofthepond.</p><p>Sobegin:Chooseanarbitraryvalueforr,say,2.7,andastartingpopulationof.02.Oneminus.02is.98.</p><p>Multiplyby0.02andyouget.0196.Multiplythatby2.7andyouget.0529.Theverysmallstarting</p><p>populationhasmorethandoubled.Repeattheprocess,usingthenewpopulationastheseed,andyouget</p><p>.1353.Withacheapprogrammablecalculator,thisiterationisjustamatterofpushingonebuttonoverand</p><p>overagain.Thepopulationrisesto.3159,then.5835,then.6562—therateofincreaseisslowing.Then,as</p><p>starvationovertakesreproduction,.6092.Then.6428,then.6199,then.6362,then.6249.Thenumbers</p><p>seemtobebouncingbackandforth,butclosinginonafixednumber:.6328,.6273,.6312,.6285,.6304,</p><p>.6291,.6300,.6294,.6299,.6295,.6297,.6296,.6297,.6296,.6296,.6296,.6296,.6296,.6296,.6296.</p><p>Success!</p><p>Inthedaysofpencil-and–paperarithmetic,andinthedaysofmechanicaladdingmachineswithhand</p><p>cranks,numericalexplorationneverwentmuchfurther.</p><p>*Withaparameterof3.5,say,andastartingvalueof.4,hewouldseeastringofnumberslikethis:</p><p>.4000,.8400,.4704,.8719,</p><p>.3908,.8332,.4862,.8743,</p><p>.3846,.8284,.4976,.8750,</p><p>.3829,.8270,.4976,.8750,</p><p>.3829,.8270,.5008,.8750,</p><p>.3828,.8269,.5009,.8750,</p><p>.3828,.8269,.5009,.8750,etc.</p><p>AGeometry</p><p>ofNature</p><p>Andyetrelationappears,</p><p>Asmallrelationexpandingliketheshade</p><p>Ofacloudonsand,ashapeonthesideofahill.</p><p>—WALLACESTEVENS</p><p>“ConnoisseurofChaos”</p><p>APICTUREOFREALITYbuiltupovertheyearsinBenoitMandelbrot’smind.</p><p>In 1960, itwas a ghost of an idea, a faint, unfocused image. ButMandelbrot</p><p>recognized it when he saw it, and there it was on the blackboard in Hendrik</p><p>Houthakker’soffice.</p><p>Mandelbrotwas amathematical jack-of–all-tradeswho had been adopted</p><p>andshelteredbythepureresearchwingoftheInternationalBusinessMachines</p><p>Corporation.He had been dabbling in economics, studying the distribution of</p><p>large and small incomes in an economy. Houthakker, a Harvard economics</p><p>professor, had invited Mandelbrot to give a talk, and when the young</p><p>mathematician arrived at Littauer Center, the stately economics building just</p><p>northofHarvardYard,hewasstartledtoseehisfindingsalreadychartedonthe</p><p>older man’s blackboard. Mandelbrot made a querulous joke—how should my</p><p>diagramhavematerializedaheadofmylecture?—butHouthakkerdidn’tknow</p><p>whatMandelbrotwastalkingabout.Thediagramhadnothingtodowithincome</p><p>distribution;itrepresentedeightyearsofcottonprices.</p><p>FromHouthakker’spointofview, too, therewassomethingstrangeabout</p><p>this chart. Economists generally assumed that the price of a commodity like</p><p>cottondancedtotwodifferentbeats,oneorderlyandonerandom.Overthelong</p><p>term, priceswould be driven steadily by real forces in the economy—the rise</p><p>andfalloftheNewEnglandtextileindustry,ortheopeningofinternationaltrade</p><p>routes.Overtheshortterm,priceswouldbouncearoundmoreorlessrandomly.</p><p>Unfortunately,Houthakker’s data failed tomatchhis</p><p>expectations.Therewere</p><p>toomanylargejumps.Mostpricechangesweresmall,ofcourse,buttheratioof</p><p>smallchangestolargewasnotashighashehadexpected.Thedistributiondid</p><p>notfalloffquicklyenough.Ithadalongtail.</p><p>Thestandardmodelforplottingvariationwasandisthebell-shapedcurve.</p><p>In themiddle,where the hump of the bell rises,most data cluster around the</p><p>average.Onthesides,thelowandhighextremesfalloffrapidly.Astatistician</p><p>uses a bell-shaped curve the way an internist uses a stethoscope, as the</p><p>instrument of first resort. It represents the standard, so-called Gaussian</p><p>distributionofthings—or,simply,thenormaldistribution.Itmakesastatement</p><p>aboutthenatureofrandomness.Thepointisthatwhenthingsvary,theytryto</p><p>staynearanaveragepoint and theymanage to scatter around theaverage ina</p><p>reasonablysmoothway.Butasameansoffindingpathsthroughtheeconomic</p><p>wilderness, the standard notions left something to be desired. As the Nobel</p><p>laureateWassilyLeontiefputit,“Innofieldofempiricalinquiryhassomassive</p><p>andsophisticatedastatisticalmachinerybeenusedwithsuchindifferentresults.”</p><p>Nomatterhowheplottedthem,Houthakkercouldnotmakethechangesin</p><p>cottonpricesfitthebell-shapedmodel.Buttheymadeapicturewhosesilhouette</p><p>Mandelbrotwasbeginning to see in surprisinglydisparateplaces.Unlikemost</p><p>mathematicians, he confronted problems by depending on his intuition about</p><p>patternsandshapes.Hemistrustedanalysis,buthe trustedhismentalpictures.</p><p>And he already had the idea that other laws, with different behavior, could</p><p>govern random, stochastic phenomena.When he went back to the giant IBM</p><p>research center in Yorktown Heights, New York, in the hills of northern</p><p>WestchesterCounty,hecarriedHouthakker’scottondata inaboxofcomputer</p><p>cards.Thenhesent to theDepartmentofAgriculture inWashington formore,</p><p>datingbackto1900.</p><p>THEBELL-SHAPEDCURVE.</p><p>Likescientists inotherfields,economistswerecrossingthe thresholdinto</p><p>thecomputerera,slowlyrealizingthattheywouldhavethepowertocollectand</p><p>organize and manipulate information on a scale that had been unimaginable</p><p>before.Notallkindsofinformationwereavailable,though,andinformationthat</p><p>couldberoundedupstillhadtobeturnedintosomeusableform.Thekeypunch</p><p>erawasjustbeginning,too.Inthehardsciences,investigatorsfounditeasierto</p><p>amass their thousands or millions of data points. Economists, like biologists,</p><p>dealtwithaworldofwillfullivingbeings.Economistsstudiedthemostelusive</p><p>creaturesofall.</p><p>But at least the economists’ environment produced a constant supply of</p><p>numbers. FromMandelbrot’s point of view, cotton prices made an ideal data</p><p>source.Therecordswerecompleteandtheywereold,datingbackcontinuously</p><p>acenturyormore.Cottonwasapieceofthebuying-and–sellinguniversewitha</p><p>centralized market—and therefore centralized record-keeping—because at the</p><p>turn of the century all the South’s cotton flowed through the New York</p><p>exchangeonroutetoNewEngland,andLiverpool’spriceswerelinkedtoNew</p><p>York’saswell.</p><p>Although economists had little to go on when it came to analyzing</p><p>commoditypricesorstockprices,thatdidnotmeantheylackedafundamental</p><p>viewpointabouthowpricechangesworked.Onthecontrary,theysharedcertain</p><p>articlesoffaith.Onewasaconvictionthatsmall,transientchangeshadnothing</p><p>incommonwithlarge,longtermchanges.Fastfluctuationscomerandomly.The</p><p>small-scale ups and downs during a day’s transactions are just noise,</p><p>unpredictable and uninteresting. Longterm changes, however, are a different</p><p>speciesentirely.Thebroadswingsofpricesovermonthsoryearsordecadesare</p><p>determinedbydeepmacroeconomicforces,thetrendsofwarorrecession,forces</p><p>thatshould in theorygiveway tounderstanding.On theonehand, thebuzzof</p><p>short-termfluctuation;ontheother,thesignaloflongtermchange.</p><p>As it happened, thatdichotomyhadnoplace in thepictureof reality that</p><p>Mandelbrotwasdeveloping.Insteadofseparatingtinychangesfromgrandones,</p><p>hispictureboundthemtogether.Hewaslookingforpatternsnotatonescaleor</p><p>another,butacrosseveryscale.Itwasfarfromobvioushowtodrawthepicture</p><p>hehadinmind,butheknewtherewouldhavetobeakindofsymmetry,nota</p><p>symmetry of right and left or top and bottom but rather a symmetry of large</p><p>scalesandsmall.</p><p>Indeed, when Mandelbrot sifted the cotton-price data through IBM’s</p><p>computers, he found the astonishing results hewas seeking.Thenumbers that</p><p>produced aberrations from the point of view of normal distribution produced</p><p>symmetry from thepointofviewof scaling.Eachparticularpricechangewas</p><p>random and unpredictable. But the sequence of changes was independent of</p><p>scale: curves for daily price changes and monthly price changes matched</p><p>perfectly. Incredibly, analyzedMandelbrot’s way, the degree of variation had</p><p>remainedconstantoveratumultuoussixty-yearperiodthatsawtwoWorldWars</p><p>andadepression.</p><p>Withinthemostdisorderlyreamsofdatalivedanunexpectedkindoforder.</p><p>Given the arbitrariness of the numbers he was examining, why, Mandelbrot</p><p>askedhimself,shouldanylawholdatall?Andwhyshoulditapplyequallywell</p><p>topersonalincomesandcottonprices?</p><p>In truth, Mandelbrot’s background in economics was as meager as his</p><p>ability to communicatewith economists.When he published an article on his</p><p>findings, itwasprecededbyanexplanatoryarticlebyoneofhisstudents,who</p><p>repeatedMandelbrot’smaterialineconomists’English.Mandelbrotmovedonto</p><p>other interests. But he took with him a growing determination to explore the</p><p>phenomenon of scaling. It seemed to be a quality with a life of its own—a</p><p>signature.</p><p>INTRODUCED FOR A LECTURE years later (“…taught economics at Harvard,</p><p>engineering at Yale, physiology at the Einstein School of Medicine…”), he</p><p>remarked proudly: “Very oftenwhen I listen to the list ofmy previous jobs I</p><p>wonderifIexist.Theintersectionofsuchsetsissurelyempty.”Indeed,sincehis</p><p>earlydaysatIBM,Mandelbrothasfailedtoexistinalonglistofdifferentfields.</p><p>Hewasalwaysanoutsider,takinganunorthodoxapproachtoanunfashionable</p><p>cornerofmathematics,exploringdisciplinesinwhichhewasrarelywelcomed,</p><p>hidinghisgrandestideasineffortstogethispaperspublished,survivingmainly</p><p>ontheconfidenceofhisemployers inYorktownHeights.Hemadeforays into</p><p>fields like economics and thenwithdrew, leaving behind tantalizing ideas but</p><p>rarelywell-foundedbodiesofwork.</p><p>Inthehistoryofchaos,Mandelbrotmadehisownway.Yet thepictureof</p><p>realitythatwasforminginhismindin1960evolvedfromanoddityintoafull-</p><p>fledged geometry. To the physicists expanding on the work of people like</p><p>Lorenz, Smale, Yorke, and May, this prickly mathematician remained a</p><p>sideshow—but his techniques and his language became an inseparable part of</p><p>theirnewscience.</p><p>Thedescriptionwouldnothaveseemedapttoanyonewhoknewhiminhis</p><p>later years,with his high imposing brow and his list of titles and honors, but</p><p>BenoitMandelbrot isbestunderstoodasa refugee.Hewasborn inWarsawin</p><p>1924toaLithuanianJewishfamily,hisfatheraclothingwholesaler,hismother</p><p>adentist.Alerttogeopoliticalreality,thefamilymovedtoParisin1936,drawn</p><p>in part by the presence of Mandelbrot’s uncle, Szolem Mandelbrojt, a</p><p>mathematician.When thewar came, the family stayed just aheadof</p><p>theNazis</p><p>onceagain,abandoningeverythingbutafewsuitcasesandjoiningthestreamof</p><p>refugeeswhocloggedtheroadssouthfromParis.Theyfinallyreachedthetown</p><p>ofTulle.</p><p>For awhileBenoitwent around as an apprentice toolmaker, dangerously</p><p>conspicuous by his height and his educated background. It was a time of</p><p>unforgettable sights and fears, yet later he recalled little personal hardship,</p><p>remembering instead the times he was befriended in Tulle and elsewhere by</p><p>schoolteachers,someofthemdistinguishedscholars,themselvesstrandedbythe</p><p>war.Inall,hisschoolingwasirregularanddiscontinuous.Heclaimedneverto</p><p>have learned the alphabet or,more significantly,multiplication tables past the</p><p>fives.Still,hehadagift.</p><p>When Paris was liberated, he took and passed the month-long oral and</p><p>written admissions examination for École Normale and École Polytechnique,</p><p>despitehis lackofpreparation.Amongother elements, the test had avestigial</p><p>examinationindrawing,andMandelbrotdiscoveredalatentfacilityforcopying</p><p>the Venus de Milo. On the mathematical sections of the test—exercises in</p><p>formalalgebraandintegratedanalysis—hemanagedtohidehislackoftraining</p><p>withthehelpofhisgeometricalintuition.Hehadrealizedthat,givenananalytic</p><p>problem,hecouldalmostalwaysthinkofitintermsofsomeshapeinhismind.</p><p>Given a shape, he could findways of transforming it, altering its symmetries,</p><p>makingitmoreharmonious.Oftenhistransformationsleddirectlytoasolution</p><p>of theanalogousproblem. Inphysicsandchemistry,wherehecouldnotapply</p><p>geometry, he got poor grades. But in mathematics, questions he could never</p><p>have answered using proper techniques melted away in the face of his</p><p>manipulationsofshapes.</p><p>The École Normale and École Polytechnique were elite schools with no</p><p>parallelinAmericaneducation.Togethertheypreparedfewerthan300students</p><p>ineachclassforcareersintheFrenchuniversitiesandcivilservice.Mandelbrot</p><p>began inNormale, thesmallerandmoreprestigiousof the two,but leftwithin</p><p>daysforPolytechnique.HewasalreadyarefugeefromBourbaki.</p><p>PerhapsnowherebutinFrance,withitsloveofauthoritarianacademiesand</p><p>received rules for learning, could Bourbaki have arisen. It began as a club,</p><p>founded in the unsettled wake ofWorldWar I by SzolemMandelbrot and a</p><p>handfulofother insouciantyoungmathematicianslookingforawaytorebuild</p><p>French mathematics. The vicious demographics of war had left an age gap</p><p>betweenuniversityprofessorsandstudents,disruptingthetraditionofacademic</p><p>continuity,andthesebrilliantyoungmensetouttoestablishnewfoundationsfor</p><p>thepracticeofmathematics.Thenameof theirgroupwasitselfaninsidejoke,</p><p>borrowedfor itsstrangeandattractivesound—soitwas laterguessed—froma</p><p>nineteenth-century French general of Greek origin. Bourbakiwas bornwith a</p><p>playfulnessthatsoondisappeared.</p><p>Itsmembersmet in secrecy. Indeed,notall theirnamesareknown.Their</p><p>numberwas fixed.Whenonemember left, aswas required at age50, another</p><p>wouldbeelectedbytheremaininggroup.Theywerethebestandthebrightestof</p><p>mathematicians,andtheirinfluencesoonspreadacrossthecontinent.</p><p>Inpart,Bourbakibegan in reaction toPoincaré, thegreatmanof the late</p><p>nineteenth century, a phenomenally prolific thinker andwriterwho cared less</p><p>thansomeforrigor.Poincaréwouldsay,Iknowitmustberight,sowhyshould</p><p>I prove it? Bourbaki believed that Poincaré had left a shaky basis for</p><p>mathematics,andthegroupbegantowriteanenormoustreatise,moreandmore</p><p>fanatical in style, meant to set the discipline straight. Logical analysis was</p><p>central.Amathematicianhadtobeginwithsolidfirstprinciplesanddeduceall</p><p>the rest from them. The group stressed the primacy of mathematics among</p><p>sciences,andalsoinsisteduponadetachmentfromothersciences.Mathematics</p><p>was mathematics—it could not be valued in terms of its application to real</p><p>physical phenomena. And above all, Bourbaki rejected the use of pictures. A</p><p>mathematician could always be fooled by his visual apparatus.Geometrywas</p><p>untrustworthy.Mathematicsshouldbepure,formal,andaustere.</p><p>Nor was this strictly a French development. In the United States, too,</p><p>mathematicianswerepullingawayfromthedemandsofthephysicalsciencesas</p><p>firmly as artists and writers were pulling away from the demands of popular</p><p>taste. A hermetic sensibility prevailed.Mathematicians’ subjects became self-</p><p>contained;theirmethodbecameformallyaxiomatic.Amathematiciancouldtake</p><p>prideinsayingthathisworkexplainednothingintheworldorinscience.Much</p><p>goodcameof thisattitude,andmathematicians treasured it.SteveSmale,even</p><p>whilehewasworking to reunitemathematicsandnatural science,believed,as</p><p>deeply as he believed anything, thatmathematics should be something all by</p><p>itself.With self-containment cameclarity.Andclarity, too,wenthand inhand</p><p>with the rigor of the axiomatic method. Every serious mathematician</p><p>understandsthatrigoristhedefiningstrengthofthediscipline,thesteelskeleton</p><p>withoutwhichallwouldcollapse.Rigoriswhatallowsmathematicianstopick</p><p>up a line of thought that extends over centuries and continue it, with a firm</p><p>guarantee.</p><p>Even so, the demands of rigor had unintended consequences for</p><p>mathematicsinthetwentiethcentury.Thefielddevelopsthroughaspecialkind</p><p>ofevolution.Aresearcherpicksupaproblemandbeginsbymakingadecision</p><p>about whichway to continue. It happened that often that decision involved a</p><p>choice between a path that was mathematically feasible and a path that was</p><p>interestingfromthepointofviewofunderstandingnature.Foramathematician,</p><p>thechoicewasclear:hewouldabandonanyobviousconnectionwithnaturefor</p><p>awhile.Eventuallyhisstudentswouldfaceasimilarchoiceandmakeasimilar</p><p>decision.</p><p>Nowhere were these values as severely codified as in France, and there</p><p>Bourbakisucceededasitsfounderscouldnothaveimagined.Itsprecepts,style,</p><p>and notation became mandatory. It achieved the unassailable Tightness that</p><p>comes from controlling all the best students and producing a steady flow of</p><p>successful mathematics. Its dominance over École Normale was total and, to</p><p>Mandelbrot, unbearable. He fled Normale because of Bourbaki, and a decade</p><p>laterhefledFranceforthesamereason,takingupresidenceintheUnitedStates.</p><p>Withinafewdecades,therelentlessabstractionofBourbakiwouldbegintodie</p><p>of a shock brought on by the computer, with its power to feed a new</p><p>mathematicsoftheeye.ButthatwastoolateforMandelbrot,unabletoliveby</p><p>Bourbaki’sformalismsandunwillingtoabandonhisgeometricalintuition.</p><p>ALWAYSABELIEVER increatinghisownmythology,Mandelbrotappended</p><p>this statement to his entry inWho’sWho: “Science would be ruined if (like</p><p>sports)itweretoputcompetitionaboveeverythingelse,andifitweretoclarify</p><p>the rules of competition by withdrawing entirely into narrowly defined</p><p>specialties. The rare scholars who are nomads-by–choice are essential to the</p><p>intellectualwelfareofthesettleddisciplines.”Thisnomad-by–choice,whoalso</p><p>called himself a pioneer-by–necessity, withdrew from academe when he</p><p>withdrew from France, accepting the shelter of IBM’s Thomas J. Watson</p><p>ResearchCenter.Inathirty-yearjourneyfromobscuritytoeminence,henever</p><p>saw his work embraced by themany disciplines towardwhich he directed</p><p>it.</p><p>Even mathematicians would say, without apparent malice, that whatever</p><p>Mandelbrotwas,hewasnotoneofthem.</p><p>Hefoundhiswayslowly,alwaysabettedbyanextravagantknowledgeof</p><p>the forgotten byways of scientific history. He ventured into mathematical</p><p>linguistics, explaining a lawof thedistributionofwords. (Apologizing for the</p><p>symbolism, he insisted that the problem came to his attention from a book</p><p>reviewthathe retrievedfromapuremathematician’swastebasketsohewould</p><p>havesomethingtoreadontheParissubway.)Heinvestigatedgametheory.He</p><p>workedhiswayinandoutofeconomics.Hewroteaboutscalingregularitiesin</p><p>the distribution of large and small cities. The general framework that tied his</p><p>worktogetherremainedinthebackground,incompletelyformed.</p><p>EarlyinhistimeatIBM,soonafterhisstudyofcommodityprices,hecame</p><p>uponapracticalproblemof intenseconcern tohiscorporatepatron.Engineers</p><p>were perplexed by the problem of noise in telephone lines used to transmit</p><p>informationfromcomputertocomputer.Electriccurrentcarriestheinformation</p><p>indiscretepackets,andengineersknewthatthestrongertheymadethecurrent</p><p>the better it would be at drowning out noise. But they found that some</p><p>spontaneousnoisecouldneverbeeliminated.Onceinawhileitwouldwipeout</p><p>apieceofsignal,creatinganerror.</p><p>Although by its nature the transmission noise was random, it was well</p><p>known to come in clusters. Periods of errorless communication would be</p><p>followed by periods of errors. By talking to the engineers, Mandelbrot soon</p><p>learned that therewasapieceof folklore about the errors thathadneverbeen</p><p>written down, because it matched none of the standard ways of thinking: the</p><p>more closely they looked at the clusters, themore complicated the patterns of</p><p>errors seemed. Mandelbrot provided a way of describing the distribution of</p><p>errors that predicted exactly the observed patterns. Yet it was exceedingly</p><p>peculiar. For one thing, it made it impossible to calculate an average rate of</p><p>errors—anaveragenumberoferrorsperhour,orperminute,orpersecond.On</p><p>average,inMandelbrot’sscheme,errorsapproachedinfinitesparseness.</p><p>Hisdescriptionworkedbymakingdeeperanddeeperseparationsbetween</p><p>periodsofcleantransmissionandperiodsoferrors.Supposeyoudividedaday</p><p>intohours.Anhourmightpasswithnoerrorsatall.Thenanhourmightcontain</p><p>errors.Thenanhourmightpasswithnoerrors.</p><p>Butsupposeyou thendivided thehourwitherrors intosmallerperiodsof</p><p>twenty minutes. You would find that here, too, some periods would be</p><p>completely clean, while some would contain a burst of errors. In fact,</p><p>Mandelbrot argued—contrary to intuition—that you could never find a time</p><p>duringwhicherrorswerescatteredcontinuously.Withinanyburstoferrors,no</p><p>matter how short, there would always be periods of completely error-free</p><p>transmission. Furthermore, he discovered a consistent geometric relationship</p><p>betweentheburstsoferrorsandthespacesofcleantransmission.Onscalesof</p><p>anhourorasecond,theproportionoferror-freeperiodstoerror-riddenperiods</p><p>remained constant. (Once, toMandelbrot’s horror, a batch of data seemed to</p><p>contradicthisscheme—butit turnedout that theengineershadfailedtorecord</p><p>themostextremecases,ontheassumptionthattheywereirrelevant.)</p><p>Engineers had no framework for understandingMandelbrot’s description,</p><p>but mathematicians did. In effect, Mandelbrot was duplicating an abstract</p><p>constructionknownastheCantorset,afterthenineteenth-centurymathematician</p><p>GeorgCantor.TomakeaCantorset,youstartwiththeintervalofnumbersfrom</p><p>zero toone, representedbya linesegment.Thenyouremovethemiddle third.</p><p>Thatleavestwosegments,andyouremovethemiddlethirdofeach(fromone-</p><p>ninth to two-ninths and from seven-ninths to eight-ninths). That leaves four</p><p>segments,andyouremovethemiddlethirdofeach—andsoontoinfinity.What</p><p>remains? A strange “dust” of points, arranged in clusters, infinitelymany yet</p><p>infinitelysparse.MandelbrotwasthinkingoftransmissionerrorsasaCantorset</p><p>arrangedintime.</p><p>Thishighlyabstractdescriptionhadpracticalweightforscientiststryingto</p><p>decide between different strategies of controlling error. In particular, itmeant</p><p>that, insteadof trying to increase signal strength to drownoutmore andmore</p><p>noise, engineers should settle for a modest signal, accept the inevitability of</p><p>errorsanduseastrategyof redundancy tocatchandcorrect them.Mandelbrot</p><p>alsochangedthewayIBM’sengineersthoughtaboutthecauseofnoise.Bursts</p><p>oferrorshadalwayssenttheengineerslookingforamanstickingascrewdriver</p><p>somewhere. ButMandelbrot’s scaling patterns suggested that the noisewould</p><p>neverbeexplainedonthebasisofspecificlocalevents.</p><p>Mandelbrotturnedtootherdata,drawnfromtheworld’srivers.Egyptians</p><p>havekeptrecordsoftheheightoftheNileformillennia.Itisamatterofmore</p><p>than passing concern. The Nile suffers unusually great variation, flooding</p><p>heavily in some years and subsiding in others. Mandelbrot classified the</p><p>variationintermsoftwokindsofeffects,commonineconomicsaswell,which</p><p>hecalledtheNoahandJosephEffects.</p><p>The Noah Effect means discontinuity: when a quantity changes, it can</p><p>change almost arbitrarily fast. Economists traditionally imagined that prices</p><p>change smoothly—rapidly or slowly, as the casemay be, but smoothly in the</p><p>sense that they pass through all the intervening levels on their way from one</p><p>pointtoanother.Thatimageofmotionwasborrowedfromphysics,likemuchof</p><p>themathematicsapplied toeconomics.But itwaswrong.Pricescanchange in</p><p>instantaneous jumps, as swiftly as a piece of news can flash across a teletype</p><p>wire and a thousand brokers can change theirminds.A stockmarket strategy</p><p>wasdoomedtofail,Mandelbrotargued,ifitassumedthatastockwouldhaveto</p><p>sellfor$50atsomepointonitswaydownfrom$60to$10.</p><p>THECANTORDUST.Beginwitha line; remove themiddle third; then remove themiddle thirdof the</p><p>remainingsegments;andsoon.TheCantorsetisthedustofpointsthatremains.Theyareinfinitelymany,</p><p>buttheirtotallengthis0.</p><p>The paradoxical qualities of such constructions disturbed nineteenth-century mathematicians, but</p><p>MandelbrotsawtheCantorsetasamodelfor theoccurrenceoferrors inanelectronic transmission line.</p><p>Engineers sawperiodsof error-free transmission,mixedwithperiodswhenerrorswouldcome inbursts.</p><p>Lookedatmore closely, thebursts, too, contained error-freeperiodswithin them.And soon—itwas an</p><p>example of fractal time. At every time scale, from hours to seconds, Mandelbrot discovered that the</p><p>relationshipoferrorstocleantransmissionremainedconstant.Suchdusts,hecontended,areindispensable</p><p>inmodelingintermittency.</p><p>The Joseph Effect means persistence. There came seven years of great</p><p>plentythroughoutthelandofEgypt.Andthereshallariseafterthemsevenyears</p><p>of famine. If the Biblical legend meant to imply periodicity, it was</p><p>oversimplified, of course. But floods and droughts do persist. Despite an</p><p>underlyingrandomness,thelongeraplacehassuffereddrought,thelikelieritis</p><p>tosuffermore.Furthermore,mathematicalanalysisoftheNile’sheightshowed</p><p>that persistence appliedover centuries aswell as over decades.TheNoah and</p><p>Joseph Effects push in different directions, but they add up to this: trends</p><p>in</p><p>naturearereal,buttheycanvanishasquicklyastheycome.</p><p>Discontinuity,burstsofnoise,Cantordusts—phenomenalikethesehadno</p><p>placein thegeometriesof thepast twothousandyears.Theshapesofclassical</p><p>geometry are lines and planes, circles and spheres, triangles and cones. They</p><p>represent a powerful abstraction of reality, and they inspired a powerful</p><p>philosophy of Platonic harmony. Euclidmade of them a geometry that lasted</p><p>twomillennia,theonlygeometrystillthatmostpeopleeverlearn.Artistsfound</p><p>anidealbeautyinthem,Ptolemaicastronomersbuiltatheoryoftheuniverseout</p><p>ofthem.Butforunderstandingcomplexity,theyturnouttobethewrongkindof</p><p>abstraction.</p><p>Clouds are not spheres,Mandelbrot is fond of saying.Mountains are not</p><p>cones.Lightningdoesnottravelinastraightline.Thenewgeometrymirrorsa</p><p>universethatisrough,notrounded,scabrous,notsmooth.Itisageometryofthe</p><p>pitted, pocked, and broken up, the twisted, tangled, and intertwined. The</p><p>understanding of nature’s complexity awaited a suspicion that the complexity</p><p>was not just random, not just accident. It required a faith that the interesting</p><p>featureofa lightningbolt’spath, forexample,wasnot itsdirection,but rather</p><p>the distribution of zigs and zags.Mandelbrot’s work made a claim about the</p><p>world, and the claim was that such odd shapes carry meaning. The pits and</p><p>tangles are more than blemishes distorting the classic shapes of Euclidian</p><p>geometry.Theyareoftenthekeystotheessenceofathing.</p><p>What is the essence of a coastline, for example? Mandelbrot asked this</p><p>questioninapaperthatbecameaturningpointforhisthinking:“HowLongIs</p><p>theCoastofBritain?”</p><p>Mandelbrot had come across the coastline question in an obscure</p><p>posthumous article by an English scientist, Lewis F. Richardson, who groped</p><p>withasurprisingnumberoftheissuesthatlaterbecamepartofchaos.Hewrote</p><p>about numerical weather prediction in the 1920s, studied fluid turbulence by</p><p>throwingasackofwhiteparsnipsintotheCapeCodCanal,andaskedina1926</p><p>paper, “Does the Wind Possess a Velocity?” (“The question, at first sight</p><p>foolish,improvesonacquaintance,”hewrote.)Wonderingaboutcoastlinesand</p><p>wiggly national borders, Richardson checked encyclopedias in Spain and</p><p>Portugal,Belgiumand theNetherlandsanddiscovereddiscrepanciesof twenty</p><p>percentintheestimatedlengthsoftheircommonfrontiers.</p><p>Mandelbrot’s analysis of this question struck listeners as either painfully</p><p>obviousorabsurdlyfalse.Hefoundthatmostpeopleanswered thequestion in</p><p>oneof twoways: “I don’t know, it’s notmy field,” or “I don’t know,but I’ll</p><p>lookitupintheencyclopedia.”</p><p>Infact,heargued,anycoastlineis—inasense—infinitelylong.Inanother</p><p>sense, the answer depends on the length of your ruler.Consider one plausible</p><p>methodofmeasuring.Asurveyortakesasetofdividers,opensthemtoalength</p><p>ofoneyard,andwalksthemalongthecoastline.Theresultingnumberofyards</p><p>isjustanapproximationofthetruelength,becausethedividersskipovertwists</p><p>and turns smaller than one yard, but the surveyor writes the number down</p><p>anyway. Then he sets the dividers to a smaller length—say, one foot—and</p><p>repeats the process. He arrives at a somewhat greater length, because the</p><p>dividerswillcapturemoreofthedetailanditwilltakemorethanthreeone-foot</p><p>stepstocoverthedistancepreviouslycoveredbyaone-yardstep.Hewritesthis</p><p>newnumberdown,setsthedividersatfourinches,andstartsagain.Thismental</p><p>experiment, using imaginary dividers, is a way of quantifying the effect of</p><p>observing an object from different distances, at different scales. An observer</p><p>tryingtoestimatethelengthofEngland’scoastlinefromasatellitewillmakea</p><p>smallerguess thananobserver trying towalk its covesandbeaches,whowill</p><p>makeasmallerguessinturnthanasnailnegotiatingeverypebble.</p><p>AFRACTALSHORE.Acomputer-generatedcoastline:thedetailsarerandom,butthefractaldimensionis</p><p>constant, so the degree of roughness or irregularity looks the same no matter how much the image is</p><p>magnified.</p><p>Commonsensesuggeststhat,althoughtheseestimateswillcontinuetoget</p><p>larger, they will approach some particular final value, the true length of the</p><p>coastline.Themeasurementsshouldconverge,inotherwords.Andinfact, ifa</p><p>coastlineweresomeEuclideanshape,suchasacircle,thismethodofsumming</p><p>finer and finer straight-line distanceswould indeed converge. ButMandelbrot</p><p>foundthatasthescaleofmeasurementbecomessmaller,themeasuredlengthof</p><p>a coastline rises without limit, bays and peninsulas revealing ever-smaller</p><p>subbaysand subpeninsulas—at leastdown toatomic scales,where theprocess</p><p>doesfinallycometoanend.Perhaps.</p><p>SINCE EUCLIDEAN MEASUREMENTS—length, depth, thickness—failed to</p><p>capturetheessenceofirregularshapes,Mandelbrotturnedtoadifferentidea,the</p><p>ideaofdimension.Dimensionisaqualitywithamuchricherlifeforscientists</p><p>than for nonscientists.We live in a three-dimensionalworld,meaning thatwe</p><p>need three numbers to specify a point: for example, longitude, latitude, and</p><p>altitude.Thethreedimensionsareimaginedasdirectionsatrightanglestoone</p><p>another. This is still the legacy ofEuclidean geometry,where space has three</p><p>dimensions,aplanehastwo,alinehasone,andapointhaszero.</p><p>TheprocessofabstractionthatallowedEuclidtoconceiveofone–ortwo-</p><p>dimensionalobjects spillsovereasily intoouruseofeverydayobjects.Aroad</p><p>map, for all practical purposes, is a quintessentially two-dimensional thing, a</p><p>pieceofaplane. Ituses its twodimensions tocarry informationofaprecisely</p><p>two-dimensionalkind.Inreality,ofcourse,roadmapsareasthree-dimensional</p><p>as everything else, but their thickness is so slight (and so irrelevant to their</p><p>purpose) that it can be forgotten. Effectively, a road map remains two-</p><p>dimensional,evenwhenitisfoldedup.Inthesameway,athreadiseffectively</p><p>one-dimensionalandaparticlehaseffectivelynodimensionatall.</p><p>Thenwhat is the dimension of a ball of twine?Mandelbrot answered, It</p><p>dependsonyourpointofview.Fromagreatdistance,theballisnomorethana</p><p>point,withzerodimensions.Fromcloser,theballisseentofillsphericalspace,</p><p>takingupthreedimensions.Fromcloserstill,thetwinecomesintoview,andthe</p><p>object becomes effectively one-dimensional, though the one dimension is</p><p>certainlytangleduparounditselfinawaythatmakesuseofthree-dimensional</p><p>space. The notion of how many numbers it takes to specify a point remains</p><p>useful. From far away, it takes none—the point is all there is. From closer, it</p><p>takesthree.Fromcloserstill,oneisenough—anygivenpositionalongthelength</p><p>oftwineisunique,whetherthetwineisstretchedoutortangledupinaball.</p><p>Andontowardmicroscopicperspectives:twineturnstothree-dimensional</p><p>columns,thecolumnsresolvethemselvesintoone-dimensionalfibers, thesolid</p><p>material dissolves into zero-dimensional points. Mandelbrot appealed,</p><p>unmathematically, to relativity: “The notion that a numerical result should</p><p>depend on the relation of object to observer is in the spirit of physics in this</p><p>centuryandisevenanexemplaryillustrationofit.”</p><p>Butphilosophyaside,theeffectivedimensionofanobjectdoesturnoutto</p><p>be different from itsmundane three dimensions. Aweakness inMandelbrot’s</p><p>verbalargumentseemedtobeitsreliance</p><p>onvaguenotions,“fromfaraway”and</p><p>“alittlecloser.”Whataboutinbetween?Surelytherewasnoclearboundaryat</p><p>which a ball of twine changes from a three-dimensional to a one-dimensional</p><p>object.Yet,farfrombeingaweakness,theill-definednatureofthesetransitions</p><p>ledtoanewideaabouttheproblemofdimensions.</p><p>Mandelbrotmovedbeyonddimensions0,1,2,3…toaseemingimpossibility:</p><p>fractional dimensions. The notion is a conceptual high-wire act. For</p><p>nonmathematicians it requires a willing suspension of disbelief. Yet it proves</p><p>extraordinarilypowerful.</p><p>Fractionaldimensionbecomesawayofmeasuringqualitiesthatotherwise</p><p>havenocleardefinition:thedegreeofroughnessorbrokennessorirregularityin</p><p>anobject.Atwistingcoastline,forexample,despiteitsimmeasurabilityinterms</p><p>of length, nevertheless has a certain characteristic degree of roughness.</p><p>Mandelbrot specified ways of calculating the fractional dimension of real</p><p>objects,givensometechniqueofconstructingashapeorgivensomedata,and</p><p>he allowed his geometry tomake a claim about the irregular patterns he had</p><p>studiedinnature.Theclaimwasthatthedegreeofirregularityremainsconstant</p><p>overdifferentscales.Surprisinglyoften,theclaimturnsouttobetrue.Overand</p><p>overagain,theworlddisplaysaregularirregularity.</p><p>Onewintry afternoon in 1975, aware of the parallel currents emerging in</p><p>physics,preparinghisfirstmajorworkforpublicationinbookform,Mandelbrot</p><p>decidedheneededanameforhisshapes,hisdimensions,andhisgeometry.His</p><p>sonwashomefromschool,andMandelbrotfoundhimselfthumbingthroughthe</p><p>boy’s Latin dictionary. He came across the adjective fractus, from the verb</p><p>frangere, to break.The resonance of themainEnglish cognates—fracture and</p><p>fraction—seemedappropriate.Mandelbrotcreatedtheword(nounandadjective,</p><p>EnglishandFrench)fractal.</p><p>INTHEMIND’SEYE,afractalisawayofseeinginfinity.</p><p>Imaginea triangle,eachof itssidesonefoot long.Nowimagineacertain</p><p>transformation—aparticular,well-defined,easilyrepeatedsetofrules.Takethe</p><p>middleone-thirdof each sideandattachanew triangle, identical in shapebut</p><p>one-thirdthesize.</p><p>TheresultisastarofDavid.Insteadofthreeone-footsegments,theoutline</p><p>ofthisshapeisnowtwelvefour-inchsegments.Insteadofthreepoints,thereare</p><p>six.</p><p>THEKOCH SNOWFLAKE. “A rough but vigorous model of a coastline,” inMandelbrot’s words. To</p><p>constructaKochcurve,beginwithatrianglewithsidesoflength1.Atthemiddleofeachside,addanew</p><p>triangleone-thirdthesize;andsoon.Thelengthoftheboundaryis3×4/3×4/3×4/3…—infinity.Yetthe</p><p>arearemainsless thantheareaofacircledrawnaroundtheoriginal triangle.Thusaninfinitelylongline</p><p>surroundsafinitearea.</p><p>Nowtakeeachofthetwelvesidesandrepeatthetransformation,attaching</p><p>asmaller triangleonto themiddle third.Nowagain,andsoon to infinity.The</p><p>outlinebecomesmoreandmoredetailed,justasaCantorsetbecomesmoreand</p><p>moresparse.Itresemblesasortofidealsnowflake.ItisknownasaKochcurve</p><p>—acurvebeinganyconnectedline,whetherstraightorround—afterHelgevon</p><p>Koch,theSwedishmathematicianwhofirstdescribeditin1904.</p><p>Onreflection,itbecomesapparentthattheKochcurvehassomeinteresting</p><p>features.Foronething,itisacontinuousloop,neverintersectingitself,because</p><p>thenewtrianglesoneachsidearealwayssmallenough toavoidbumping into</p><p>eachother.Eachtransformationaddsalittleareatotheinsideofthecurve,but</p><p>thetotalarearemainsfinite,notmuchbiggerthantheoriginaltriangle,infact.If</p><p>you drew a circle around the original triangle, the Koch curve would never</p><p>extendbeyondit.</p><p>Yet the curve itself is infinitely long, as long as aEuclidean straight line</p><p>extendingtotheedgesofanunboundeduniverse.Justasthefirsttransformation</p><p>replacesaone-footsegmentwithfourfour-inchsegments,everytransformation</p><p>multipliesthetotallengthbyfour-thirds.Thisparadoxicalresult,infinitelength</p><p>inafinitespace,disturbedmanyoftheturn-of–the-centurymathematicianswho</p><p>thoughtaboutit.TheKochcurvewasmonstrous,disrespectfultoallreasonable</p><p>intuition about shapes and—it almost went without saying—pathologically</p><p>unlikeanythingtobefoundinnature.</p><p>Under the circumstances, theirworkmade little impact at the time, but a</p><p>few equally perversemathematicians imagined other shapeswith some of the</p><p>bizarre qualities of the Koch curve. There were Peano curves. There were</p><p>SierpińskicarpetsandSierpińskigaskets.Tomakeacarpet,startwithasquare,</p><p>divide it three-by–three into nine equal squares, and remove the central one.</p><p>Thenrepeattheoperationontheeightremainingsquares,puttingasquarehole</p><p>inthecenterofeach.Thegasketisthesamebutwithequilateraltrianglesinstead</p><p>of squares; it has the hard-to–imagine property that any arbitrary point is a</p><p>branchingpoint,aforkinthestructure.Hardtoimagine,thatis,untilyouhave</p><p>thought about the Eiffel Tower, a good three-dimensional approximation, its</p><p>beamsandtrussesandgirdersbranchingintoalatticeofever-thinnermembers,a</p><p>shimmeringnetworkoffinedetail.Eiffel,ofcourse,couldnotcarrythescheme</p><p>to infinity,butheappreciated the subtleengineeringpoint thatallowedhim to</p><p>removeweightwithoutalsoremovingstructuralstrength.</p><p>Themindcannotvisualizethewholeinfiniteself-embeddingofcomplexity.</p><p>But to someone with a geometer’s way of thinking about form, this kind of</p><p>repetition of structure on finer and finer scales can open a whole world.</p><p>Exploringtheseshapes,pressingone’smentalfingersintotherubberyedgesof</p><p>theirpossibilities,wasakindofplaying,andMandelbrottookachildlikedelight</p><p>inseeingvariationsthatnoonehadseenorunderstoodbefore.Whentheyhad</p><p>no names, he named them: ropes and sheets, sponges and foams, curds and</p><p>gaskets.</p><p>Fractionaldimensionprovedtobepreciselytherightyardstick.Inasense,</p><p>thedegreeof irregularitycorrespondedto theefficiencyof theobject in taking</p><p>upspace.Asimple,Euclidean,one-dimensionallinefillsnospaceatall.Butthe</p><p>outlineoftheKochcurve,withinfinitelengthcrowdingintofinitearea,doesfill</p><p>space. It is more than a line, yet less than a plane. It is greater than one-</p><p>dimensional,yetlessthanatwo-dimensionalform.Usingtechniquesoriginated</p><p>bymathematicians early in the century and then all but forgotten,Mandelbrot</p><p>could characterize the fractional dimension precisely. For theKoch curve, the</p><p>infinitelyextendedmultiplicationbyfour-thirdsgivesadimensionof1.2618.</p><p>CONSTRUCTING WITH HOLES. A few mathematicians in the early twentieth century conceived</p><p>monstrous-seemingobjectsmadebythetechniqueofaddingorremovinginfinitelymanyparts.Onesuch</p><p>shapeistheSierpinskicarpet,constructedbycuttingthecenterone-ninthofasquare;thencuttingoutthe</p><p>centersoftheeightsmallersquaresthatremain;andsoon.Thethree-dimensionalanalogueistheMenger</p><p>sponge,asolid-lookinglatticethathasaninfinitesurfacearea,yetzerovolume.</p><p>In pursuing this path,Mandelbrot had two great advantages over the few</p><p>othermathematicianswhohadthoughtaboutsuchshapes.Onewashisaccessto</p><p>thecomputingresources thatgowith thenameofIBM.Herewasanother task</p><p>ideally suited to the computer’s particular form of high-speed idiocy. Just as</p><p>meteorologists needed to perform the same few calculations at millions of</p><p>neighboringpoints in theatmosphere,Mandelbrotneeded toperformaneasily</p><p>programmed</p><p>transformation again and again and again and again. Ingenuity</p><p>could conceive of transformations. Computers could draw them—sometimes</p><p>with unexpected results. The early twentieth-century mathematicians quickly</p><p>reached a barrier of hard calculation, like the barrier faced by early pro-</p><p>tobiologistswithoutmicroscopes. In looking into a universe of finer and finer</p><p>detail,theimaginationcancarryoneonlysofar.</p><p>InMandelbrot’swords:“Therewasalonghiatusofahundredyearswhere</p><p>drawingdidnotplayanyroleinmathematicsbecausehandandpencilandruler</p><p>wereexhausted.Theywerewellunderstoodandnolongerintheforefront.And</p><p>thecomputerdidnotexist.</p><p>“WhenIcameinthisgame,therewasatotalabsenceofintuition.Onehad</p><p>tocreateanintuitionfromscratch.Intuitionasitwastrainedbytheusualtools—</p><p>the hand, the pencil, and the ruler—found these shapes quite monstrous and</p><p>pathological. The old intuition was misleading. The first pictures were to me</p><p>quite a surprise; then Iwould recognize somepictures frompreviouspictures,</p><p>andsoon.</p><p>“Intuitionisnotsomethingthatisgiven.I’vetrainedmyintuitiontoaccept</p><p>asobvious shapeswhichwere initially rejectedas absurd, and I findeveryone</p><p>elsecandothesame.”</p><p>Mandelbrot’s other advantage was the picture of reality he had begun</p><p>forminginhisencounterswithcottonprices,withelectronictransmissionnoise,</p><p>andwith river floods.Thepicturewasbeginning tocome into focusnow.His</p><p>studiesofirregularpatternsinnaturalprocessesandhisexplorationofinfinitely</p><p>complex shapes had an intellectual intersection: a quality of self-similarity.</p><p>Aboveall,fractalmeantself-similar.</p><p>Self-similarityissymmetryacrossscale.Itimpliesrecursion,patterninside</p><p>of pattern.Mandelbrot’s price charts and river charts displayed self-similarity,</p><p>because not only did they produce detail at finer and finer scales, they also</p><p>produceddetailwithcertainconstantmeasurements.Monstrousshapes like the</p><p>Koch curve display self-similarity because they look exactly the same even</p><p>under high magnification. The self-similarity is built into the technique of</p><p>constructing the curves—the same transformation is repeated at smaller and</p><p>smaller scales. Self-similarity is an easily recognizable quality. Its images are</p><p>everywhereintheculture: intheinfinitelydeepreflectionofapersonstanding</p><p>between two mirrors, or in the cartoon notion of a fish eating a smaller fish</p><p>eating a smaller fish eating a smaller fish.Mandelbrot likes to quote Jonathan</p><p>Swift:“So,Nat’ralistsobserve,aFlea/HathsmallerFleasthatonhimprey,/And</p><p>thesehavesmallerFleastobite’em,/Andsoproceedadinfinitum.”</p><p>INTHENORTHEASTERNUnitedStates,thebestplacetostudyearthquakesis</p><p>the Lamont-Doherty Geophysical Observatory, a group of unprepossessing</p><p>buildings hidden in the woods of southern New York State, just west of the</p><p>Hudson River. Lamont-Doherty is where Christopher Scholz, a Columbia</p><p>Universityprofessorspecializingintheformandstructureofthesolidearth,first</p><p>startedthinkingaboutfractals.</p><p>Whilemathematiciansand theoreticalphysicistsdisregardedMandelbrot’s</p><p>work,Scholzwaspreciselythekindofpragmatic,workingscientistmostready</p><p>to pick up the tools of fractal geometry. He had stumbled across Benoit</p><p>Mandelbrot’sname in the1960s,whenMandelbrotwasworking ineconomics</p><p>andScholzwasanM.I.T.graduate student spendingagreatdealof timeona</p><p>stubbornquestionaboutearthquakes. Ithadbeenwellknown for twentyyears</p><p>that the distribution of large and small earthquakes followed a particular</p><p>mathematicalpattern,precisely thesamescalingpattern that seemed togovern</p><p>thedistributionofpersonalincomesinafree-marketeconomy.Thisdistribution</p><p>was observed everywhere on earth, wherever earthquakes were counted and</p><p>measured. Considering how irregular and unpredictable earthquakes were</p><p>otherwise, it was worthwhile to ask what sort of physical processes might</p><p>explainthisregularity.OrsoitseemedtoScholz.Mostseismologistshadbeen</p><p>contenttonotethefactandmoveon.</p><p>ScholzrememberedMandelbrot’sname,andin1978heboughtaprofusely</p><p>illustrated, bizarrely erudite, equation-studded book called Fractals: Form,</p><p>ChanceandDimension. Itwasas ifMandelbrothadcollected inone rambling</p><p>volumeeverythinghekneworsuspectedabouttheuniverse.Withinafewyears</p><p>this book and its expanded and refined replacement,TheFractalGeometry of</p><p>Nature,hadsoldmorecopiesthananyotherbookofhighmathematics.Itsstyle</p><p>wasabstruseandexasperating,byturnswitty,literary,andopaque.Mandelbrot</p><p>himselfcalledit“amanifestoandacasebook.”</p><p>Likea fewcounterparts inahandfulofother fields,particularlyscientists</p><p>whoworkedonthematerialpartsofnature,Scholzspentseveralyearstryingto</p><p>figureoutwhattodowiththisbook.Itwasfarfromobvious.Fractalswas,as</p><p>Scholz put it, “not a how-to book but a gee-whiz book.” Scholz, however,</p><p>happened to care deeply about surfaces, and surfaceswere everywhere in this</p><p>book. He found that he could not stop thinking about the promise of</p><p>Mandelbrot’s ideas.Hebegan toworkout awayofusing fractals todescribe,</p><p>classify,andmeasurethepiecesofhisscientificworld.</p><p>Hesoonrealizedthathewasnotalone,althoughitwasseveralmoreyears</p><p>beforefractalsconferencesandseminarsbeganmultiplying.Theunifyingideas</p><p>of fractal geometry brought together scientists who thought their own</p><p>observations were idiosyncratic and who had no systematic way of</p><p>understandingthem.Theinsightsoffractalgeometryhelpedscientistswhostudy</p><p>thewaythingsmeldtogether,thewaytheybranchapart,orthewaytheyshatter.</p><p>It is amethodof looking atmaterials—themicroscopically jagged surfaces of</p><p>metals, the tinyholes andchannelsofporousoil-bearing rock, the fragmented</p><p>landscapesofanearthquakezone.</p><p>As Scholz saw it, it was the business of geophysicists to describe the</p><p>surfaceof theearth, the surfacewhose intersectionwith the flatoceansmakes</p><p>coastlines.Withinthetopofthesolideartharesurfacesofanotherkind,surfaces</p><p>of cracks.Faults and fractures sodominate the structureof the earth’s surface</p><p>that they become the key to any good description,more important on balance</p><p>thanthematerialtheyrunthrough.Thefracturescrisscrosstheearth’ssurfacein</p><p>three dimensions, creatingwhatScholzwhimsically called the “schizosphere.”</p><p>Theycontroltheflowoffluidthroughtheground—theflowofwater,theflow</p><p>of oil, and the flow of natural gas. They control the behavior of earthquakes.</p><p>Understandingsurfaceswasparamount,yetScholzbelievedthathisprofession</p><p>wasinaquandary.Intruth,noframeworkexisted.</p><p>Geophysicists looked at surfaces the way anyone would, as shapes. A</p><p>surfacemightbeflat.Oritmighthaveaparticularshape.Youcouldlookatthe</p><p>outlineofaVolkswagenBeetle,forexample,anddrawthatsurfaceasacurve.</p><p>The curvewould bemeasurable in familiar Euclideanways.You could fit an</p><p>equation to it. But in Scholz’s description, youwould only be looking at that</p><p>surfacethroughanarrowspectralband.Itwouldbelikelookingattheuniverse</p><p>througharedfilter—youseewhatishappeningatthatparticularwavelengthof</p><p>light,butyoumisseverythinghappeningatthewavelengthsofothercolors,not</p><p>tomentionthatvastrangeofactivityatpartsof thespectrumcorrespondingto</p><p>infraredradiationorradiowaves.Thespectrum,inthisanalogy,</p><p>correspondsto</p><p>scale.TothinkofthesurfaceofaVolkswagenintermsofitsEuclideanshapeis</p><p>toseeitonlyonthescaleofanobservertenmetersoronehundredmetersaway.</p><p>Whataboutanobserveronekilometeraway,oronehundredkilometers?What</p><p>aboutanobserveronemillimeteraway,oronemicron?</p><p>Imaginetracingthesurfaceoftheearthasitwouldlookfromadistanceof</p><p>onehundredkilometersoutinspace.Thelinegoesupanddownovertreesand</p><p>hillocks, buildings and—in a parking lot somewhere—aVolkswagen.On that</p><p>scale,thesurfaceisjustabumpamongmanyotherbumps,abitofrandomness.</p><p>OrimaginelookingattheVolkswagenfromcloserandcloser,zoomingin</p><p>with magnifying glass and microscope. At first the surface seems to get</p><p>smoother, as the roundnessofbumpersandhoodpassesoutofview.But then</p><p>themicroscopicsurfaceofthesteelturnsouttobebumpyitself,inanapparently</p><p>randomway.Itseemschaotic.</p><p>Scholzfoundthatfractalgeometryprovidedapowerfulwayofdescribing</p><p>theparticularbumpinessoftheearth’ssurface,andmetallurgistsfoundthesame</p><p>for the surfaces of different kinds of steel. The fractal dimension of ametal’s</p><p>surface,forexample,oftenprovidesinformationthatcorrespondstothemetal’s</p><p>strength.And the fractal dimensionof the earth’s surfaceprovides clues to its</p><p>importantqualitiesaswell.Scholzthoughtaboutaclassicgeologicalformation,</p><p>a talus slope on a mountainside. From a distance it is a Euclidean shape,</p><p>dimensiontwo.Asageologistapproaches,though,hefindshimselfwalkingnot</p><p>somuchonitasinit—thetalushasresolveditselfintobouldersthesizeofcars.</p><p>Its effective dimension has become about 2.7, because the rock surfaces hook</p><p>overandwraparoundandnearlyfillthree-dimensionalspace,likethesurfaceof</p><p>asponge.</p><p>Fractal descriptions found immediate application in a series of problems</p><p>connectedtothepropertiesofsurfacesincontactwithoneanother.Thecontact</p><p>between tire treads and concrete is such a problem. So is contact inmachine</p><p>joints, or electrical contact. Contacts between surfaces have properties quite</p><p>independent of the materials involved. They are properties that turn out to</p><p>dependonthefractalqualityofthebumpsuponbumpsuponbumps.Onesimple</p><p>butpowerfulconsequenceofthefractalgeometryofsurfacesisthatsurfacesin</p><p>contactdonottoucheverywhere.Thebumpinessatallscalespreventsthat.Even</p><p>in rock under enormous pressure, at some sufficiently small scale it becomes</p><p>clear that gaps remain, allowing fluid to flow. To Scholz, it is the Humpty-</p><p>DumptyEffect. It iswhytwopiecesofabrokenteacupcanneverberejoined,</p><p>eventhoughtheyappeartofit togetheratsomegrossscale.Atasmallerscale,</p><p>irregularbumpsarefailingtocoincide.</p><p>Scholzbecameknowninhisfieldasoneofafewpeopletakingupfractal</p><p>techniques. He knew that some of his colleagues viewed this small group as</p><p>freaks. If he used the word fractal in the title of a paper, he felt that he was</p><p>regardedeitherasbeingadmirablycurrentornot-so–admirablyonabandwagon.</p><p>Eventhewritingofpapersforceddifficultdecisions,betweenwritingforasmall</p><p>audience of fractal aficionados or writing for a broader geophysical audience</p><p>thatwouldneedexplanationsofthebasicconcepts.Still,Scholzconsideredthe</p><p>toolsoffractalgeometryindispensable.</p><p>“It’s a single model that allows us to cope with the range of changing</p><p>dimensions of the earth,” he said. “It gives youmathematical and geometrical</p><p>tools todescribe andmakepredictions.Onceyouget over thehump, andyou</p><p>understand theparadigm,youcan start actuallymeasuring things and thinking</p><p>aboutthingsinanewway.Youseethemdifferently.Youhaveanewvision.It’s</p><p>notthesameastheoldvisionatall—it’smuchbroader.”</p><p>HOWBIGISIT?Howlongdoesitlast?Thesearethemostbasicquestionsa</p><p>scientist can ask about a thing. They are so basic to the way people</p><p>conceptualizetheworldthatit isnoteasytoseethattheyimplyacertainbias.</p><p>Theysuggestthatsizeandduration,qualitiesthatdependonscale,arequalities</p><p>withmeaning,qualities that canhelpdescribe anobjector classify it.Whena</p><p>biologistdescribesahumanbeing,oraphysicistdescribesaquark,howbigand</p><p>how long are indeed appropriate questions. In their gross physical structure,</p><p>animalsareverymuchtiedtoaparticularscale.Imagineahumanbeingscaled</p><p>up to twice its size, keeping all proportions the same, and you imagine a</p><p>structurewhoseboneswillcollapseunderitsweight.Scaleisimportant.</p><p>Thephysicsofearthquakebehaviorismostlyindependentofscale.Alarge</p><p>earthquakeisjustascaled-upversionofasmallearthquake.Thatdistinguishes</p><p>earthquakes from animals, for example—a ten-inch animalmust be structured</p><p>quite differently from a one-inch animal, and a hundred-inch animal needs a</p><p>differentarchitecturestill,ifitsbonesarenottosnapundertheincreasedmass.</p><p>Clouds, on the other hand, are scaling phenomena like earthquakes. Their</p><p>characteristic irregularity—describable in terms of fractal dimension—changes</p><p>notatallastheyareobservedondifferentscales.Thatiswhyairtravelerslose</p><p>all perspective on how far away a cloud is.Without help from cues such as</p><p>haziness,acloudtwentyfeetawaycanbeindistinguishablefromtwothousand</p><p>feet away. Indeed, analysis of satellite pictures has shown an invariant fractal</p><p>dimensionincloudsobservedfromhundredsofmilesaway.</p><p>Itishardtobreakthehabitofthinkingofthingsintermsofhowbigthey</p><p>areandhowlong they last.But theclaimof fractalgeometry is that, forsome</p><p>elements of nature, looking for a characteristic scale becomes a distraction.</p><p>Hurricane. By definition, it is a storm of a certain size. But the definition is</p><p>imposedbypeopleonnature.Inreality,atmosphericscientistsarerealizingthat</p><p>tumult in theair formsacontinuum,fromthegustyswirlingof litteronacity</p><p>streetcornertothevastcyclonicsystemsvisiblefromspace.Categoriesmislead.</p><p>Theendsofthecontinuumareofapiecewiththemiddle.</p><p>It happens that the equations of fluid flow are in many contexts</p><p>dimensionless, meaning that they apply without regard to scale. Scaled-down</p><p>airplanewingsandshippropellerscanbetestedinwindtunnelsandlaboratory</p><p>basins.And,withsomelimitations,smallstormsactlikelargestorms.</p><p>Blood vessels, from aorta to capillaries, form another kind of continuum.</p><p>Theybranchanddivideandbranchagainuntiltheybecomesonarrowthatblood</p><p>cells are forced to slide through single file. The nature of their branching is</p><p>fractal. Their structure resembles one of the monstrous imaginary objects</p><p>conceivedbyMandelbrot’s turn-of–the-centurymathematicians.Asamatterof</p><p>physiologicalnecessity,bloodvesselsmustperformabitofdimensionalmagic.</p><p>Just as the Koch curve, for example, squeezes a line of infinite length into a</p><p>small area, the circulatory system must squeeze a huge surface area into a</p><p>limitedvolume.Intermsofthebody’sresources,bloodisexpensiveandspaceis</p><p>atapremium.Thefractalstructurenaturehasdevisedworkssoefficientlythat,</p><p>inmost tissue,nocell isevermore than threeorfourcellsawayfromablood</p><p>vessel.Yet thevesselsandblood takeup little space,nomore thanabout five</p><p>percent of the body. It is, as Mandelbrot put it, the Merchant of Venice</p><p>Syndrome—notonlycan’tyoutakeapoundoffleshwithoutspillingblood,you</p><p>can’ttakeamilligram.</p><p>This exquisite</p><p>structure—actually, two intertwining trees of veins and</p><p>arteries—isfarfromexceptional.Thebodyisfilledwithsuchcomplexity.Inthe</p><p>digestive tract, tissue reveals undulations within undulations. The lungs, too,</p><p>need topack thegreatestpossiblesurface into thesmallest space.Ananimal’s</p><p>abilitytoabsorboxygenisroughlyproportionaltothesurfaceareaofitslungs.</p><p>Typicalhumanlungspackinasurfacebiggerthanatenniscourt.Asanadded</p><p>complication,thelabyrinthofwindpipesmustmergeefficientlywiththearteries</p><p>andveins.</p><p>Every medical student knows that lungs are designed to accommodate a</p><p>hugesurfacearea.Butanatomistsaretrainedtolookatonescaleatatime—for</p><p>example, at themillionsof alveoli,microscopic sacs, that end the sequenceof</p><p>branching pipes. The language of anatomy tends to obscure the unity across</p><p>scales.Thefractalapproach,bycontrast,embracesthewholestructureinterms</p><p>ofthebranchingthatproducesit,branchingthatbehavesconsistentlyfromlarge</p><p>scales to small.Anatomists study thevas-culatory systembyclassifyingblood</p><p>vesselsintocategoriesbasedonsize—arteriesandarterioles,veinsandvenules.</p><p>Forsomepurposes, thosecategoriesproveuseful.But forothers theymislead.</p><p>Sometimes the textbook approach seems to dance around the truth: “In the</p><p>gradualtransitionfromonetypeofarterytoanotheritissometimesdifficultto</p><p>classifytheintermediateregion.Somearteriesofintermediatecaliberhavewalls</p><p>that suggest larger arteries,while some large arteries havewalls like those of</p><p>medium-sizedarteries.Thetransitionalregions…areoftendesignatedarteriesof</p><p>mixedtype.”</p><p>Notimmediately,butadecadeafterMandelbrotpublishedhisphysiological</p><p>speculations, some theoretical biologists began to find fractal organization</p><p>controlling structures all through the body. The standard “exponential”</p><p>description of bronchial branching proved to be quite wrong; a fractal</p><p>description turned out to fit the data. The urinary collecting system proved</p><p>fractal.Thebiliaryduct in the liver.Thenetworkof special fibers in theheart</p><p>thatcarrypulsesofelectriccurrenttothecontractingmuscles.Thelaststructure,</p><p>known to heart specialists as theHis-Purkinje network, inspired a particularly</p><p>important line of research.Considerablework on healthy and abnormal hearts</p><p>turnedout tohingeon thedetailsofhow themusclecellsof the leftand right</p><p>pumpingchambersallmanagetocoordinatetheirtiming.Severalchaos-minded</p><p>cardiologists found that the frequency spectrum of heartbeat timing, like</p><p>earthquakes and economic phenomena, followed fractal laws, and they argued</p><p>thatonekeytounderstandingheartbeattimingwasthefractalorganizationofthe</p><p>His-Purkinje network, a labyrinth of branchingpathwaysorganized to be self-</p><p>similaronsmallerandsmallerscales.</p><p>How did nature manage to evolve such complicated architecture?</p><p>Mandelbrot’s point is that the complications exist only in the context of</p><p>traditional Euclidean geometry. As fractals, branching structures can be</p><p>described with transparent simplicity, with just a few bits of information.</p><p>PerhapsthesimpletransformationsthatgaverisetotheshapesdevisedbyKoch,</p><p>Peano, and Sierpiński have their analogue in the coded instructions of an</p><p>organism’s genes. DNA surely cannot specify the vast number of bronchi,</p><p>bronchioles, and alveoli or the particular spatial structure of the resulting tree,</p><p>but it can specify a repeating process of bifurcation and development. Such</p><p>processes suitnature’spurposes.WhenE. I.DuPontdeNemours&Company</p><p>andtheUnitedStatesArmyfinallybegantoproduceasyntheticmatchforgoose</p><p>down,itwasbyfinallyrealizingthatthephenomenalair-trappingabilityofthe</p><p>naturalproductcamefromthefractalnodesandbranchesofdown’skeyprotein,</p><p>keratin.Mandelbrotglidedmatter-of-factlyfrompulmonaryandvasculartreesto</p><p>realbotanical trees, trees thatneed tocapture sunand resistwind,with fractal</p><p>branches and fractal leaves.And theoretical biologists began to speculate that</p><p>fractal scaling was not just common but universal in morphogenesis. They</p><p>argued that understanding how such patternswere encoded and processed had</p><p>becomeamajorchallengetobiology.</p><p>“I STARTED LOOKING in the trash cans of science for such phenomena,</p><p>becauseIsuspectedthatwhatIwasobservingwasnotanexceptionbutperhaps</p><p>very widespread. I attended lectures and looked in unfashionable periodicals,</p><p>mostof themof littleornoyield,butonceinawhilefindingsomeinteresting</p><p>things.Inawayitwasanaturalist’sapproach,notatheoretician’sapproach.But</p><p>mygamblepaidoff.”</p><p>Having consolidated a life’s collection of ideas about nature and</p><p>mathematical history into one book, Mandelbrot found an unaccustomed</p><p>measure of academic success. He became a fixture of the scientific lecture</p><p>circuit,withhisindispensabletraysofcolorslidesandhiswispywhitehair.He</p><p>begantowinprizesandotherprofessionalhonors,andhisnamebecameaswell</p><p>known to the nonscientific public as any mathematician’s. In part that was</p><p>becauseoftheaestheticappealofhisfractalpictures;inpartbecausethemany</p><p>thousands of hobbyists with microcomputers could begin exploring his world</p><p>themselves.Inpartitwasbecauseheputhimselfforward.Hisnameappearedon</p><p>a little list compiled by the Harvard historian of science I. Bernard Cohen.</p><p>Cohenhadscouredtheannalsofdiscoveryforyears,lookingforscientistswho</p><p>haddeclaredtheirownworktobe“revolutions.”Alltold,hefoundjustsixteen.</p><p>RobertSymmer,aScotscontemporaryofBenjaminFranklinwhoseideasabout</p><p>electricitywere indeed radical,butwrong. Jean-PaulMarat,known todayonly</p><p>for his bloody contribution to the French Revolution. Von Liebig. Hamilton.</p><p>CharlesDarwin, of course.Virchow.Cantor. Einstein.Minkowski.VonLaue.</p><p>AlfredWegener—continentaldrift.Compton.Just.JamesWatson—thestructure</p><p>ofDNA.AndBenoitMandelbrot.</p><p>To pure mathematicians, however, Mandelbrot remained an outsider,</p><p>contendingasbitterlyas everwith thepoliticsof science.At theheightofhis</p><p>success, he was reviled by some colleagues, who thought he was unnaturally</p><p>obsessedwithhisplaceinhistory.Theysaidhehectoredthemaboutgivingdue</p><p>credit. Unquestionably, in his years as a professional heretic he honed an</p><p>appreciation for the tactics aswell as the substance of scientific achievement.</p><p>Sometimeswhenarticlesappearedusing ideasfromfractalgeometryhewould</p><p>callorwrite theauthors tocomplain thatnoreferencewasmade tohimorhis</p><p>book.</p><p>Hisadmirers foundhisegoeasy toforgive,considering thedifficultieshe</p><p>hadovercome in getting recognition for hiswork. “Of course, he is a bit of a</p><p>megalomaniac, he has this incredible ego, but it’s beautiful stuff he does, so</p><p>mostpeoplelethimgetawaywithit,”onesaid.Inthewordsofanother:“Hehad</p><p>so many difficulties with his fellow mathematicians that simply in order to</p><p>survive he had to develop this strategy of boosting his own ego. If he hadn’t</p><p>donethat, ifhehadn’tbeensoconvincedthathehadtherightvisions, thenhe</p><p>wouldneverhavesucceeded.”</p><p>Thebusinessoftakingandgivingcreditcanbecomeobsessiveinscience.</p><p>Mandelbrotdidplentyofboth.Hisbookringswith thefirstperson:Iclaim…I</p><p>conceived and developed…and implemented…I have confirmed…I show…I</p><p>coined…Inmy travels through newly opened or newly settled territory,</p><p>ofacoursechange</p><p>forallofphysics.Thefieldhadbeendominated longenough, theyfelt,by the</p><p>glitteringabstractionsofhigh-energyparticlesandquantummechanics.</p><p>The cosmologist Stephen Hawking, occupant of Newton’s chair at</p><p>Cambridge University, spoke for most of physics when he took stock of his</p><p>scienceina1980lecturetitled“IstheEndinSightforTheoreticalPhysics?”</p><p>“Wealreadyknowthephysicallawsthatgoverneverythingweexperience</p><p>ineverydaylife….Itisatributetohowfarwehavecomeintheoreticalphysics</p><p>that itnowtakesenormousmachinesandagreatdealofmoney toperforman</p><p>experimentwhoseresultswecannotpredict.”</p><p>YetHawkingrecognized thatunderstandingnature’s lawson the termsof</p><p>particlephysicsleftunansweredthequestionofhowtoapplythoselawstoany</p><p>butthesimplestofsystems.Predictabilityisonethinginacloudchamberwhere</p><p>twoparticlescollideat theendofaracearoundanaccelerator.It issomething</p><p>elsealtogetherinthesimplesttubofroilingfluid,orintheearth’sweather,orin</p><p>thehumanbrain.</p><p>Hawking’s physics, efficiently gathering upNobel Prizes and bigmoney</p><p>for experiments, hasoftenbeen called a revolution.At times it seemedwithin</p><p>reach of that grail of science, the Grand Unified Theory or “theory of</p><p>everything.”Physicshadtracedthedevelopmentofenergyandmatterinallbut</p><p>the first eyeblinkof theuniverse’shistory.Butwaspostwarparticlephysics a</p><p>revolution? Or was it just the fleshing out of the framework laid down by</p><p>Einstein, Bohr, and the other fathers of relativity and quantum mechanics?</p><p>Certainly,theachievementsofphysics,fromtheatomicbombtothetransistor,</p><p>changed the twentieth-century landscape.Yet ifanything, thescopeofparticle</p><p>physics seemed to have narrowed.Twogenerations hadpassed since the field</p><p>producedanewtheoreticalideathatchangedthewaynonspecialistsunderstand</p><p>theworld.</p><p>The physics described by Hawking could complete its mission without</p><p>answeringsomeofthemostfundamentalquestionsaboutnature.Howdoeslife</p><p>begin?What is turbulence?Aboveall, inauniverseruledbyentropy,drawing</p><p>inexorably toward greater and greater disorder, how does order arise? At the</p><p>same time, objects of everyday experience like fluids andmechanical systems</p><p>cametoseemsobasicandsoordinarythatphysicistshadanaturaltendencyto</p><p>assumetheywerewellunderstood.Itwasnotso.</p><p>As the revolution in chaos runs its course, the best physicists find</p><p>themselves returningwithout embarrassment to phenomena on a human scale.</p><p>They study not just galaxies but clouds. They carry out profitable computer</p><p>research not just on Crays but on Macintoshes. The premier journals print</p><p>articlesonthestrangedynamicsofaballbouncingonatablesidebysidewith</p><p>articles on quantum physics. The simplest systems are now seen to create</p><p>extraordinarily difficult problems of predictability. Yet order arises</p><p>spontaneouslyinthosesystems—chaosandordertogether.Onlyanewkindof</p><p>sciencecouldbegintocrossthegreatgulfbetweenknowledgeofwhatonething</p><p>does—one water molecule, one cell of heart tissue, one neuron—and what</p><p>millionsofthemdo.</p><p>Watch twobitsof foamflowingsidebysideat thebottomofawaterfall.</p><p>Whatcanyouguessabouthowclose theywereat the top?Nothing.As faras</p><p>standard physics was concerned, Godmight just as well have taken all those</p><p>water molecules under the table and shuffled them personally. Traditionally,</p><p>when physicists saw complex results, they looked for complex causes.When</p><p>they saw a random relationship between what goes into a system and what</p><p>comes out, they assumed that they would have to build randomness into any</p><p>realistictheory,byartificiallyaddingnoiseorerror.Themodernstudyofchaos</p><p>beganwiththecreepingrealizationinthe1960sthatquitesimplemathematical</p><p>equations could model systems every bit as violent as a waterfall. Tiny</p><p>differencesininputcouldquicklybecomeoverwhelmingdifferencesinoutput—</p><p>aphenomenongiven thename“sensitivedependenceon initial conditions.” In</p><p>weather,forexample,thistranslatesintowhatisonlyhalf-jokinglyknownasthe</p><p>ButterflyEffect—thenotionthatabutterflystirringtheairtodayinPekingcan</p><p>transformstormsystemsnextmonthinNewYork.</p><p>Whentheexplorersofchaosbegantothinkbackonthegenealogyoftheir</p><p>newscience,theyfoundmanyintellectualtrailsfromthepast.Butonestoodout</p><p>clearly. For the youngphysicists andmathematicians leading the revolution, a</p><p>startingpointwastheButterflyEffect.</p><p>TheButterfly</p><p>Effect</p><p>Physicists like to think thatallyouhave todo issay, theseare theconditions,</p><p>nowwhathappensnext?</p><p>—RICHARDP.FEYNMAN</p><p>THESUNBEATDOWN throughasky thathadneverseenclouds.Thewinds</p><p>sweptacrossanearthassmoothasglass.Nightnevercame,andautumnnever</p><p>gavewaytowinter.Itneverrained.ThesimulatedweatherinEdwardLorenz’s</p><p>new electronic computer changed slowly but certainly, drifting through a</p><p>permanentdrymiddaymidseason, as if theworldhad turned intoCamelot, or</p><p>someparticularlyblandversionofsouthernCalifornia.</p><p>Outside his window Lorenz could watch real weather, the early-morning</p><p>fogcreepingalongtheMassachusettsInstituteofTechnologycampusorthelow</p><p>cloudsslippingovertherooftopsfromtheAtlantic.Fogandcloudsneverarose</p><p>in the model running on his computer. The machine, a RoyalMcBee, was a</p><p>thicket of wiring and vacuum tubes that occupied an ungainly portion of</p><p>Lorenz’s office,made a surprising and irritating noise, and broke down every</p><p>week or so. It had neither the speed nor the memory to manage a realistic</p><p>simulation of the earth’s atmosphere and oceans. Yet Lorenz created a toy</p><p>weatherin1960thatsucceededinmesmerizinghiscolleagues.Everyminutethe</p><p>machinemarked the passing of a day by printing a row of numbers across a</p><p>page.Ifyouknewhowtoreadtheprintouts,youwouldseeaprevailingwesterly</p><p>windswingnowtothenorth,nowtothesouth,nowbacktothenorth.Digitized</p><p>cyclones spun slowly around an idealized globe. As word spread through the</p><p>department, the other meteorologists would gather around with the graduate</p><p>students, making bets on what Lorenz’s weather would do next. Somehow,</p><p>nothingeverhappenedthesamewaytwice.</p><p>Lorenz enjoyed weather—by no means a prerequisite for a research</p><p>meteorologist. He savored its changeability. He appreciated the patterns that</p><p>comeandgointheatmosphere,familiesofeddiesandcyclones,alwaysobeying</p><p>mathematicalrules,yetneverrepeatingthemselves.Whenhe lookedatclouds,</p><p>hethoughthesawakindofstructureinthem.Oncehehadfearedthatstudying</p><p>the science of weather would be like prying a jack-in–the-box apart with a</p><p>screwdriver.Nowhewonderedwhethersciencewouldbeable topenetrate the</p><p>magicatall.Weatherhadaflavorthatcouldnotbeexpressedbytalkingabout</p><p>averages.ThedailyhightemperatureinCambridge,Massachusetts,averages75</p><p>degrees in June.Thenumberof rainydays inRiyadh,SaudiArabia,averages</p><p>ten a year. Those were statistics. The essence was the way patterns in the</p><p>atmospherechangedovertime,andthatwaswhatLorenzcapturedontheRoyal</p><p>McBee.</p><p>Hewasthegodofthismachineuniverse,freetochoosethelawsofnature</p><p>as he pleased. After a certain amount of undivine trial and error, he chose</p><p>twelve.Theywere numerical rules—equations that expressed the relationships</p><p>between temperature</p><p>1was</p><p>oftenmovedtoexerttherightofnamingitslandmarks.</p><p>Many scientists failed to appreciate this kind of style. Nor were they</p><p>mollified that Mandelbrot was equally copious with his references to</p><p>predecessors, some thoroughlyobscure. (Andall, ashisdetractorsnoted,quite</p><p>safelydeceased.)Theythoughtitwasjusthiswayoftryingtopositionhimself</p><p>squarelyinthecenter,settinghimselfuplikethePope,castinghisbenedictions</p><p>fromonesideofthefieldtotheother.Theyfoughtback.Scientistscouldhardly</p><p>avoid the word fractal, but if they wanted to avoid Mandelbrot’s name they</p><p>couldspeakoffractionaldimensionasHausdorff-Besicovitchdimension.They</p><p>also—particularly mathematicians—resented the way he moved in and out of</p><p>different disciplines, making his claims and conjectures and leaving the real</p><p>workofprovingthemtoothers.</p><p>It was a legitimate question. If one scientist announces that a thing is</p><p>probablytrue,andanotherdemonstratesitwithrigor,whichonehasdonemore</p><p>toadvancescience?Is themakingofaconjectureanactofdiscovery?Oris it</p><p>justacold-bloodedstakingofaclaim?Mathematicianshavealwaysfacedsuch</p><p>issues,butthedebatebecamemoreintenseascomputersbegantoplaytheirnew</p><p>role. Those who used computers to conduct experiments became more like</p><p>laboratory scientists,playingby rules thatalloweddiscoverywithout theusual</p><p>theorem-proof,theorem-proofofthestandardmathematicspaper.</p><p>Mandelbrot’s book was wide-ranging and stuffed with the minutiae of</p><p>mathematicalhistory.Whereverchaosled,Mandelbrothadsomebasistoclaim</p><p>that he had been there first. Little did it matter that most readers found his</p><p>referencesobscureorevenuseless.Theyhadtoacknowledgehisextraordinary</p><p>intuition for the direction of advances in fields he had never actually studied,</p><p>from seismology to physiology. It was sometimes uncanny, and sometimes</p><p>irritating. Even an admirer would cry with exasperation, “Mandelbrot didn’t</p><p>haveeverybody’sthoughtsbeforetheydid.”</p><p>It hardlymatters. The face of genius need not alwayswear anEinstein’s</p><p>saintlikemien.Yetfordecades,Mandelbrotbelieves,hehadtoplaygameswith</p><p>hiswork.Hehad tocouchoriginal ideas in terms thatwouldnotgiveoffense.</p><p>He had to delete his visionary-sounding prefaces to get his articles published.</p><p>Whenhewrotethefirstversionofhisbook,publishedinFrenchin1975,hefelt</p><p>he was forced to pretend it contained nothing too startling. That waswhy he</p><p>wrote the latest version explicitly as “a manifesto and a casebook.” He was</p><p>copingwiththepoliticsofscience.</p><p>THECOMPLEXBOUNDARIESOFNEWTON’SMETHOD.Theattractingpullof fourpoints—in the</p><p>fourdarkholes—creates“basinsofattraction,”eachadifferentcolor,withacomplicatedfractalboundary.</p><p>TheimagerepresentsthewayNewton’smethodforsolvingequationsleadsfromdifferentstartingpointsto</p><p>oneoffourpossiblesolutions(inthiscasetheequationisx4-1=0).</p><p>FRACTALCLUSTERS.Arandomclusteringofpraticlesgeneratedbyacomputerproducesa“percolation</p><p>network,”oneofmanyvisualmodelsinspiredbyfactalgeometry.Appliedphysicistsdiscoveredthatsuch</p><p>modelsimitateavarietyofreal-worldprocesses,suchastheformationofpolymersandthediffusionofoil</p><p>through factured rock. Each color in the percolation network represents a grouping that is connected</p><p>throughout.</p><p>THEGREATREDSPOT:REALANDSIMULATED.TheVoyagersatelliterevealedJupiter’ssurfaceisa</p><p>seething,turbulentfluid,withhorizontalbandsofeast-westflow.TheGreatRedSpotisseenfromabove</p><p>theplanet’sequatorandalsoinaviewlookingdownontheSouthPole.</p><p>ComputergraphicsfromPhillipMarcus’ssimulationpresenttheSouthPoleview.Thecolorshowsthe</p><p>directionofspinforparticularpiecesoffluid:piecesturningcounterclockwisearered,andpiecesturning</p><p>clockwiseareblue.Nomatterwhatthestaringconfiguration,clumpsofbluetendtobreadup,whilethered</p><p>tendsotmergeintoasinglespot,stableandcoherentamitthesurroundingtumult.</p><p>“ThepoliticsaffectedthestyleinasensewhichIlatercametoregret.Iwas</p><p>saying,‘It’snaturalto…,It’saninterestingobservationthat….’Now,infact,it</p><p>wasanythingbutnatural,andtheinterestingobservationwasinfacttheresultof</p><p>very long investigations and search for proof and self-criticism. It had a</p><p>philosophicalandremovedattitudewhichIfeltwasnecessarytogetitaccepted.</p><p>Thepoliticswas that, if I said Iwasproposinga radicaldeparture, thatwould</p><p>havebeentheendofthereaders’interest.</p><p>“Lateron,Igotbacksomesuchstatements,peoplesaying,‘Itisnaturalto</p><p>observe…’ThatwasnotwhatIhadbargainedfor.”</p><p>Looking back, Mandelbrot saw that scientists in various disciplines</p><p>respondedtohisapproachinsadlypredictablestages.Thefirststagewasalways</p><p>thesame:Whoareyouandwhyareyou interested inour field?Second:How</p><p>doesitrelatetowhatwehavebeendoing,andwhydon’tyouexplainitonthe</p><p>basisofwhatweknow?Third:Areyou sure it’s standardmathematics? (Yes,</p><p>I’msure.)Thenwhydon’tweknowit?(Becauseit’sstandardbutveryobscure.)</p><p>Mathematicsdiffersfromphysicsandotherappliedsciencesinthisrespect.</p><p>A branch of physics, once it becomes obsolete or unproductive, tends to be</p><p>foreverpart of thepast. Itmaybe ahistorical curiosity, perhaps the sourceof</p><p>someinspirationtoamodernscientist,butdeadphysicsisusuallydeadforgood</p><p>reason.Mathematics, by contrast, is full of channels and byways that seem to</p><p>lead nowhere in one era and become major areas of study in another. The</p><p>potentialapplicationofapieceofpure thoughtcanneverbepredicted.That is</p><p>why mathematicians value work in an aesthetic way, seeking elegance and</p><p>beauty as artists do. It is alsowhyMandelbrot, inhis antiquarianmode, came</p><p>acrosssomuchgoodmathematicsthatwasreadytobedustedoff.</p><p>So the fourth stage was this: What do people in these branches of</p><p>mathematicsthinkaboutyourwork?(Theydon’tcare,becauseitdoesn’taddto</p><p>themathematics.Infact,theyaresurprisedthattheirideasrepresentnature.)</p><p>In the end, the word fractal came to stand for a way of describing,</p><p>calculating,andthinkingaboutshapesthatareirregularandfragmented,jagged</p><p>and broken-up—shapes from the crystalline curves of snowflakes to the</p><p>discontinuousdustsofgalaxies.Afractalcurveimpliesanorganizingstructure</p><p>that lies hidden among the hideous complication of such shapes. High school</p><p>studentscouldunderstandfractalsandplaywiththem;theywereasprimaryas</p><p>theelementsofEuclid.Simplecomputerprogramstodrawfractalpicturesmade</p><p>theroundsofpersonalcomputerhobbyists.</p><p>Mandelbrot found his most enthusiastic acceptance among applied</p><p>scientistsworkingwithoilor rockormetals,particularly incorporate research</p><p>centers.Bythemiddleofthe1980s,vastnumbersofscientistsatExxon’shuge</p><p>researchfacility,forexample,workedonfractalproblems.AtGeneralElectric,</p><p>fractals became an organizing principle in the study of polymers and also—</p><p>thoughthisworkwasconductedsecretly—inproblemsofnuclearreactorsafety.</p><p>InHollywood,fractalsfoundtheirmostwhimsicalapplicationinthecreationof</p><p>phenomenallyrealisticlandscapes,earthlyandextraterrestrial,inspecialeffects</p><p>formovies.</p><p>Thepatterns that people likeRobertMayand JamesYorkediscovered in</p><p>the early 1970s, with their complex boundaries between orderly and chaotic</p><p>behavior,hadunsuspectedregularities thatcould</p><p>onlybedescribed in termsof</p><p>the relation of large scales to small. The structures that provided the key to</p><p>nonlineardynamicsproved tobe fractal.Andon themost immediatepractical</p><p>level, fractal geometry also provided a set of tools that were taken up by</p><p>physicists, chemists, seismologists, metallurgists, probability theorists and</p><p>physiologists. These researchers were convinced, and they tried to convince</p><p>others,thatMandelbrot’snewgeometrywasnature’sown.</p><p>Theymadean irrefutable impactonorthodoxmathematicsandphysicsas</p><p>well,butMandelbrothimselfnevergainedthefullrespectofthosecommunities.</p><p>Evenso,theyhadtoacknowledgehim.Onemathematiciantoldfriendsthathe</p><p>had awakened one night still shaking from a nightmare. In this dream, the</p><p>mathematician was dead, and suddenly heard the unmistakable voice of God.</p><p>“Youknow,”Heremarked,“therereallywassomethingtothatMandelbrot.”</p><p>THENOTIONOFSELF-SIMILARITYstrikesancientchordsinourculture.Anold</p><p>straininWesternthoughthonorstheidea.Leibnizimaginedthatadropofwater</p><p>contained awhole teeming universe, containing, in turn,water drops and new</p><p>universeswithin.“Toseetheworldinagrainofsand,”Blakewrote,andoften</p><p>scientists were predisposed to see it.When spermwere first discovered, each</p><p>wasthoughttobeahomunculus,ahuman,tinybutfullyformed.</p><p>But self-similaritywithered as a scientific principle, for a good reason. It</p><p>did not fit the facts. Sperm are notmerely scaled-downhumans—they are far</p><p>more interesting than that—and the process of ontogenetic development is far</p><p>moreinterestingthanmereenlargement.Theearlysenseofself-similarityasan</p><p>organizingprinciplecamefromthelimitationsonthehumanexperienceofscale.</p><p>Howelsetoimaginetheverygreatandverysmall,theveryfastandveryslow,</p><p>butasextensionsoftheknown?</p><p>Themythdiedhard as thehumanvisionwas extendedby telescopes and</p><p>microscopes. The first discoverieswere realizations that each change of scale</p><p>brought new phenomena and new kinds of behavior. For modern particle</p><p>physicists,theprocesshasneverended.Everynewaccelerator,withitsincrease</p><p>inenergyandspeed,extendsscience’sfieldofviewtotinierparticlesandbriefer</p><p>timescales,andeveryextensionseemstobringnewinformation.</p><p>Atfirstblush,theideaofconsistencyonnewscalesseemstoprovideless</p><p>information.Inpart, thatisbecauseaparallel trendinsciencehasbeentoward</p><p>reductionism.Scientistsbreakthingsapartandlookatthemoneatatime.Ifthey</p><p>want to examine the interaction of subatomic particles, they put two or three</p><p>together. There is complication enough. The power of self-similarity, though,</p><p>begins at much greater levels of complexity. It is a matter of looking at the</p><p>whole.</p><p>AlthoughMandelbrot made themost comprehensive geometric use of it,</p><p>the return of scaling ideas to science in the 1960s and 1970s became an</p><p>intellectual current that made itself felt simultaneously in many places. Self-</p><p>similarity was implicit in Edward Lorenz’s work. It was part of his intuitive</p><p>understandingofthefinestructureofthemapsmadebyhissystemofequations,</p><p>a structure he could sense but not see on the computers available in 1963.</p><p>Scalingalsobecamepartofamovementinphysicsthatled,moredirectlythan</p><p>Mandelbrot’sownwork,tothedisciplineknownaschaos.Evenindistantfields,</p><p>scientistswere beginning to think in termsof theories that usedhierarchies of</p><p>scales,asinevolutionarybiology,whereitbecameclearthatafulltheorywould</p><p>havetorecognizepatternsofdevelopmentingenes,inindividualorganisms,in</p><p>species,andinfamiliesofspecies,allatonce.</p><p>Paradoxically, perhaps, the appreciation of scaling phenomenamust have</p><p>come from the same kind of expansion of human vision that had killed the</p><p>earliernaïveideasofself-similarity.Bythelatetwentiethcentury,inwaysnever</p><p>beforeconceivable,imagesoftheincomprehensiblysmallandtheunimaginably</p><p>large became part of everyone’s experience. The culture saw photographs of</p><p>galaxiesandofatoms.Noonehadtoimagine,withLeibniz,whattheuniverse</p><p>mightbelikeonmicroscopicor telescopicscales—microscopesandtelescopes</p><p>madethoseimagespartofeverydayexperience.Giventheeagernessofthemind</p><p>to find analogies in experience, new kinds of comparison between large and</p><p>smallwereinevitable—andsomeofthemwereproductive.</p><p>Often the scientists drawn to fractal geometry felt emotional parallels</p><p>betweentheirnewmathematicalaestheticandchangesintheartsinthesecond</p><p>half of the century. They felt that they were drawing some inner enthusiasm</p><p>fromthecultureatlarge.ToMandelbrottheepitomeoftheEuclideansensibility</p><p>outsidemathematicswas thearchitectureof theBauhaus. Itmight just aswell</p><p>have been the style of painting best exemplified by the color squares of Josef</p><p>Albers:spare,orderly, linear,reductionist,geometrical.Geometrical—theword</p><p>means what it has meant for thousands of years. Buildings that are called</p><p>geometrical are composed of simple shapes, straight lines and circles,</p><p>describablewithjustafewnumbers.Thevogueforgeometricalarchitectureand</p><p>paintingcameandwent.Architectsnolongercaretobuildblockishskyscrapers</p><p>like the Seagram Building in New York, once much hailed and copied. To</p><p>Mandelbrot and his followers the reason is clear. Simple shapes are inhuman.</p><p>Theyfailtoresonatewiththewaynatureorganizesitselforwiththewayhuman</p><p>perceptionseestheworld.InthewordsofGertEilenberger,aGermanphysicist</p><p>whotookupnonlinearscienceafterspecializinginsuperconductivity:“Whyisit</p><p>thatthesilhouetteofastorm-bentleaflesstreeagainstaneveningskyinwinter</p><p>isperceivedasbeautiful,butthecorrespondingsilhouetteofanymulti-purpose</p><p>university building is not, in spite of all efforts of the architect? The answer</p><p>seemstome,evenifsomewhatspeculative,tofollowfromthenewinsightsinto</p><p>dynamical systems. Our feeling for beauty is inspired by the harmonious</p><p>arrangement of order and disorder as it occurs in natural objects—in clouds,</p><p>trees,mountainranges,orsnowcrystals.Theshapesofallthesearedynamical</p><p>processes jelled into physical forms, and particular combinations of order and</p><p>disorderaretypicalforthem.”</p><p>Ageometrical shapehas a scale, a characteristic size.ToMandelbrot, art</p><p>that satisfies lacks scale, in the sense that it contains important elements at all</p><p>sizes. Against the Seagram Building, he offers the architecture of the Beaux-</p><p>Arts,withitssculpturesandgargoyles,itsquoinsandjambstones,itscartouches</p><p>decorated with scrollwork, its cornices topped with cheneaux and lined with</p><p>dentils.ABeaux-Artsparagonlike theParisOperahasnoscalebecause ithas</p><p>everyscale.Anobserverseeingthebuildingfromanydistancefindssomedetail</p><p>that draws the eye. The composition changes as one approaches and new</p><p>elementsofthestructurecomeintoplay.</p><p>Appreciating the harmonious structure of any architecture is one thing;</p><p>admiringthewildnessofnatureisquiteanother.Intermsofaestheticvalues,the</p><p>new mathematics of fractal geometry brought hard science in tune with the</p><p>peculiarlymodern feeling for untamed, uncivilized, undomesticated nature.At</p><p>onetimerainforests,deserts,bush,andbadlandsrepresentedallthatsocietywas</p><p>strivingtosubdue.Ifpeoplewantedaestheticsatisfactionfromvegetation,they</p><p>lookedat</p><p>gardens.AsJohnFowlesputit,writingofeighteenth-centuryEngland:</p><p>“The period had no sympathy with unregulated or primordial nature. It was</p><p>aggressivewilderness, an ugly and all-invasive reminder of theFall, ofman’s</p><p>eternal exile from theGardenofEden….Even its natural sciences…remained</p><p>essentially hostile to wild nature, seeing it only as something to be tamed,</p><p>classified,utilised, exploited.”By theendof the twentiethcentury, theculture</p><p>hadchanged,andnowsciencewaschangingwithit.</p><p>Sosciencefoundauseafterallfortheobscureandfancifulcousinsofthe</p><p>CantorsetandtheKochcurve.Atfirst,theseshapescouldhaveservedasitems</p><p>of evidence in the divorce proceedings betweenmathematics and the physical</p><p>sciences at the turn of the century, the end of a marriage that had been the</p><p>dominating theme of science since Newton. Mathematicians like Cantor and</p><p>Koch had delighted in their originality. They thought they were outsmarting</p><p>nature—when actually they had not yet caught upwith nature’s creation. The</p><p>prestigiousmainstreamofphysics,too,turnedawayfromtheworldofeveryday</p><p>experience. Only later, after Steve Smale brought mathematicians back to</p><p>dynamical systems, could a physicist say, “We have the astronomers and</p><p>mathematicians to thank for passing the field on to us, physicists, in a much</p><p>bettershapethanweleftittothem,70yearsago.”</p><p>Yet,despiteSmaleanddespiteMandelbrot,itwastobethephysicistsafter</p><p>all whomade a new science of chaos.Mandelbrot provided an indispensable</p><p>languageandacatalogueofsurprisingpicturesofnature.AsMandelbrothimself</p><p>acknowledged, his program described better than it explained. He could list</p><p>elements of nature along with their fractal dimensions—seacoasts, river</p><p>networks, treebark,galaxies—andscientists coulduse thosenumbers tomake</p><p>predictions.But physicistswanted to knowmore. Theywanted to knowwhy.</p><p>There were forms in nature—not visible forms, but shapes embedded in the</p><p>fabricofmotion—waitingtoberevealed.</p><p>StrangeAttractors</p><p>Bigwhorlshavelittlewhorls</p><p>Whichfeedontheirvelocity,</p><p>Andlittlewhorlshavelesserwhorls</p><p>Andsoontoviscosity.</p><p>—LEWISF.RICHARDSON</p><p>TURBULENCEWASAPROBLEMwithpedigree.Thegreatphysicistsallthought</p><p>about it, formally or informally. A smooth flow breaks up into whorls and</p><p>eddies. Wild patterns disrupt the boundary between fluid and solid. Energy</p><p>drains rapidly from large-scale motions to small. Why? The best ideas came</p><p>from mathematicians; for most physicists, turbulence was too dangerous to</p><p>waste time on. It seemed almost unknowable. There was a story about the</p><p>quantum theorist Werner Heisenberg, on his deathbed, declaring that he will</p><p>have two questions for God: why relativity, and why turbulence. Heisenberg</p><p>says,“IreallythinkHemayhaveananswertothefirstquestion.”</p><p>Theoreticalphysicshadreachedakindofstandoffwiththephenomenonof</p><p>turbulence.Ineffect,sciencehaddrawnalineonthegroundandsaid,Beyond</p><p>thiswecannotgo.Onthenearsideof the line,wherefluidsbehave inorderly</p><p>ways, therewasplenty toworkwith.Fortunately,asmooth-flowingfluiddoes</p><p>notactasthoughithasanearlyinfinitenumberofindependentmolecules,each</p><p>capable of independentmotion. Instead, bits of fluid that start nearby tend to</p><p>remainnearby, likehorses inharness.Engineershaveworkable techniques for</p><p>calculating flow, as long as it remains calm. They use a body of knowledge</p><p>datingbacktothenineteenthcentury,whenunderstandingthemotionsofliquids</p><p>andgaseswasaproblemonthefrontlinesofphysics.</p><p>By themodern era, however, it was on the front lines no longer. To the</p><p>deeptheorists,fluiddynamicsseemedtoretainnomysterybuttheonethatwas</p><p>unapproachableeveninheaven.Thepracticalsidewassowellunderstoodthatit</p><p>could be left to the technicians. Fluid dynamics was no longer really part of</p><p>physics, the physicists would say. It was mere engineering. Bright young</p><p>physicists had better things to do. Fluid dynamicists were generally found in</p><p>universityengineeringdepartments.Apracticalinterestinturbulencehasalways</p><p>beenintheforeground,andthepracticalinterestisusuallyone-sided:makethe</p><p>turbulencegoaway. Insomeapplications, turbulence isdesirable—insidea jet</p><p>engine, for example,where efficient burning depends on rapidmixing.But in</p><p>most, turbulence means disaster. Turbulent airflow over a wing destroys lift.</p><p>Turbulent flow in an oil pipe creates stupefying drag. Vast amounts of</p><p>government and corporatemoney are staked on the design of aircraft, turbine</p><p>engines,propellers,submarinehulls,andothershapesthatmovethroughfluids.</p><p>Researchers must worry about flow in blood vessels and heart valves. They</p><p>worryabout the shapeandevolutionofexplosions.Theyworryaboutvortices</p><p>andeddies, flamesandshockwaves. In theory theWorldWarIIatomicbomb</p><p>projectwasaprobleminnuclearphysics.Inrealitythenuclearphysicshadbeen</p><p>mostly solved before the project began, and the business that occupied the</p><p>scientistsassembledatLosAlamoswasaprobleminfluiddynamics.</p><p>Whatisturbulencethen?Itisamessofdisorderatallscales,smalleddies</p><p>withinlargeones.Itisunstable.Itishighlydissipative,meaningthatturbulence</p><p>drainsenergyandcreatesdrag.It ismotionturnedrandom.Buthowdoesflow</p><p>change from smooth to turbulent? Suppose you have a perfectly smooth pipe,</p><p>withaperfectlyevensourceofwater,perfectlyshieldedfromvibrations—how</p><p>cansuchaflowcreatesomethingrandom?</p><p>Alltherulesseemtobreakdown.Whenflowissmooth,orlaminar,small</p><p>disturbances die out. But past the onset of turbulence, disturbances grow</p><p>catastrophically. This onset—this transition—became a critical mystery in</p><p>science.Thechannelbelowa rock ina streambecomesawhirlingvortex that</p><p>grows, splits off and spins downstream. A plume of cigarette smoke rises</p><p>smoothly from an ashtray, accelerating until it passes a critical velocity and</p><p>splintersintowildeddies.Theonsetofturbulencecanbeseenandmeasuredin</p><p>laboratory experiments; it can be tested for any new wing or propeller by</p><p>experimentalworkinawindtunnel;butitsnatureremainselusive.Traditionally,</p><p>knowledgegainedhasalwaysbeenspecial,notuniversal.Researchbytrialand</p><p>erroronthewingofaBoeing707aircraftcontributesnothingtoresearchbytrial</p><p>and error on the wing of an F–16 fighter. Even supercomputers are close to</p><p>helplessinthefaceofirregularfluidmotion.</p><p>Somethingshakesafluid,excitingit.Thefluid isviscous—sticky,so that</p><p>energy drains out of it, and if you stopped shaking, the fluidwould naturally</p><p>come to rest.Whenyou shake it, you add energy at low frequencies, or large</p><p>wavelengths, and the first thing to notice is that the large wavelengths</p><p>decomposeintosmallones.Eddiesform,andsmallereddieswithinthem,each</p><p>dissipatingthefluid’senergyandeachproducingacharacteristicrhythm.Inthe</p><p>1930sA.N.Kolmogorovputforwardamathematicaldescriptionthatgavesome</p><p>feeling for how these eddieswork.He imagined thewhole cascade of energy</p><p>downthroughsmallerandsmallerscalesuntilfinallyalimitisreached,whenthe</p><p>eddiesbecomesotinythattherelativelylargereffectsofviscositytakeover.</p><p>Forthesakeofacleandescription,Kolmogorovimaginedthattheseeddies</p><p>fill thewhole space of the fluid,making the fluid everywhere</p><p>the same. This</p><p>assumption, theassumptionofhomogeneity, turnsoutnot tobe true,andeven</p><p>Poincaréknewitfortyyearsearlier,havingseenattheroughsurfaceofariver</p><p>that the eddies always mix with regions of smooth flow. The vorticity is</p><p>localized.Energyactuallydissipatesonlyinpartofthespace.Ateachscale,as</p><p>youlookcloserataturbulenteddy,newregionsofcalmcomeintoview.Thus</p><p>the assumption of homogeneity givesway to the assumption of intermittency.</p><p>The intermittent picture, when idealized somewhat, looks highly fractal, with</p><p>intermixed regionsof roughnessandsmoothnessonscales runningdownfrom</p><p>thelargetothesmall.Thispicture, too,turnsouttofallsomewhatshortofthe</p><p>reality.</p><p>Closelyrelated,butquitedistinct,wasthequestionofwhathappenswhen</p><p>turbulence begins. How does a flow cross the boundary from smooth to</p><p>turbulent?Beforeturbulencebecomesfullydeveloped,whatintermediatestages</p><p>might exist? For these questions, a slightly stronger theory existed. This</p><p>orthodoxparadigmcamefromLevD.Landau,thegreatRussianscientistwhose</p><p>textonfluiddynamicsremainsastandard.TheLandaupictureisapilingupof</p><p>competing rhythms.Whenmore energy comes into a system, he conjectured,</p><p>new frequencies begin one at a time, each incompatible with the last, as if a</p><p>violin string responds to harder bowing by vibratingwith a second, dissonant</p><p>tone, and then a third, and a fourth, until the sound becomes an</p><p>incomprehensiblecacophony.</p><p>Anyliquidorgasisacollectionofindividualbits,somanythattheymayas</p><p>wellbe infinite. Ifeachpiecemoved independently, then the fluidwouldhave</p><p>infinitelymanypossibilities,infinitelymany“degreesoffreedom”inthejargon,</p><p>andtheequationsdescribingthemotionwouldhavetodealwithinfinitelymany</p><p>variables.But eachparticledoesnotmove independently—itsmotiondepends</p><p>verymuchonthemotionofitsneighbors—andinasmoothflow,thedegreesof</p><p>freedomcanbe few.Potentiallycomplexmovements remaincoupled together.</p><p>Nearbybits remainnearbyordriftapart inasmooth, linearway thatproduces</p><p>neatlinesinwind-tunnelpictures.Theparticlesinacolumnofcigarettesmoke</p><p>riseasone,forawhile.</p><p>Then confusion appears, a menagerie of mysterious wild motions.</p><p>Sometimesthesemotionsreceivednames: theoscillatory, theskewedvaricose,</p><p>the cross-roll, the knot, the zigzag. In Landau’s view, these unstable new</p><p>motions simply accumulated, one on top of another, creating rhythms with</p><p>overlapping speeds and sizes. Conceptually, this orthodox idea of turbulence</p><p>seemedtofit thefacts,andif the theorywasmathematicallyuseless—whichit</p><p>was—well, so be it. Landau’s paradigmwas away of retaining dignitywhile</p><p>throwingupthehands.</p><p>Watercoursesthroughapipe,oraroundacylinder,makingafaintsmooth</p><p>hiss. Inyourmind,you turnup thepressure.Aback-and–forth rhythmbegins.</p><p>Like a wave, it knocks slowly against the pipe. Turn the knob again. From</p><p>somewhere,asecondfrequencyenters,outofsynchronizationwiththefirst.The</p><p>rhythmsoverlap, compete, jar against one another.Already they create such a</p><p>complicated motion, waves banging against the walls, interfering with one</p><p>another, thatyoualmostcannotfollowit.Nowturnuptheknobagain.Athird</p><p>frequencyenters,thenafourth,afifth,asixth,allincommensurate.Theflowhas</p><p>become extremely complicated.Perhaps this is turbulence.Physicists accepted</p><p>thispicture,butnoonehadanyideahowtopredictwhenanincreaseinenergy</p><p>wouldcreateanewfrequency,orwhatthenewfrequencywouldbe.Noonehad</p><p>seen thesemysteriouslyarriving frequencies inanexperimentbecause, in fact,</p><p>noonehadevertestedLandau’stheoryfortheonsetofturbulence.</p><p>THEORISTSCONDUCTEXPERIMENTSwith theirbrains.Experimentershave to</p><p>use their hands, too. Theorists are thinkers, experimenters are craftsmen. The</p><p>theoristneedsnoaccomplice.Theexperimenterhastomustergraduatestudents,</p><p>cajolemachinists,flatter labassistants.Thetheoristoperates inapristineplace</p><p>freeofnoise,ofvibration,ofdirt.Theexperimenterdevelopsanintimacywith</p><p>matterasasculptordoeswithclay,battlingit,shapingit,andengagingit.The</p><p>theoristinventshiscompanions,asanaiveRomeoimaginedhisidealJuliet.The</p><p>experimenter’sloverssweat,complain,andfart.</p><p>Theyneedeachother,buttheoristsandexperimentershaveallowedcertain</p><p>inequitiestoentertheirrelationshipssincetheancientdayswheneveryscientist</p><p>wasboth.Thoughthebestexperimentersstillhavesomeofthetheoristinthem,</p><p>the converse does not hold.Ultimately, prestige accumulates on the theorist’s</p><p>sideofthetable.Inhighenergyphysics,especially,glorygoestothetheorists,</p><p>while experimenters have become highly specialized technicians, managing</p><p>expensive and complicated equipment. In the decades sinceWorldWar II, as</p><p>physics came to be defined by the study of fundamental particles, the best</p><p>publicized experimentswere those carried outwith particle accelerators. Spin,</p><p>symmetry, color, flavor—these were the glamorous abstractions. To most</p><p>laymenfollowingscience,andtomorethanafewscientists,thestudyofatomic</p><p>particles was physics. But studying smaller particles, on shorter time scales,</p><p>meanthigher levelsof energy.So themachineryneeded forgoodexperiments</p><p>grew with the years, and the nature of experimentation changed for good in</p><p>particle physics. The field was crowded, and the big experiment encouraged</p><p>teams.TheparticlephysicspapersoftenstoodoutinPhysicalReviewLetters:a</p><p>typicalauthorslistcouldtakeupnearlyone-quarterofapaper’slength.</p><p>Some experimenters, however, preferred to work alone or in pairs. They</p><p>workedwithsubstancesclosertohand.Whilesuchfieldsashydrodynamicshad</p><p>lost status, solid-state physics had gained, eventually expanding its territory</p><p>enoughtorequireamorecomprehensivename,“condensedmatterphysics”:the</p><p>physicsof stuff. Incondensedmatterphysics, themachinerywassimpler.The</p><p>gapbetweentheoristandexperimenterremainednarrower.Theoristsexpresseda</p><p>littlelesssnobbery,experimentersalittlelessdefensiveness.</p><p>Even so, perspectives differed. It was fully in character for a theorist to</p><p>interrupt anexperimenter’s lecture to ask:Wouldn’tmoredatapointsbemore</p><p>convincing?Isn’tthatgraphalittlemessy?Shouldn’tthosenumbersextendup</p><p>anddownthescaleforafewmoreordersofmagnitude?</p><p>Andinreturn,itwasfullyincharacterforHarrySwinneytodrawhimself</p><p>up to his maximum height, something around five and a half feet, and say,</p><p>“That’strue,”withamixtureofinnateLouisianacharmandacquiredNewYork</p><p>irascibility.“That’strueifyouhaveaninfiniteamountofnoise-freedata.”And</p><p>wheel dismissively back toward the blackboard, adding, “In reality, of course,</p><p>youhavealimitedamountofnoisydata.”</p><p>Swinneywasexperimentingwithstuff.Forhimtheturningpointhadcome</p><p>when hewas a graduate student at JohnsHopkins. The excitement of particle</p><p>physicswaspalpable.The inspiringMurrayGell-Manncame to talkonce,and</p><p>Swinneywascaptivated.Butwhenhelookedintowhatgraduatestudentsdid,he</p><p>discovered that they were all writing computer programs or soldering spark</p><p>chambers.Itwasthenthathebegantalkingtoanolderphysiciststartingtowork</p><p>onphasetransitions—changesfromsolidtoliquid,fromnonmagnettomagnet,</p><p>fromconductortosuperconductor.BeforelongSwinneyhadanemptyroom—</p><p>not much bigger than a closet, but it</p><p>was his alone. He had an equipment</p><p>catalogue, and he began ordering. Soon he had a table and a laser and some</p><p>refrigeratingequipmentandsomeprobes.Hedesignedanapparatustomeasure</p><p>howwellcarbondioxideconductedheataroundthecriticalpointwhereitturned</p><p>fromvapor to liquid.Mostpeople thought that the thermalconductivitywould</p><p>changeslightly.Swinney found that it changedbya factorof1,000.Thatwas</p><p>exciting—aloneinatinyroom,discoveringsomethingthatnooneelseknew.He</p><p>sawtheother-worldlylightthatshinesfromavapor,anyvapor,nearthecritical</p><p>point,thelightcalled“opalescence”becausethesoftscatteringofraysgivesthe</p><p>whiteglowofanopal.</p><p>Like so much of chaos itself, phase transitions involve a kind of</p><p>macroscopicbehavior thatseemshard topredictby lookingat themicroscopic</p><p>details. When a solid is heated, its molecules vibrate with the added energy.</p><p>Theypushoutwardagainst theirbondsandforce thesubstance toexpand.The</p><p>more heat, themore expansion.Yet at a certain temperature and pressure, the</p><p>changebecomessuddenanddiscontinuous.Aropehasbeenstretching;nowit</p><p>breaks. Crystalline form dissolves, and the molecules slide away from one</p><p>another.Theyobeyfluidlawsthatcouldnothavebeeninferredfromanyaspect</p><p>ofthesolid.Theaverageatomicenergyhasbarelychanged,butthematerial—</p><p>nowaliquid,oramagnet,orasuperconductor—hasenteredanewrealm.</p><p>GünterAhlers, atAT&TBell Laboratories inNew Jersey, had examined</p><p>the so-called superfluid transition in liquid helium, in which, as temperature</p><p>falls,thematerialbecomesasortofmagicalflowingliquidwithnoperceptible</p><p>viscosityorfriction.Othershadstudiedsuperconductivity.Swinneyhadstudied</p><p>the critical point where matter changes between liquid and vapor. Swinney,</p><p>Ahlers,PierreBergé, JerryGollub,MarzioGiglio—by themiddle1970s these</p><p>experimenters and others in the United States, France, and Italy, all from the</p><p>youngtraditionofexploringphase transitions,were lookingfornewproblems.</p><p>Asintimatelyasamailcarrierlearnsthestoopsandalleysofhisneighborhood,</p><p>theyhadlearnedthepeculiarsignpostsofsubstanceschangingtheirfundamental</p><p>state.Theyhadstudiedabrinkuponwhichmatterstandspoised.</p><p>Themarchofphasetransitionresearchhadproceededalongsteppingstones</p><p>of analogy: a nonmagnet-magnet phase transition proved to be like a liquid-</p><p>vaporphasetransition.Thefluid-superfluidphasetransitionprovedtobelikethe</p><p>conductor-superconductorphasetransition.Themathematicsofoneexperiment</p><p>appliedtomanyotherexperiments.Bythe1970stheproblemhadbeenlargely</p><p>solved. A question, though, was how far the theory could be extended.What</p><p>other changes in theworld,when examined closely,would prove to be phase</p><p>transitions?</p><p>Itwasneither themostoriginal ideanor themostobvious toapplyphase</p><p>transition techniques to flow in fluids.Not themostoriginalbecause thegreat</p><p>hydrodynamicpioneers,ReynoldsandRayleighandtheirfollowersintheearly</p><p>twentiethcentury,hadalreadynotedthatacarefullycontrolledfluidexperiment</p><p>produces a change in the quality of motion—in mathematical terms a</p><p>bifurcation.Inafluidcell,forexample,liquidheatedfromthebottomsuddenly</p><p>goesfrommotionlessnesstomotion.Physicistsweretemptedtosupposethatthe</p><p>physicalcharacterofthatbifurcationresembledthechangesinasubstancethat</p><p>fellundertherubricofphasetransitions.</p><p>Itwasnot themostobvioussortofexperimentbecause,unlike realphase</p><p>transitions, these fluid bifurcations entailed no change in the substance itself.</p><p>Instead they added a new element: motion. A still liquid becomes a flowing</p><p>liquid. Why should the mathematics of such a change correspond to the</p><p>mathematicsofacondensingvapor?</p><p>IN 1973 SWINNEY was teaching at the City College of New York. Jerry</p><p>Gollub, a serious and boyish graduate ofHarvard,was teaching atHaverford.</p><p>Haverford, amildly bucolic liberal arts college nearPhiladelphia, seemed less</p><p>thananidealplaceforaphysicisttoendup.Ithadnograduatestudentstohelp</p><p>with laboratoryworkandotherwise fill in thebottomhalfof the all-important</p><p>mentor-protégépartnership.Gollub, though, lovedteachingundergraduatesand</p><p>beganbuildingupthecollege’sphysicsdepartmentintoacenterwidelyknown</p><p>forthequalityofitsexperimentalwork.Thatyear,hetookasabbaticalsemester</p><p>andcametoNewYorktocollaboratewithSwinney.</p><p>Withtheanalogyinmindbetweenphasetransitionsandfluidinstabilities,</p><p>thetwomendecidedtoexamineaclassicsystemofliquidconfinedbetweentwo</p><p>verticalcylinders.Onecylinderrotatedinsidetheother,pullingtheliquidaround</p><p>with it. The system enclosed its flow between surfaces. Thus it restricted the</p><p>possiblemotionoftheliquidinspace,unlikejetsandwakesinopenwater.The</p><p>rotatingcylindersproducedwhatwasknownasCouette-Taylorflow.Typically,</p><p>theinnercylinderspinsinsideastationaryshell,asamatterofconvenience.As</p><p>the rotation begins and picks up speed, the first instability occurs: the liquid</p><p>formsanelegantpattern resemblinga stackof inner tubesat a service station.</p><p>Doughnut-shapedbandsappeararoundthecylinder,stackedoneatopanother.A</p><p>speckinthefluidrotatesnotjusteasttowestbutalsoupandinanddownand</p><p>outaroundthedoughnuts.Thismuchwasalreadyunderstood.G.I.Taylorhad</p><p>seenitandmeasureditin1923.</p><p>FLOWBETWEENROTATINGCYLINDERS.Thepatternedflowofwaterbetween twocylindersgave</p><p>HarrySwinneyandJerryGollubawaytolookattheonsetofturbulence.Astherateofspinisincreased,</p><p>thestructuregrowsmorecomplex.Firstthewaterformsacharacteristicpatternofflowresemblingstacked</p><p>doughnuts.Thenthedoughnutsbegintoripple.Thephysicistsusedalasertomeasurethewater’schanging</p><p>velocityaseachnewinstabilityappeared.</p><p>TostudyCouetteflow,SwinneyandGollubbuiltanapparatusthatfitona</p><p>desktop,anouterglasscylinderthesizeofaskinnycanoftennisballs,abouta</p><p>foot high and two inches across.An inner cylinder of steel slid neatly inside,</p><p>leaving just one-eighth of an inch between for water. “It was a string-and–</p><p>sealing-wax affair,” said Freeman Dyson, one of an unexpected series of</p><p>prominentsightseersinthemonthsthatfollowed.“Youhadthesetwogentlemen</p><p>in a poky little lab with essentially no money doing an absolutely beautiful</p><p>experiment.Itwasthebeginningofgoodquantitativeworkonturbulence.”</p><p>The twohad inmind a legitimate scientific task thatwouldhavebrought</p><p>them a standard bit of recognition for their work and would then have been</p><p>forgotten.SwinneyandGollubintendedtoconfirmLandau’sideafortheonset</p><p>ofturbulence.Theexperimentershadnoreasontodoubtit.Theyknewthatfluid</p><p>dynamicistsbelievedtheLandaupicture.Asphysiciststheylikeditbecauseitfit</p><p>the general picture of phase transitions, and Landau himself had provided the</p><p>most workable early framework for studying phase transitions, based on his</p><p>insight that such phenomenamight obey universal laws, with regularities that</p><p>overrodedifferencesinparticularsubstances.WhenHarrySwinneystudiedthe</p><p>liquid-vaporcriticalpointincarbondioxide,hedidsowithLandau’sconviction</p><p>thathis findingswouldcarryover to the liquid-vaporcriticalpoint inxenon—</p><p>and indeed they did. Why shouldn’t turbulence prove to be a steady</p><p>accumulationofconflictingrhythmsinamovingfluid?</p><p>Swinney and Gollub prepared to combat the messiness</p><p>of moving fluids</p><p>withanarsenalofneatexperimentaltechniquesbuiltupoveryearsofstudying</p><p>phase transitions in the most delicate of circumstances. They had laboratory</p><p>styles and measuring equipment that a fluid dynamicist would never have</p><p>imagined.Toprobe the rolling currents, theyused laser light.Abeamshining</p><p>through the water would produce a deflection, or scattering, that could be</p><p>measuredina techniquecalledlaserdoppler interferometry.Andthestreamof</p><p>datacouldbe storedandprocessedbyacomputer—adevice that in1975was</p><p>rarelyseeninatabletoplaboratoryexperiment.</p><p>Landau had said new frequencieswould appear, one at a time, as a flow</p><p>increased.“Sowereadthat,”Swinneyrecalled,“andwesaid,fine,wewilllook</p><p>at the transitions where these frequencies come in. So we looked, and sure</p><p>enough there was a very well-defined transition. We went back and forth</p><p>throughthetransition,bringingtherotationspeedofthecylinderupanddown.It</p><p>wasverywelldefined.”</p><p>When they began reporting results, Swinney and Gollub confronted a</p><p>sociologicalboundaryinscience,betweenthedomainofphysicsandthedomain</p><p>offluiddynamics.Theboundaryhadcertainvividcharacteristics.Inparticular,</p><p>it determined which bureaucracy within the National Science Foundation</p><p>controlled their financing. By the 1980s a Couette-Taylor experiment was</p><p>physicsagain,butin1973itwasjustplainfluiddynamics,andforpeoplewho</p><p>wereaccustomedtofluiddynamics, thefirstnumberscomingoutof thissmall</p><p>CityCollegelaboratoryweresuspiciouslyclean.Fluiddynamicists justdidnot</p><p>believe them.Theywerenotaccustomedtoexperiments in theprecisestyleof</p><p>phase-transitionphysics.Furthermore, intheperspectiveoffluiddynamics, the</p><p>theoreticalpointofsuchanexperimentwashardtosee.ThenexttimeSwinney</p><p>andGollub tried to getNationalScienceFoundationmoney, theywere turned</p><p>down.Somerefereesdidnotcredittheirresults,andsomesaidtherewasnothing</p><p>new.</p><p>Buttheexperimenthadneverstopped.“Therewasthetransition,verywell</p><p>defined,”Swinney said. “So thatwas great.Thenwewent on, to look for the</p><p>nextone.”</p><p>There the expected Landau sequence broke down. Experiment failed to</p><p>confirmtheory.Atthenexttransitiontheflowjumpedallthewaytoaconfused</p><p>state with no distinguishable cycles at all. No new frequencies, no gradual</p><p>buildupofcomplexity.“Whatwefoundwas,itbecamechaotic.”Afewmonths</p><p>later,alean,intenselycharmingBelgianappearedatthedoortotheirlaboratory.</p><p>DAVIDRUELLESOMETIMESSAIDthereweretwokindsofphysicists,thekind</p><p>thatgrewuptakingapartradios—thisbeinganerabeforesolid-state,whenyou</p><p>could still look at wires and orange-glowing vacuum tubes and imagine</p><p>somethingabouttheflowofelectrons—andthekindthatplayedwithchemistry</p><p>sets.Ruelleplayedwithchemistry sets,ornotquite sets in the laterAmerican</p><p>sense,butchemicals,explosiveorpoisonous,cheerfullydispensedinhisnative</p><p>northern Belgium by the local pharmacist and then mixed, stirred, heated,</p><p>crystallized,andsometimesblownupbyRuellehimself.HewasborninGhent</p><p>in1935,thesonofagymnasticsteacherandauniversityprofessoroflinguistics,</p><p>andthoughhemadehiscareer inanabstractrealmofsciencehealwayshada</p><p>taste for a dangerous side of nature that hid its surprises in cryptogamous</p><p>fungoidmushroomsorsaltpeter,sulfur,andcharcoal.</p><p>It was in mathematical physics, though, that Ruelle made his lasting</p><p>contributiontotheexplorationofchaos.By1970hehadjoinedtheInstitutdes</p><p>HautesÉtudesScientifiques,an instituteoutsideParismodeledon the Institute</p><p>for Advanced Study in Princeton. He had already developed what became a</p><p>lifelonghabitofleavingtheinstituteandhisfamilyperiodicallytotakesolitary</p><p>walks, weeks long, carrying only a backpack through empty wildernesses in</p><p>IcelandorruralMexico.Oftenhesawnoone.Whenhecameacrosshumansand</p><p>acceptedtheirhospitality—perhapsamealofmaizetortillas,withnofat,animal</p><p>or vegetable—he felt that hewas seeing theworld as it existed twomillennia</p><p>before.Whenhereturnedtotheinstitutehewouldbeginhisscientificexistence</p><p>again,hisfacejustalittlemoregaunt,theskinstretchedalittlemoretightlyover</p><p>hisroundbrowandsharpchin.RuellehadheardtalksbySteveSmaleaboutthe</p><p>horseshoemapand thechaoticpossibilitiesofdynamicalsystems.Hehadalso</p><p>thoughtaboutfluidturbulenceandtheclassicLandaupicture.Hesuspectedthat</p><p>theseideaswererelated—andcontradictory.</p><p>Ruellehadnoexperiencewithfluidflows,butthatdidnotdiscouragehim</p><p>anymorethanithaddiscouragedhismanyunsuccessfulpredecessors.“Always</p><p>nonspecialistsfindthenewthings,”hesaid.“Thereisnotanaturaldeeptheory</p><p>of turbulence. All the questions you can ask about turbulence are of a more</p><p>general nature, and therefore accessible to nonspecialists.” It was easy to see</p><p>why turbulence resisted analysis. The equations of fluid flow are nonlinear</p><p>partial differential equations, unsolvable except in special cases. Yet Ruelle</p><p>workedoutanabstractalternativetoLandau’spicture,couchedinthelanguage</p><p>ofSmale,with imagesofspaceasapliablematerial tobesqueezed,stretched,</p><p>andfoldedintoshapeslikehorseshoes.Hewroteapaperathisinstitutewitha</p><p>visitingDutchmathematician,FlorisTakens,andtheypublisheditin1971.The</p><p>style was unmistakably mathematics—physicists, beware!—meaning that</p><p>paragraphswouldbeginwithDefinitionorPropositionorProof,followedbythe</p><p>inevitablethrust:Let….</p><p>“Proposition(5.2).LetXµbeaone-parameterfamilyofCkvectorfieldson</p><p>aHilbertspaceHsuchthat…”</p><p>Yet the title claimedaconnectionwith the realworld: “On theNatureof</p><p>Turbulence,” a deliberate echo of Landau’s famous title, “On the Problem of</p><p>Turbulence.”TheclearpurposeofRuelleandTakens’sargumentwentbeyond</p><p>mathematics;theymeanttoofferasubstituteforthetraditionalviewoftheonset</p><p>of turbulence. Insteadofapilingupof frequencies, leading toan infinitudeof</p><p>independent overlapping motions, they proposed that just three independent</p><p>motions would produce the full complexity of turbulence. Mathematically</p><p>speaking,someoftheirlogicturnedouttobeobscure,wrong,borrowed,orall</p><p>three—opinionsstillvariedfifteenyearslater.</p><p>Buttheinsight,thecommentary,themarginalia,andthephysicswoveninto</p><p>the paper made it a lasting gift.Most seductive of all was an image that the</p><p>authors called a strange attractor. This phrase was psychoanalytically</p><p>“suggestive,”Ruellefelt later. Itsstatus in thestudyofchaoswassuchthathe</p><p>andTakens jousted below a polite surface for the honor of having chosen the</p><p>words.The truthwas thatneitherquite remembered,butTakens,a tall, ruddy,</p><p>fiercelyNordicman,mightsay,“DidyoueveraskGodwhetherhecreatedthis</p><p>damned universe?…I don’t remember anything…. I often create without</p><p>remembering it,”whileRuelle, thepaper’s seniorauthor,would remarksoftly,</p><p>“TakenshappenedtobevisitingIHES.Differentpeopleworkdifferently.Some</p><p>peoplewouldtrytowriteapaperallbythemselvessotheykeepallthecredit.”</p><p>The strange attractor lives in phase space, one of the most powerful</p><p>inventionsofmodernscience.Phasespacegivesawayofturningnumbersinto</p><p>pictures,abstractingeverybitofessentialinformationfromasystemofmoving</p><p>parts,mechanicalorfluid,andmakingaflexibleroadmaptoallitspossibilities.</p><p>Physicists alreadyworkedwith two simpler</p><p>kinds of “attractors”: fixed points</p><p>and limit cycles, representing behavior that reached a steady state or repeated</p><p>itselfcontinuously.</p><p>Inphasespacethecompletestateofknowledgeaboutadynamicalsystem</p><p>atasingleinstantintimecollapsestoapoint.Thatpointisthedynamicalsystem</p><p>—atthatinstant.Atthenextinstant,though,thesystemwillhavechanged,ever</p><p>so slightly, and so the point moves. The history of the system time can be</p><p>charted by the moving point, tracing its orbit through phase space with the</p><p>passageoftime.</p><p>How can all the information about a complicated system be stored in a</p><p>point? If thesystemhasonly twovariables, theanswer is simple. It is straight</p><p>from the Cartesian geometry taught in high school—one variable on the</p><p>horizontalaxis,theotheronthevertical.Ifthesystemisaswinging,frictionless</p><p>pendulum, one variable is position and the other velocity, and they change</p><p>continuously,makingalineofpointsthattracesaloop,repeatingitselfforever,</p><p>around and around. The same system with a higher energy level—swinging</p><p>fasterandfarther—formsaloopinphasespacesimilartothefirst,butlarger.</p><p>Alittlerealism,intheformoffriction,changesthepicture.Wedonotneed</p><p>theequationsofmotion toknowthedestinyofapendulumsubject to friction.</p><p>Every orbit must eventually end up at the same place, the center: position 0,</p><p>velocity 0. This central fixed point “attracts” the orbits. Instead of looping</p><p>aroundforever, theyspiral inward.The frictiondissipates thesystem’senergy,</p><p>andinphasespacethedissipationshowsitselfasapulltowardthecenter,from</p><p>theouterregionsofhighenergytotheinnerregionsoflowenergy.Theattractor</p><p>—the simplest kindpossible—is like apinpointmagnet embedded in a rubber</p><p>sheet.</p><p>One advantage of thinking of states as points in space is that it makes</p><p>change easier towatch.A systemwhose variables change continuously up or</p><p>down becomes a moving point, like a fly moving around a room. If some</p><p>combinationsofvariablesneveroccur, thenascientistcansimplyimaginethat</p><p>partoftheroomasoutofbounds.Theflynevergoesthere.Ifasystembehaves</p><p>periodically, coming around to the same state again and again, then the fly</p><p>moves in a loop, passing through the same position in phase space again and</p><p>again.Phase-spaceportraitsofphysicalsystemsexposedpatternsofmotionthat</p><p>wereinvisibleotherwise,asaninfraredlandscapephotographcanrevealpatterns</p><p>and details that exist just beyond the reach of perception. When a scientist</p><p>looked at a phase portrait, he could use his imagination to think back to the</p><p>systemitself.Thisloopcorrespondstothatperiodicity.Thistwistcorrespondsto</p><p>thatchange.Thisemptyvoidcorrespondstothatphysicalimpossibility.</p><p>Evenintwodimensions,phase-spaceportraitshadmanysurprisesinstore,</p><p>and even desktop computers could easily demonstrate some of them, turning</p><p>equations into colorful moving trajectories. Some physicists began making</p><p>movies and videotapes to show their colleagues, and somemathematicians in</p><p>California published bookswith a series of green, blue, and red cartoon-style</p><p>drawings—“chaoscomics,” someof their colleagues said,with just a touchof</p><p>malice. Two dimensions did not begin to cover the kinds of systems that</p><p>physicistsneededtostudy.Theyhadtoshowmorevariablesthantwo,andthat</p><p>meant more dimensions. Every piece of a dynamical system that can move</p><p>independently is another variable, another degreeof freedom.Everydegreeof</p><p>freedomrequiresanotherdimension inphase space, tomakesure thata single</p><p>pointcontainsenoughinformationtodeterminethestateofthesystemuniquely.</p><p>The simple equations Robert May studied were one-dimensional—a single</p><p>numberwasenough,anumber thatmight stand for temperatureorpopulation,</p><p>and that number defined the position of a point on a one-dimensional line.</p><p>Lorenz’s stripped-down systemof fluid convectionwas three-dimensional, not</p><p>because the fluid moved through three dimensions, but because it took three</p><p>distinctnumberstonaildownthestateofthefluidatanyinstant.</p><p>Spacesoffour,five,ormoredimensionstaxthevisualimaginationofeven</p><p>the most agile topologist. But complex systems have many independent</p><p>variables. Mathematicians had to accept the fact that systems with infinitely</p><p>manydegreesof freedom—untrammelednatureexpressing itself ina turbulent</p><p>waterfall or an unpredictable brain—required a phase space of infinite</p><p>dimensions.Butwhocouldhandlesucha thing?Itwasahydra,mercilessand</p><p>uncontrollable, and it was Landau’s image for turbulence: infinite modes,</p><p>infinitedegreesoffreedom,infinitedimensions.</p><p>Velocityiszeroasthependulumstartsitsswing.Positionisanegativenumber,thedistancetotheleftof</p><p>thecenter.</p><p>Thetwonumbersspecifyasinglepointintwo-dimensionalphasespace.</p><p>Velocityreachesitsmaximumasthependulum’spositionpassesthroughzero.</p><p>Velocitydeclinesagaintozero,andthenbecomesnegativetorepresentleftwardmotion.</p><p>ANOTHERWAYTOSEEAPENDULUM.Onepointinphasespace(right)containsalltheinformation</p><p>aboutthestateofadynamicalsystematanyinstant(left).Forasimplependulum,twonumbers—velocity</p><p>andposition—areallyouneedtoknow.</p><p>The points trace a trajectory that provides a way of visualizing the continuous longterm behavior of a</p><p>dynamicalsystem.Arepeatinglooprepresentsasystemthatrepeatsitselfatregularintervalsforever.</p><p>Iftherepeatingbehaviorisstable,asinapendulumclock,thenthesystemreturnstothisorbitafter</p><p>smallperturbations.Inphasespace,trajectoriesneartheorbitaredrawnintoit;theorbitisanattractor.</p><p>Anattractorcanbeasinglepoint.Forapendulumsteadilylosingenergytofriction,alltrajectoriesspiral</p><p>inwardtowardapointthatrepresentsasteadystate—inthiscase,thesteadystateofnomotionatall.</p><p>APHYSICISTHADGOODREASONtodislikeamodelthatfoundsolittleclarity</p><p>in nature. Using the nonlinear equations of fluid motion, the world’s fastest</p><p>supercomputerswereincapableofaccuratelytrackingaturbulentflowofevena</p><p>cubiccentimeterformorethanafewseconds.Theblameforthiswascertainly</p><p>nature’smore thanLandau’s, but even so theLandaupicturewent against the</p><p>grain. Absent any knowledge, a physicist might be permitted to suspect that</p><p>some principlewas evading discovery. The great quantum theorist Richard P.</p><p>Feynman expressed this feeling. “It always bothers me that, according to the</p><p>laws as we understand them today, it takes a computing machine an infinite</p><p>numberoflogicaloperationstofigureoutwhatgoesoninnomatterhowtinya</p><p>regionof space,andnomatterhow tinya regionof time.Howcanall thatbe</p><p>goingon in that tiny space?Whyshould it takean infinite amountof logic to</p><p>figureoutwhatonetinypieceofspace/timeisgoingtodo?”</p><p>Likesomanyofthosewhobeganstudyingchaos,DavidRuellesuspected</p><p>that the visible patterns in turbulent flow—self-entangled stream lines, spiral</p><p>vortices,whorlsthatrisebeforetheeyeandvanishagain—mustreflectpatterns</p><p>explainedbylawsnotyetdiscovered.Inhismind,thedissipationofenergyina</p><p>turbulentflowmuststillleadtoakindofcontractionofthephasespace,apull</p><p>towardanattractor.Certainly theattractorwouldnotbea fixedpoint,because</p><p>theflowwouldnevercometorest.Energywaspouringintothesystemaswell</p><p>asdrainingout.</p><p>Whatotherkindofattractorcould itbe?According todogma,</p><p>only one other kind existed, a periodic attractor, or limit cycle—an orbit that</p><p>attractedallothernearbyorbits.Ifapendulumgainsenergyfromaspringwhile</p><p>itlosesitthroughfriction—thatis,ifthependulumisdrivenaswellasdamped</p><p>—astableorbitmaybetheclosedloopinphasespacethatrepresentstheregular</p><p>swingingmotionofagrandfatherclock.Nomatterwherethependulumstarts,it</p><p>willsettleintothatoneorbit.Orwillit?Forsomeinitialconditions—thosewith</p><p>thelowestenergy—thependulumwillstillsettletoastop,sothesystemactually</p><p>hastwoattractors,oneaclosedloopandtheotherafixedpoint.Eachattractor</p><p>hasits“basin,”justastwonearbyrivershavetheirownwatershedregions.</p><p>Intheshorttermanypointinphasespacecanstandforapossiblebehavior</p><p>of the dynamical system. In the long term the only possible behaviors are the</p><p>attractors themselves. Other kinds of motion are transient. By definition,</p><p>attractors had the important property of stability—in a real system, where</p><p>moving parts are subject to bumps and jiggles from real-world noise, motion</p><p>tendstoreturntotheattractor.Abumpmayshoveatrajectoryawayforabrief</p><p>time,buttheresultingtransientmotionsdieout.Evenifthecatknocksintoit,a</p><p>pendulumclockdoesnotswitchtoasixty-two–secondminute.Turbulenceina</p><p>fluidwasabehaviorofadifferentorder,neverproducinganysinglerhythmto</p><p>theexclusionofothers.Awell-knowncharacteristicof turbulencewasthat the</p><p>wholebroadspectrumofpossiblecycleswaspresentatonce.Turbulenceislike</p><p>white noise, or static. Could such a thing arise from a simple, deterministic</p><p>systemofequations?</p><p>Ruelle and Takenswonderedwhether some other kind of attractor could</p><p>have the right set of properties. Stable—representing the final state of a</p><p>dynamicalsysteminanoisyworld.Low-dimensional—anorbitinaphasespace</p><p>that might be a rectangle or a box, with just a few degrees of freedom.</p><p>Nonperiodic—neverrepeatingitself,andneverfallingintoasteadygrandfather-</p><p>clockrhythm.Geometricallythequestionwasapuzzle:Whatkindoforbitcould</p><p>bedrawninalimitedspacesothatitwouldneverrepeatitselfandnevercross</p><p>itself—becauseonceasystemreturnstoastateithasbeeninbefore,itthereafter</p><p>mustfollowthesamepath.Toproduceeveryrhythm,theorbitwouldhavetobe</p><p>aninfinitelylonglineinafinitearea.Inotherwords—butthewordhadnotbeen</p><p>invented—itwouldhavetobefractal.</p><p>Bymathematical reasoning,Ruelle andTakens claimed that such a thing</p><p>mustexist.Theyhadneverseenone,andtheydidnotdrawone.Buttheclaim</p><p>wasenough.Later,deliveringaplenaryaddresstotheInternationalCongressof</p><p>MathematiciansinWarsaw,withthecomfortableadvantageofhindsight,Ruelle</p><p>declared:“Thereactionofthescientificpublictoourproposalwasquitecold.In</p><p>particular, thenotion thatcontinuousspectrumwouldbeassociatedwitha few</p><p>degrees of freedom was viewed as heretical by many physicists.” But it was</p><p>physicists—ahandful, tobesure—whorecognized the importanceof the1971</p><p>paperandwenttoworkonitsimplications.</p><p>ACTUALLY,BY1971thescientificliteraturealreadycontainedonesmallline</p><p>drawingoftheunimaginablebeastthatRuelleandTakensweretryingtobring</p><p>alive.EdwardLorenzhadattachedittohis1963paperondeterministicchaos,a</p><p>picturewith just twocurveson the right,one inside theother, and fiveon the</p><p>left.Toplot just theseseven loops required500successivecalculationson the</p><p>computer.Apointmovingalongthistrajectoryinphasespace,aroundtheloops,</p><p>illustrated the slow, chaotic rotation of a fluid as modeled by Lorenz’s three</p><p>equations for convection.Because the systemhad three independent variables,</p><p>thisattractorlayinathree-dimensionalphasespace.AlthoughLorenzdrewonly</p><p>afragmentofit,hecouldseemorethanhedrew:asortofdoublespiral,likea</p><p>pairofbutterflywings interwovenwith infinitedexterity.Whentherisingheat</p><p>ofhissystempushedthefluidaroundinonedirection,thetrajectorystayedon</p><p>the right wing; when the rolling motion stopped and reversed itself, the</p><p>trajectorywouldswingacrosstotheotherwing.</p><p>Theattractorwasstable,low-dimensional,andnonperiodic.Itcouldnever</p><p>intersectitself,becauseifitdid,returningtoapointalreadyvisited,fromthenon</p><p>themotionwouldrepeatitselfinaperiodicloop.Thatneverhappened—thatwas</p><p>thebeautyof theattractor.Those loopsandspiralswere infinitelydeep,never</p><p>quitejoining,neverintersecting.Yettheystayedinsideafinitespace,confined</p><p>by a box.How could that be?How could infinitelymany paths lie in a finite</p><p>space?</p><p>InanerabeforeMandelbrot’spicturesoffractalshadfloodedthescientific</p><p>marketplace,thedetailsofconstructingsuchashapewerehardtoimagine,and</p><p>Lorenz acknowledged an “apparent contradiction” in his tentative description.</p><p>“It is difficult to reconcile the merging of two surfaces, one containing each</p><p>spiral,withtheinabilityoftwotrajectoriestomerge,”hewrote.Buthesawan</p><p>answer too delicate to appear in the few calculations within range of his</p><p>computer.Wherethespiralsappeartojoin,thesurfacesmustdivide,herealized,</p><p>formingseparatelayersinthemannerofaflakymille-feuille.“Weseethateach</p><p>surfaceisreallyapairofsurfaces,sothat,wheretheyappeartomerge,thereare</p><p>reallyfoursurfaces.Continuingthisprocessforanothercircuit,weseethatthere</p><p>are really eight surfaces, etc., andwe finally conclude that there is an infinite</p><p>complexof surfaces, each extremely close tooneor theotherof twomerging</p><p>surfaces.” It was nowonder thatmeteorologists in 1963 left such speculation</p><p>alone,nor thatRuelleadecadelaterfeltastonishmentandexcitementwhenhe</p><p>finallylearnedofLorenz’swork.HewenttovisitLorenzonce,intheyearsthat</p><p>followed,andleftwithasmallsenseofdisappointmentthattheyhadnottalked</p><p>moreoftheircommonterritoryinscience.Withcharacteristicdiffidence,Lorenz</p><p>made the occasion a social one, and they went with their wives to an art</p><p>museum.</p><p>THEFIRSTSTRANGEATTRACTOR.In1963EdwardLorenzcouldcomputeonlythefirstfewstrands</p><p>oftheattractorforhissimplesystemofequations.Buthecouldseethattheinterleavingofthetwospiral</p><p>wingsmusthaveanextraordinarystructureoninvisiblysmallscales.</p><p>Theeffort topursuethehintsputforwardbyRuelleandTakenstooktwo</p><p>paths.Onewas the theoretical struggle tovisualize strange attractors.Was the</p><p>Lorenz attractor typical?What other sorts of shapeswere possible?The other</p><p>was a line of experimental work meant to confirm or refute the highly</p><p>unmathematicalleapoffaiththatsuggestedtheapplicabilityofstrangeattractors</p><p>tochaosinnature.</p><p>In Japan the study of electrical circuits that imitated the behavior of</p><p>mechanical springs—but much faster—led Yoshisuke Ueda to discover an</p><p>extraordinarilybeautifulsetofstrangeattractors.(HemetanEasternversionof</p><p>thecoolnessthatgreetedRuelle:“Yourresultisnomorethananalmostperiodic</p><p>oscillation. Don’t form a selfish concept of steady states.”) In Germany Otto</p><p>Rössler,anonpracticingmedicaldoctorwhocametochaosbywayofchemistry</p><p>and theoretical biology, beganwith an odd ability to see strange attractors as</p><p>philosophical objects, letting the mathematics follow along behind. Rössler’s</p><p>namebecameattachedtoaparticularlysimpleattractorintheshapeofabandof</p><p>ribbonwithafoldin it,muchstudied</p><p>becauseitwaseasytodraw,buthealso</p><p>visualizedattractorsinhigherdimensions—“asausageinasausageinasausage</p><p>inasausage,”hewouldsay,“takeitout,foldit,squeezeit,putitback.”Indeed,</p><p>thefoldingandsqueezingofspacewasakeytoconstructingstrangeattractors,</p><p>andperhaps a key to the dynamics of the real systems that gave rise to them.</p><p>Rösslerfeltthattheseshapesembodiedaself-organizingprincipleintheworld.</p><p>Hewouldimaginesomethinglikeawindsockonanairfield,“anopenhosewith</p><p>ahole in the end, and thewind forces itsway in,”he said. “Then thewind is</p><p>trapped.Againstitswill,energyisdoingsomethingproductive,likethedevilin</p><p>medieval history. The principle is that nature does something against its own</p><p>willand,byself-entanglement,producesbeauty.”</p><p>Making pictures of strange attractors was not a trivial matter. Typically,</p><p>orbitswouldwindtheirever-more–complicatedpathsthroughthreedimensions</p><p>ormore, creatingadark scribble in spacewith an internal structure that could</p><p>notbeseenfromtheoutside.Toconvertthesethree-dimensionalskeinsintoflat</p><p>pictures, scientists first used the technique of projection, in which a drawing</p><p>represented the shadow that an attractor would cast on a surface. But with</p><p>complicated strange attractors, projection just smears the detail into an</p><p>indecipherablemess.Amorerevelatorytechniquewastomakeareturnmap,or</p><p>aPoincarémap, ineffect,takingaslicefromthetangledheartoftheattractor,</p><p>removinga two-dimensional section just as apathologistpreparesa sectionof</p><p>tissueforamicroscopeslide.</p><p>The Poincaré map removes a dimension from an attractor and turns a</p><p>continuouslineintoacollectionofpoints.InreducinganattractortoitsPoincaré</p><p>map, a scientist implicitly assumes that he can preservemuch of the essential</p><p>movement. He can imagine, for example, a strange attractor buzzing around</p><p>beforehis eyes, its orbits carryingupanddown, left and right, and to and fro</p><p>throughhis computer screen.Each time the orbit passes through the screen, it</p><p>leavesaglowingpointattheplaceofintersection,andthepointseitherforma</p><p>randomblotchorbegintotracesomeshapeinphosphorus.</p><p>Theprocesscorresponds tosampling thestateofasystemeverysooften,</p><p>insteadofcontinuously.Whentosample—wheretotaketheslicefromastrange</p><p>attractor—is a question that gives an investigator some flexibility. The most</p><p>informativeintervalmightcorrespondtosomephysicalfeatureofthedynamical</p><p>system:forexample,aPoincarémapcouldsample thevelocityofapendulum</p><p>bob each time it passed through its lowest point. Or the investigator could</p><p>choosesomeregular time interval, freezingsuccessivestates in the flashofan</p><p>imaginarystrobelight.Eitherway,suchpicturesfinallybegantorevealthefine</p><p>fractalstructureguessedatbyEdwardLorenz.</p><p>EXPOSINGANATTRACTOR’S STRUCTURE. The strange attractor above—first one orbit, then ten,</p><p>then one hundred—depicts the chaotic behavior of a rotor, a pendulum swinging through a full circle,</p><p>driven by an energetic kick at regular intervals. By the time 1,000 orbits have been drawn (below), the</p><p>attractorhasbecomeanimpenetrablytangledskein.</p><p>To see the structurewithin, a computer can take a slice through an attractor, a so-calledPoincaré</p><p>section. The technique reduces a three-dimensional picture to two dimensions. Each time the trajectory</p><p>passesthroughaplane,itmarksapoint,andgraduallyaminutelydetailedpatternemerges.Thisexample</p><p>has more than 8,000 points, each standing for a full orbit around the attractor. In effect, the system is</p><p>“sampled”atregularintervals.Onekindofinformationislost;anotherisbroughtoutinhighrelief.</p><p>THEMOST ILLUMINATING STRANGEATTRACTOR, because itwas the simplest,</p><p>came from a man far removed from the mysteries of turbulence and fluid</p><p>dynamics.Hewasanastronomer,MichelHénonoftheNiceObservatoryonthe</p><p>southern coast of France. In one way, of course, astronomy gave dynamical</p><p>systems its start, the clockworkmotionsof planets providingNewtonwithhis</p><p>triumphandLaplacewithhisinspiration.Butcelestialmechanicsdifferedfrom</p><p>mostearthlysystemsinacrucialrespect.Systemsthatloseenergytofrictionare</p><p>dissipative.Astronomicalsystemsarenot:theyareconservative,orHamiltonian.</p><p>Actually,onanearlyinfinitesimalscale,evenastronomicalsystemssufferakind</p><p>of drag, with stars radiating away energy and tidal friction draining some</p><p>momentum from orbiting bodies, but for practical purposes, astronomers’</p><p>calculationscould ignoredissipation.Andwithoutdissipation, thephase space</p><p>would not fold and contract in the way needed to produce an infinite fractal</p><p>layering.Astrangeattractorcouldneverarise.Couldchaos?</p><p>Manyastronomershave longandhappycareerswithoutgivingdynamical</p><p>systemsathought,butHénonwasdifferent.HewasborninParisin1931,afew</p><p>years younger than Lorenz but, like him, a scientist with a certain unfulfilled</p><p>attraction tomathematics.Hénon liked small, concrete problems that could be</p><p>attached to physical situations—“not like the kind of mathematics people do</p><p>today,” he would say.When computers reached a size suitable for hobbyists,</p><p>Hénongotone, aHeathkit thathe soldered together andplayedwith at home.</p><p>Long before that, though, he took on a particularly baffling problem in</p><p>dynamics. It concernedglobular clusters—crowdedballs of stars, sometimes a</p><p>million in one place, that form the oldest and possibly the most breathtaking</p><p>objects in thenight sky.Globularclustersareamazinglydensewithstars.The</p><p>problemofhowtheystaytogetherandhowtheyevolveovertimehasperplexed</p><p>astronomersthroughoutthetwentiethcentury.</p><p>Dynamicallyspeaking,aglobularclusterisabigmany-bodyproblem.The</p><p>two-bodyproblemiseasy.Newtonsolveditcompletely.Eachbody—theearth</p><p>andthemoon,forexample—travelsinaperfectellipsearoundthesystem’sjoint</p><p>center of gravity. Add just one more gravitational object, however, and</p><p>everything changes.The three-bodyproblem is hard, andworse thanhard.As</p><p>Poincaré discovered, it ismost often impossible. The orbits can be calculated</p><p>numericallyforawhile,andwithpowerfulcomputerstheycanbetrackedfora</p><p>longwhilebeforeuncertaintiesbegintotakeover.Buttheequationscannotbe</p><p>solved analytically, which means that longterm questions about a three-body</p><p>systemcannotbeanswered.Isthesolarsystemstable?Itcertainlyappearstobe,</p><p>in the short term, but even today no one knows for sure that some planetary</p><p>orbitscouldnotbecomemoreandmoreeccentricuntil theplanetsflyofffrom</p><p>thesystemforever.</p><p>Asystemlikeaglobularclusterisfartoocomplextobetreateddirectlyasa</p><p>many-body problem, but its dynamics can be studiedwith the help of certain</p><p>compromises.Itisreasonable,forexample,tothinkofindividualstarswinging</p><p>theirway through an average gravitational fieldwith a particular gravitational</p><p>center. Every so often, however, two stars will approach each other closely</p><p>enough that their interaction must be treated separately. And astronomers</p><p>realizedthatglobularclustersgenerallymustnotbestable.Binarystarsystems</p><p>tendtoforminsidethem,starspairingoffintightlittleorbits,andwhenathird</p><p>star encounters a binary, one of the three tends to get a sharp kick. Every so</p><p>often, a starwill gain enoughenergy</p><p>fromsuchan interaction to reachescape</p><p>velocityanddepart theclusterforever; therestof theclusterwill thencontract</p><p>slightly.WhenHénon took on this problem for his doctoral thesis in Paris in</p><p>1960,hemadearatherarbitraryassumption:thatastheclusterchangedscale,it</p><p>would remain self-similar. Working out the calculations, he reached an</p><p>astonishingresult.Thecoreofaclusterwouldcollapse,gainingkineticenergy</p><p>andseekingastateofinfinitedensity.Thiswashardtoimagine,andfurthermore</p><p>it was not supported by the evidence of clusters so far observed. But slowly</p><p>Hénon’stheory—latergiventhename“gravothermalcollapse”—tookhold.</p><p>Thus fortified,willing to trymathematics on old problems andwilling to</p><p>pursueunexpectedresultstotheirunlikelyoutcomes,hebeganworkonamuch</p><p>easierprobleminstardynamics.</p><p>Thistime,in1962,visitingPrincetonUniversity,hehadaccessforthefirst</p><p>time to computers, just as Lorenz atM.I.T. was starting to use computers in</p><p>meteorology. Hénon began modeling the orbits of stars around their galactic</p><p>center.Inreasonablysimpleform,galacticorbitscanbetreatedliketheorbitsof</p><p>planets around a sun, with one exception: the central gravity source is not a</p><p>point,butadiskwiththicknessinthreedimensions.</p><p>He made a compromise with the differential equations. “To have more</p><p>freedom of experimentation,” as he put it, “we forget momentarily about the</p><p>astronomical origin of the problem.” Although he did not say so at the time,</p><p>“freedomofexperimentation”meant,inpart,freedomtoplaywiththeproblem</p><p>onaprimitivecomputer.Hismachinehadlessthanathousandthofthememory</p><p>onasinglechipofapersonalcomputertwenty-fiveyearslater,anditwasslow,</p><p>too.But like laterexperimenters in thephenomenaofchaos,Hénonfoundthat</p><p>theoversimplificationpaidoff.Byabstractingonlytheessenceofhissystem,he</p><p>made discoveries that applied to other systems as well, and more important</p><p>systems. Years later, galactic orbits were still a theoretical game, but the</p><p>dynamicsofsuchsystemswereunderintense,expensiveinvestigationbythose</p><p>interested in the orbits of particles in high-energy accelerators and those</p><p>interested in the confinement ofmagnetic plasmas for the creation of nuclear</p><p>fusion.</p><p>Stellarorbitsingalaxies,onatimescaleofsome200millionyears,takeon</p><p>a three-dimensional character instead of making perfect ellipses. Three-</p><p>dimensionalorbitsareashardtovisualizewhentheorbitsarerealaswhenthey</p><p>are imaginary constructions in phase space. So Hénon used a technique</p><p>comparable to themaking of Poincarémaps.He imagined a flat sheet placed</p><p>uprightononesideofthegalaxysothateveryorbitwouldsweepthroughit,as</p><p>horsesonaracetracksweepacrossthefinishline.Thenhewouldmarkthepoint</p><p>wheretheorbitcrossedthisplaneandtracethemovementofthepointfromorbit</p><p>toorbit.</p><p>Hénonhadtoplotthesepointsbyhand,buteventuallythemanyscientists</p><p>usingthistechniquewouldwatchthemappearonacomputerscreen,likedistant</p><p>streetlampscomingononebyoneatnightfall.Atypicalorbitmightbeginwith</p><p>apoint toward the lower left of thepage.Then,on thenextgo-round, apoint</p><p>wouldappearafewinchestotheright.Thenanother,moretotherightandupa</p><p>little—andsoon.At firstnopatternwouldbeobvious,butafter tenor twenty</p><p>points an egg-shaped curve would take shape. The successive points actually</p><p>makeacircuitaround thecurve,butsince theydonotcomearound toexactly</p><p>the sameplace, eventually, after hundredsor thousandsof points, the curve is</p><p>solidlyoutlined.</p><p>Such orbits are not completely regular, since they never exactly repeat</p><p>themselves, but they are certainly predictable, and they are far from chaotic.</p><p>Points never arrive inside the curve or outside it. Translated back to the full</p><p>three-dimensionalpicture, theorbitswereoutliningatorus,ordoughnutshape,</p><p>andHénon’smappingwas a cross-section of the torus. So far, hewasmerely</p><p>illustratingwhatallhispredecessorshadtakenforgranted.Orbitswereperiodic.</p><p>At the observatory in Copenhagen, from 1910 to 1930, a generation of</p><p>astronomerspainstakinglyobservedandcalculatedhundredsofsuchorbits—but</p><p>they were only interested in the ones that proved periodic. “I, too, was</p><p>convinced, likeeveryoneelseat that time, thatallorbitsshouldberegular like</p><p>this,” Hénon said. But he and his graduate student at Princeton, Carl Heiles,</p><p>continuedcomputingdifferentorbits, steadily increasing the levelofenergy in</p><p>theirabstractsystem.Soontheysawsomethingutterlynew.</p><p>First the egg-shaped curve twisted into something more complicated,</p><p>crossingitselfinfigureeightsandsplittingapartintoseparateloops.Still,every</p><p>orbit fell on some loop.Then, at evenhigher levels, another changeoccurred,</p><p>quiteabruptly.“Herecomesthesurprise,”HénonandHeileswrote.Someorbits</p><p>becamesounstable that thepointswouldscatterrandomlyacross thepaper.In</p><p>someplaces,curvescouldstillbedrawn;inothers,nocurvefitthepoints.The</p><p>picture became quite dramatic: evidence of complete disordermixedwith the</p><p>clearremnantsoforder,formingshapesthatsuggested“islands”and“chainsof</p><p>islands” to these astronomers. They tried two different computers and two</p><p>differentmethodsofintegration,buttheresultswerethesame.Theycouldonly</p><p>explore and speculate. Based solely on their numerical experimentation, they</p><p>made a guess about the deep structure of such pictures. With greater</p><p>magnification, they suggested, more islands would appear on smaller and</p><p>smaller scales,perhapsall theway to infinity.Mathematicalproofwasneeded</p><p>—“butthemathematicalapproachtotheproblemdoesnotseemtooeasy.”</p><p>ORBITS AROUND THE GALACTIC CENTER. To understand the trajectories of the stars through a</p><p>galaxy,MichelHénoncomputedtheintersectionsofanorbitwithaplane.Theresultingpatternsdepended</p><p>on the system’s total energy.Thepoints froma stable orbit gradually produced a continuous, connected</p><p>curve (left). Other energy levels, however, produced complicated mixtures of stability and chaos,</p><p>representedbyregionsofscatteredpoints.</p><p>Hénonwentontootherproblems,butfourteenyearslater,whenfinallyhe</p><p>heardabout thestrangeattractorsofDavidRuelleandEdwardLorenz,hewas</p><p>preparedto listen.By1976hehadmovedto theObservatoryofNice,perched</p><p>highabovetheMediterraneanSeaontheGrandeCorniche,andheheardatalk</p><p>byavisitingphysicistabouttheLorenzattractor.Thephysicisthadbeentrying</p><p>differenttechniquestoilluminatethefine“micro-structure”oftheattractor,with</p><p>littlesuccess.Hénon,thoughdissipativesystemswerenothisfield(“sometimes</p><p>astronomersarefearfulofdissipativesystems—they’reuntidy”),thoughthehad</p><p>anidea.</p><p>Onceagain,hedecidedtothrowoutallreferencetothephysicaloriginsof</p><p>the system and concentrate only on the geometrical essence he wanted to</p><p>explore.Where Lorenz and others had stuck to differential equations—flows,</p><p>withcontinuouschanges inspaceand time—heturned todifferenceequations,</p><p>discreteintime.Thekey,hebelieved,wastherepeatedstretchingandfoldingof</p><p>phasespaceinthemannerofapastrychefwhorollsthedough,foldsit,rollsit</p><p>out again, folds it, creating a structure that will eventually be a sheaf of thin</p><p>layers.Hénon drew a flat oval on a piece of paper. To stretch it, he picked a</p><p>shortnumericalfunction</p><p>and pressure, between pressure and wind speed. Lorenz</p><p>understood that he was putting into practice the laws of Newton, appropriate</p><p>tools for a clockmaker deity who could create a world and set it running for</p><p>eternity.Thankstothedeterminismofphysicallaw,furtherinterventionwould</p><p>thenbeunnecessary.Thosewhomadesuchmodelstookforgrantedthat,from</p><p>presenttofuture,thelawsofmotionprovideabridgeofmathematicalcertainty.</p><p>Understandthelawsandyouunderstandtheuniverse.Thatwasthephilosophy</p><p>behindmodelingweatheronacomputer.</p><p>Indeed, if the eighteenth-century philosophers imagined their creator as a</p><p>benevolentnoninterventionist, content to remainbehind the scenes, theymight</p><p>have imagined someone likeLorenz.Hewasanoddsortofmeteorologist.He</p><p>hadthewornfaceofaYankeefarmer,withsurprisingbrighteyesthatmadehim</p><p>seemtobelaughingwhetherhewasornot.Heseldomspokeabouthimselfor</p><p>his work, but he listened. He often lost himself in a realm of calculation or</p><p>dreaming that his colleagues found inaccessible. His closest friends felt that</p><p>Lorenzspentagooddealofhistimeoffinaremoteouterspace.</p><p>Asaboyhehadbeenaweatherbug,atleasttotheextentofkeepingclose</p><p>tabsonthemax-minthermometerrecordingthedays’highsandlowsoutsidehis</p><p>parents’ house inWest Hartford, Connecticut. But he spent more time inside</p><p>playing with mathematical puzzle books than watching the thermometer.</p><p>Sometimesheandhisfatherwouldworkoutpuzzlestogether.Oncetheycame</p><p>upon a particularly difficult problem that turnedout to be insoluble.Thatwas</p><p>acceptable,hisfathertoldhim:youcanalwaystrytosolveaproblembyproving</p><p>that no solution exists. Lorenz liked that, as he always liked the purity of</p><p>mathematics, and when he graduated from Dartmouth College, in 1938, he</p><p>thought thatmathematicswashiscalling.Circumstance interfered,however, in</p><p>theformofWorldWarII,whichputhimtoworkasaweatherforecasterforthe</p><p>Army Air Corps. After the war Lorenz decided to stay with meteorology,</p><p>investigating the theoryof it, pushing themathematics a little further forward.</p><p>Hemadeanameforhimselfbypublishingworkonorthodoxproblems,suchas</p><p>thegeneralcirculationoftheatmosphere.Andinthemeantimehecontinuedto</p><p>thinkaboutforecasting.</p><p>Tomostseriousmeteorologists,forecastingwaslessthanscience.Itwasa</p><p>seat-of–the-pantsbusinessperformedbytechnicianswhoneededsomeintuitive</p><p>ability toreadthenextday’sweather in the instrumentsandtheclouds. Itwas</p><p>guesswork. At centers like M.I.T., meteorology favored problems that had</p><p>solutions. Lorenz understood the messiness of weather prediction as well as</p><p>anyone,havingtrieditfirsthandforthebenefitofmilitarypilots,butheharbored</p><p>aninterestintheproblem—amathematicalinterest.</p><p>Notonlydidmeteorologistsscornforecasting,butinthe1960svirtuallyall</p><p>serious scientists mistrusted computers. These souped-up calculators hardly</p><p>seemed like tools for theoretical science. So numericalweathermodelingwas</p><p>something of a bastard problem. Yet the time was right for it. Weather</p><p>forecasting had been waiting two centuries for a machine that could repeat</p><p>thousandsofcalculationsoverandoveragainbybruteforce.Onlyacomputer</p><p>could cash in the Newtonian promise that the world unfolded along a</p><p>deterministic path, rule-bound like the planets, predictable like eclipses and</p><p>tides. In theory a computer could letmeteorologists dowhat astronomers had</p><p>been able to dowith pencil and slide rule: reckon the future of their universe</p><p>from its initial conditions and the physical laws that guide its evolution. The</p><p>equationsdescribing themotionof air andwaterwereaswellknownas those</p><p>describing themotion of planets. Astronomers did not achieve perfection and</p><p>neverwould,notinasolarsystemtuggedbythegravitiesofnineplanets,scores</p><p>ofmoonsandthousandsofasteroids,butcalculationsofplanetarymotionwere</p><p>so accurate that people forgot they were forecasts.When an astronomer said,</p><p>“CometHalleywillbebackthiswayinseventy-sixyears,”itseemedlikefact,</p><p>not prophecy.Deterministic numerical forecasting figured accurate courses for</p><p>spacecraftandmissiles.Whynotwindsandclouds?</p><p>Weather was vastly more complicated, but it was governed by the same</p><p>laws. Perhaps a powerful enough computer could be the supreme intelligence</p><p>imagined by Laplace, the eighteenth-century philosopher-mathematician who</p><p>caught theNewtonian fever like no one else: “Such an intelligence,” Laplace</p><p>wrote, “would embrace in the same formula the movements of the greatest</p><p>bodiesof theuniverse and thoseof the lightest atom; for it, nothingwouldbe</p><p>uncertainandthefuture,asthepast,wouldbepresenttoitseyes.”Inthesedays</p><p>of Einstein’s relativity and Heisenberg’s uncertainty, Laplace seems almost</p><p>buffoon-like in his optimism, but much of modern science has pursued his</p><p>dream.Implicitly,themissionofmanytwentieth-centuryscientists—biologists,</p><p>neurologists, economists—has been to break their universes down into the</p><p>simplest atoms that will obey scientific rules. In all these sciences, a kind of</p><p>Newtonian determinism has been brought to bear. The fathers of modern</p><p>computing always hadLaplace inmind, and the history of computing and the</p><p>historyofforecastingwereintermingledeversinceJohnvonNeumanndesigned</p><p>hisfirstmachinesattheInstituteforAdvancedStudyinPrinceton,NewJersey,</p><p>inthe1950s.VonNeumannrecognizedthatweathermodelingcouldbeanideal</p><p>taskforacomputer.</p><p>Therewasalwaysonesmallcompromise,sosmall thatworkingscientists</p><p>usually forgot it was there, lurking in a corner of their philosophies like an</p><p>unpaid bill. Measurements could never be perfect. Scientists marching under</p><p>Newton’s banner actually waved another flag that said something like this:</p><p>Given an approximate knowledge of a system’s initial conditions and an</p><p>understandingofnaturallaw,onecancalculatetheapproximatebehaviorofthe</p><p>system. This assumption lay at the philosophical heart of science. As one</p><p>theoreticianlikedtotellhisstudents:“ThebasicideaofWesternscienceisthat</p><p>you don’t have to take into account the falling of a leaf on some planet in</p><p>anothergalaxywhenyou’retryingtoaccountforthemotionofabilliardballon</p><p>a pool table on earth. Very small influences can be neglected. There’s a</p><p>convergenceinthewaythingswork,andarbitrarilysmallinfluencesdon’tblow</p><p>uptohavearbitrarilylargeeffects.”Classically,thebeliefinapproximationand</p><p>convergencewaswelljustified.Itworked.Atinyerrorinfixingthepositionof</p><p>CometHalleyin1910wouldonlycausea tinyerror inpredictingitsarrival in</p><p>1986,andtheerrorwouldstaysmallformillionsofyearstocome.Computers</p><p>relyonthesameassumptioninguidingspacecraft:approximatelyaccurateinput</p><p>gives approximately accurate output. Economic forecasters rely on this</p><p>assumption,thoughtheirsuccessislessapparent.Sodidthepioneersinglobal</p><p>weatherforecasting.</p><p>Withhisprimitivecomputer,Lorenzhadboiledweatherdowntothebarest</p><p>skeleton. Yet, line by line, the winds and temperatures in Lorenz’s printouts</p><p>seemed to behave in a recognizable earthlyway. Theymatched his cherished</p><p>intuitionabout theweather,his sense that it repeated itself,displaying familiar</p><p>patternsovertime,pressurerisingandfalling,theairstream</p><p>thatwouldmoveanypointintheovaltoanewpointin</p><p>ashapethatwasstretchedupwardinthecenter,anarch.Thiswasamapping—</p><p>point by point, the entire ovalwas “mapped” onto the arch. Then he chose a</p><p>secondmapping, this time a contraction thatwould shrink the arch inward to</p><p>makeitnarrower.Andthenathirdmappingturnedthenarrowarchonitsside,</p><p>sothatitwouldlineupneatlywiththeoriginaloval.Thethreemappingscould</p><p>becombinedintoasinglefunctionforpurposesofcalculation.</p><p>InspirithewasfollowingSmale’shorseshoeidea.Numerically,thewhole</p><p>processwassosimplethatitcouldeasilybetrackedonacalculator.Anypoint</p><p>hasanxcoordinateandaycoordinatetofixitshorizontalandverticalposition.</p><p>Tofindthenewx,therulewastotaketheoldy,add1andsubtract1.4timesthe</p><p>oldxsquared.Tofindthenewy,multiply0.3bytheoldx.Thatis:xnew=y+1</p><p>–1.4x2andynew=0.3x.Hénonpickedastartingpointmoreorlessatrandom,</p><p>tookhiscalculatorandstartedplottingnewpoints,oneafteranother,untilhehad</p><p>plotted thousands. Then he used a real computer, an IBM 7040, and quickly</p><p>plotted fivemillion.Anyonewith a personal computer and a graphics display</p><p>couldeasilydothesame.</p><p>Atfirstthepointsappeartojumprandomlyaroundthescreen.Theeffectis</p><p>that of a Poincaré section of a three-dimensional attractor,weaving erratically</p><p>back and forth across the display. But quickly a shape begins to emerge, an</p><p>outline curved like a banana. The longer the program runs, the more detail</p><p>appears.Partsoftheoutlineseemtohavesomethickness,butthenthethickness</p><p>resolves itself into two distinct lines, then the two into four, one pair close</p><p>together andonepair farther apart.Ongreatermagnification, eachof the four</p><p>linesturnsouttobecomposedoftwomorelines—andsoon,adinfinitum.Like</p><p>Lorenz’sattractor,Hénon’sdisplaysinfiniteregress,likeanunendingsequence</p><p>ofRussiandollsoneinsidetheother.</p><p>Thenesteddetail,lineswithinlines,canbeseeninfinalforminaseriesof</p><p>pictures with progressively greater magnification. But the eerie effect of the</p><p>strange attractor can be appreciated another way when the shape emerges in</p><p>time,pointbypoint.Itappearslikeaghostoutofthemist.Newpointsscatterso</p><p>randomlyacrossthescreenthatitseemsincrediblethatanystructureisthere,let</p><p>aloneastructuresointricateandfine.Anytwoconsecutivepointsarearbitrarily</p><p>farapart,justlikeanytwopointsinitiallynearbyinaturbulentflow.Givenany</p><p>numberofpoints,itisimpossibletoguesswherethenextwillappear—except,</p><p>ofcourse,thatitwillbesomewhereontheattractor.</p><p>Thepointswandersorandomly,thepatternappearssoethereally,thatitis</p><p>hardtorememberthattheshapeisanattractor.Itisnotjustanytrajectoryofa</p><p>dynamical system. It is the trajectory toward which all other trajectories</p><p>converge.Thatiswhythechoiceofstartingconditionsdoesnotmatter.Aslong</p><p>as thestartingpoint liessomewherenear theattractor, thenextfewpointswill</p><p>convergetotheattractorwithgreatrapidity.</p><p>YEARSBEFORE,WHENDAVIDRUELLEarrivedattheCityCollegelaboratory</p><p>ofGollub and Swinney in 1974, the three physicists found themselveswith a</p><p>slender link between theory and experiment. One piece of mathematics,</p><p>philosophically bold but technically uncertain.One cylinder of turbulent fluid,</p><p>notmuch to lookat,but clearlyoutofharmonywith theold theory.Themen</p><p>spenttheafternoontalking,andthenSwinneyandGollubleftforavacationwith</p><p>theirwivesinGollub’scabinintheAdirondackmountains.Theyhadnotseena</p><p>strange attractor, and they had not measured much of what might actually</p><p>happenat theonsetof turbulence.But theyknew thatLandauwaswrong,and</p><p>theysuspectedthatRuellewasright.</p><p>THEATTRACTOROFHÉNON.A simple combinationof folding and stretchingproduced an attractor</p><p>thateasy tocomputeyet stillpoorlyunderstoodbymathematicians.As thousands, themillionsofpoints</p><p>appear,moreandmoredetailemerges.Whatappeartobesinglelinesprove,onmagnification,tobepairs,</p><p>thenpairsofpairs.Yetwhetheranytwosuccessivepointsappearnearbyorfarapartisunpredictable.</p><p>Asanelement in theworld revealedbycomputerexploration, thestrange</p><p>attractor began as a mere possibility, marking a place where many great</p><p>imaginationsinthetwentiethcenturyhadfailedtogo.Soon,whenscientistssaw</p><p>what computers had to show, it seemed like a face they had been seeing</p><p>everywhere, in the music of turbulent flows or in clouds scattered like veils</p><p>acrossthesky.Naturewasconstrained.Disorderwaschanneled,itseemed,into</p><p>patternswithsomecommonunderlyingtheme.</p><p>Later, the recognition of strange attractors fed the revolution in chaos by</p><p>givingnumericalexplorersaclearprogramtocarryout.Theylookedforstrange</p><p>attractorseverywhere,wherevernatureseemedtobebehavingrandomly.Many</p><p>arguedthattheearth’sweathermightlieonastrangeattractor.Othersassembled</p><p>millions of pieces of stock market data and began searching for a strange</p><p>attractorthere,peeringatrandomnessthroughtheadjustablelensofacomputer.</p><p>Inthemiddle1970sthesediscoverieslayinthefuture.Noonehadactually</p><p>seena strange attractor in an experiment, and itwas far fromclearhow togo</p><p>about looking for one. In theory the strange attractor could givemathematical</p><p>substance to fundamental new properties of chaos. Sensitive dependence on</p><p>initial conditions was one. “Mixing” was another, in a sense that would be</p><p>meaningful toa jetenginedesigner,forexample,concernedabout theefficient</p><p>combination of fuel and oxygen. But no one knew how to measure these</p><p>properties, how to attach numbers to them. Strange attractors seemed fractal,</p><p>implying that their true dimension was fractional, but no one knew how to</p><p>measure thedimensionor how to apply such ameasurement in the context of</p><p>engineeringproblems.</p><p>Mostimportant,nooneknewwhetherstrangeattractorswouldsayanything</p><p>aboutthedeepestproblemwithnonlinearsystems.Unlikelinearsystems,easily</p><p>calculatedandeasilyclassified,nonlinearsystemsstillseemed,intheiressence,</p><p>beyondclassification—eachdifferentfromeveryother.Scientistsmightbeginto</p><p>suspect that they shared common properties, but when it came time to make</p><p>measurementsandperformcalculations,eachnonlinearsystemwasaworldunto</p><p>itself.Understandingoneseemedtooffernohelpinunderstandingthenext.An</p><p>attractor like Lorenz’s illustrated the stability and the hidden structure of a</p><p>systemthatotherwiseseemedpatternless,buthowdidthispeculiardoublespiral</p><p>helpresearchersexploringunrelatedsystems?Nooneknew.</p><p>For now, the excitement went beyond pure science. Scientists who saw</p><p>these shapes allowed themselves to forget momentarily the rules of scientific</p><p>discourse. Ruelle, for example: “I have not spoken of the esthetic appeal of</p><p>strange attractors. These systems of curves, these clouds of points suggest</p><p>sometimes fireworks or galaxies, sometimes strange and disquieting vegetal</p><p>proliferations. A realm lies there of forms to explore, and harmonies to</p><p>discover.”</p><p>Universality</p><p>Theiteratingoftheselinesbringsgold;</p><p>Theframingofthiscircleontheground</p><p>Bringswhirlwinds,tempests,thunderandlightning.</p><p>—MARLOWE,Dr.Faustus</p><p>A FEW DOZEN YARDS upstream from awaterfall, a smooth flowing stream</p><p>seems to intuit the coming drop. The water begins to speed and</p><p>shudder.</p><p>Individual rivuletsstandout likecoarse, throbbingveins.MitchellFeigenbaum</p><p>stands at streamside.He is sweating slightly in sports coat and corduroys and</p><p>puffingonacigarette.Hehasbeenwalkingwithfriends,buttheyhavegoneon</p><p>ahead to the quieter pools upstream. Suddenly, in what might be a demented</p><p>high-speedparodyofa tennisspectator,hestarts turninghisheadfromside to</p><p>side. “You can focus on something, a bit of foam or something. If youmove</p><p>yourheadfastenough,youcanallofasuddendiscernthewholestructureofthe</p><p>surface,andyoucanfeelitinyourstomach.”Hedrawsinmoresmokefromhis</p><p>cigarette.“But foranyonewithamathematicalbackground, ifyou lookat this</p><p>stuff,oryouseecloudswithalltheirpuffsontopofpuffs,oryoustandatasea</p><p>wallinastorm,youknowthatyoureallydon’tknowanything.”</p><p>Orderinchaos.Itwasscience’soldestcliché.Theideaofhiddenunityand</p><p>common underlying form in nature had an intrinsic appeal, and it had an</p><p>unfortunatehistoryofinspiringpseudoscientistsandcranks.WhenFeigenbaum</p><p>came to Los Alamos National Laboratory in 1974, a year shy of his thirtieth</p><p>birthday, he knew that if physicistswere tomake something of the idea now,</p><p>theywouldneedapracticalframework,awaytoturnideasintocalculations.It</p><p>wasfarfromobvioushowtomakeafirstapproachtotheproblem.</p><p>Feigenbaum was hired by Peter Carruthers, a calm, deceptively genial</p><p>physicistwhocamefromCornellin1973totakeovertheTheoreticalDivision.</p><p>Hisfirstactwastodismissahalf-dozenseniorscientists—LosAlamosprovides</p><p>itsstaffwithnoequivalentofuniversitytenure—andtoreplacethemwithsome</p><p>brightyoungresearchersofhisownchoosing.Asascientificmanager,hehad</p><p>strongambition,butheknewfromexperiencethatgoodsciencecannotalways</p><p>beplanned.</p><p>“IfyouhadsetupacommitteeinthelaboratoryorinWashingtonandsaid,</p><p>‘Turbulence is really in our way, we’ve got to understand it, the lack of</p><p>understandingreallydestroysourchanceofmakingprogressinalotoffields,’</p><p>then,ofcourse,youwouldhireateam.You’dgetagiantcomputer.You’dstart</p><p>runningbigprograms.Andyouwouldnevergetanywhere.Insteadwehavethis</p><p>smartguy,sittingquietly—talkingtopeople,tobesure,butmostlyworkingall</p><p>by himself.” They had talked about turbulence, but time passed, and even</p><p>CarrutherswasnolongersurewhereFeigenbaumwasheaded.“Ithoughthehad</p><p>quitandfoundadifferentproblem.LittledidIknowthatthisotherproblemwas</p><p>the same problem. It seems to have been the issue on which many different</p><p>fields of sciencewere stuck—theywere stuck on this aspect of the nonlinear</p><p>behaviorofsystems.Now,nobodywouldhavethoughtthattherightbackground</p><p>for this problem was to know particle physics, to know something about</p><p>quantumfieldtheory,andtoknowthatinquantumfieldtheoryyouhavethese</p><p>structures known as the renormalization group. Nobody knew that youwould</p><p>need to understand the general theory of stochastic processes, and also fractal</p><p>structures.</p><p>“Mitchellhadtherightbackground.Hedidtherightthingattherighttime,</p><p>andhediditverywell.Nothingpartial.Hecleanedoutthewholeproblem.”</p><p>FeigenbaumbroughttoLosAlamosaconvictionthathissciencehadfailed</p><p>to understand hard problems—nonlinear problems.Although he had produced</p><p>almost nothing as a physicist, he had accumulated an unusual intellectual</p><p>background. He had a sharp working knowledge of the most challenging</p><p>mathematicalanalysis,newkindsofcomputational techniquethatpushedmost</p><p>scientists to their limits. He had managed not to purge himself of some</p><p>seeminglyunscientific ideas fromeighteenth-centuryRomanticism.Hewanted</p><p>to do science that would be new. He began by putting aside any thought of</p><p>understanding real complexity and instead turned to the simplest nonlinear</p><p>equationshecouldfind.</p><p>THEMYSTERY OF THE UNIVERSE first announced itself to the four-year–old</p><p>Mitchell Feigenbaum through a Silvertone radio sitting in his parents’ living</p><p>roomintheFlatbushsectionofBrooklynsoonafterthewar.Hewasdizzywith</p><p>the thoughtofmusic arriving fromno tangible cause.Thephonograph,on the</p><p>other hand, he felt he understood. His grandmother had given him a special</p><p>dispensationtoputonthe78s.</p><p>HisfatherwasachemistwhoworkedforthePortofNewYorkAuthority</p><p>andlaterforClairol.Hismothertaughtinthecity’spublicschools.Mitchellfirst</p><p>decided to become an electrical engineer, a sort of professional known in</p><p>Brooklyntomakeagoodliving.Laterherealizedthatwhathewantedtoknow</p><p>aboutaradiowasmorelikelytobefoundinphysics.Hewasoneofageneration</p><p>ofscientists raised in theouterboroughsofNewYorkwhomade theirway to</p><p>brilliantcareersviathegreatpublichighschools—inhiscase,SamuelJ.Tilden</p><p>—andthenCityCollege.</p><p>GrowingupsmartinBrooklynwasinsomemeasureamatterofsteeringan</p><p>unevencoursebetweentheworldofmindandtheworldofotherpeople.Hewas</p><p>immenselygregariouswhenveryyoung,whichheregardedasakeytonotbeing</p><p>beaten up. But something clickedwhen he realized he could learn things. He</p><p>becamemoreandmoredetachedfromhisfriends.Ordinaryconversationcould</p><p>notholdhisinterest.Sometimeinhislastyearofcollege,itstruckhimthathe</p><p>hadmissedhis adolescence, andhemadeadeliberateproject outof regaining</p><p>touchwithhumanity.Hewouldsitsilentlyinthecafeteria,listeningtostudents</p><p>chattingaboutshavingorfood,andgraduallyherelearnedmuchofthescience</p><p>oftalkingtopeople.</p><p>He graduated in 1964 and went on to the Massachusetts Institute of</p><p>Technology,wherehegothisdoctorateinelementaryparticlephysicsin1970.</p><p>Thenhe spent a fruitless fouryears atCornell andat theVirginiaPolytechnic</p><p>Institute—fruitless, that is, in terms of the steady publication of work on</p><p>manageableproblemsthatisessentialforayounguniversityscientist.Postdocs</p><p>were supposed to produce papers. Occasionally an advisor would ask</p><p>Feigenbaum what had happened to some problem, and he would say, “Oh, I</p><p>understoodit.”</p><p>Newly installed at Los Alamos, Carruthers, a formidable scientist in his</p><p>own right, prided himself on his ability to spot talent. He looked not for</p><p>intelligence but for a sort of creativity that seemed to flow from somemagic</p><p>gland.HealwaysrememberedthecaseofKennethWilson,anothersoft-spoken</p><p>Cornellphysicistwhoseemedtobeproducingabsolutelynothing.Anyonewho</p><p>talked toWilson for long realized that he had a deep capacity for seeing into</p><p>physics.SothequestionofWilson’stenurebecameasubjectofseriousdebate.</p><p>The physicists willing to gamble on his unproven potential prevailed—and it</p><p>wasasifadamburst.NotonebutafloodofpaperscameforthfromWilson’s</p><p>deskdrawers,includingworkthatwonhimtheNobelPrizein1982.</p><p>Wilson’s great contribution to physics, along with work by two other</p><p>physicists, Leo Kadanoff and Michael Fisher, was an important ancestor of</p><p>chaostheory.Thesemen,workingindependently,wereallthinkingindifferent</p><p>waysaboutwhathappenedinphasetransitions.Theywerestudyingthebehavior</p><p>ofmatternearthepointwhereitchangesfromonestatetoanother—fromliquid</p><p>to gas, or from unmagnetized tomagnetized. As singular boundaries between</p><p>two realms of existence, phase transitions tend to be highly nonlinear in their</p><p>mathematics.The smoothandpredictablebehaviorofmatter inanyonephase</p><p>tends to be little help in understanding the transitions. A pot of water on the</p><p>stoveheatsup in a regularwayuntil it reaches theboilingpoint.But then the</p><p>change in temperaturepauseswhile somethingquite interestinghappensat the</p><p>molecularinterfacebetweenliquidandgas.</p><p>AsKadanoff viewed the problem in the 1960s, phase transitions pose an</p><p>intellectualpuzzle.Thinkofablockofmetalbeingmagnetized.Asitgoesinto</p><p>anorderedstate,itmustmakeadecision.Themagnetcanbeorientedoneway</p><p>ortheother.Itisfreetochoose.Buteachtinypieceofthemetalmustmakethe</p><p>samechoice.How?</p><p>Somehow, in the process of choosing, the atoms of the metal must</p><p>communicate information to one another. Kadanoff’s insight was that the</p><p>communicationcanbemostsimplydescribed in termsofscaling. Ineffect,he</p><p>imagined dividing the metal into boxes. Each box communicates with its</p><p>immediateneighbors.Thewaytodescribethatcommunicationisthesameasthe</p><p>way todescribe thecommunicationofanyatomwith itsneighbors.Hence the</p><p>usefulnessofscaling:thebestwaytothinkofthemetalisintermsofafractal-</p><p>likemodel,withboxesofalldifferentsizes.</p><p>Muchmathematicalanalysis,andmuchexperiencewithrealsystems,was</p><p>neededtoestablishthepowerofthescalingidea.Kadanofffeltthathehadtaken</p><p>an unwieldy business and created a world of extreme beauty and self-</p><p>containedness.Partofthebeautylayinitsuniversality.Kadanoff’sideagavea</p><p>backbone to themoststrikingfactaboutcriticalphenomena,namely that these</p><p>seemingly unrelated transitions—the boiling of liquids, the magnetizing of</p><p>metals—allfollowthesamerules.</p><p>ThenWilsondidtheworkthatbroughtthewholetheorytogetherunderthe</p><p>rubricofrenormalizationgrouptheory,providingapowerfulwayofcarryingout</p><p>realcalculationsaboutrealsystems.Renormalizationhadenteredphysicsinthe</p><p>1940sasapartofquantumtheorythatmadeitpossibletocalculateinteractions</p><p>of electrons and photons. A problem with such calculations, as with the</p><p>calculationsKadanoffandWilsonworriedabout,wasthatsomeitemsseemedto</p><p>require treatment as infinite quantities, a messy and unpleasant business.</p><p>Renormalizing the system, in ways devised by Richard Feynman, Julian</p><p>Schwinger,FreemanDyson,andotherphysicists,gotridoftheinfinities.</p><p>Onlymuchlater,inthe1960s,didWilsondigdowntotheunderlyingbasis</p><p>for renormalization’s success. Like Kadanoff, he thought about scaling</p><p>principles. Certain quantities, such as themass of a particle, had always been</p><p>consideredfixed—asthemassofanyobjectineverydayexperienceisfixed.The</p><p>renormalization shortcut succeeded by acting as though a quantity like mass</p><p>werenotfixedatall.Suchquantitiesseemedtofloatupordowndependingon</p><p>thescale fromwhich theywereviewed. It seemedabsurd.Yet itwasanexact</p><p>analogueofwhatBenoitMandelbrotwasrealizingaboutgeometricalshapesand</p><p>the coastline of England. Their length could not be measured independent of</p><p>scale.Therewasakindofrelativityinwhichthepositionoftheobserver,near</p><p>orfar,onthebeachorinasatellite,affectedthemeasurement.AsMandelbrot,</p><p>too, had seen, the variation across scales was not arbitrary; it followed rules.</p><p>Variabilityinthestandardmeasuresofmassorlengthmeantthatadifferentsort</p><p>of quantity was remaining fixed. In the case of fractals, it was the fractional</p><p>dimension—a constant that could be calculated and used as a tool for further</p><p>calculations. Allowing mass to vary depending on scale meant that</p><p>mathematicianscouldrecognizesimilarityacrossscales.</p><p>Soforthehardworkofcalculation,Wilson’srenormalizationgrouptheory</p><p>providedadifferentrouteintoinfinitelydenseproblems.Untilthentheonlyway</p><p>to approach highly nonlinear problems was with a device called perturbation</p><p>theory. For purposes of calculation, you assume that the nonlinear problem is</p><p>reasonably close to some solvable, linear problem—just a small perturbation</p><p>away.You solve the linear problemandperforma complicatedbit of trickery</p><p>withtheleftoverpart,expandingitintowhatarecalledFeynmandiagrams.The</p><p>more accuracy you need, the more of these agonizing diagrams you must</p><p>produce.With luck, your calculations converge toward a solution. Luck has a</p><p>way of vanishing, however, whenever a problem is especially interesting.</p><p>Feigenbaum, like every other young particle physicist in the 1960s, found</p><p>himselfdoingendlessFeynmandiagrams.Hewas leftwith theconviction that</p><p>perturbation theory was tedious, nonilluminating, and stupid. So he loved</p><p>Wilson’snewrenormalizationgrouptheory.Byacknowledgingself-similarity,it</p><p>gaveawayofcollapsingthecomplexity,onelayeratatime.</p><p>Inpracticetherenormalizationgroupwasfarfromfoolproof.Itrequireda</p><p>gooddealof ingenuity tochoose just the rightcalculations tocapture theself-</p><p>similarity.However, itworkedwell enough and often enough to inspire some</p><p>physicists,Feigenbaumincluded,totryitontheproblemofturbulence.Afterall,</p><p>self-similarity seemed to be the signature of turbulence, fluctuations upon</p><p>fluctuations,whorlsuponwhorls.Butwhatabout theonsetof turbulence—the</p><p>mysterious moment when an orderly system turned chaotic. There was no</p><p>evidencethattherenormalizationgrouphadanythingtosayaboutthistransition.</p><p>Therewasnoevidence,forexample,thatthetransitionobeyedlawsofscaling.</p><p>AS A GRADUATE STUDENT at M.I.T., Feigenbaum had an experience that</p><p>stayedwithhimformanyyears.HewaswalkingwithfriendsaroundtheLincoln</p><p>Reservoir inBoston.Hewasdeveloping ahabit of taking four– and five-hour</p><p>walks,attuninghimselftothepanoplyofimpressionsandideasthatwouldflow</p><p>throughhismind.Onthisdayhebecamedetachedfromthegroupandwalked</p><p>alone.Hepassedsomepicnickersand,ashemovedaway,heglancedbackevery</p><p>so often, hearing the sounds of their voices, watching the motions of hands</p><p>gesticulatingorreachingforfood.Suddenlyhefeltthatthetableauhadcrossed</p><p>somethresholdintoincomprehensibility.Thefiguresweretoosmalltobemade</p><p>out. The actions seemed disconnected, arbitrary, random. What faint sounds</p><p>reachedhimhadlostmeaning.</p><p>The ceaseless motion and incomprehensible bustle of life. Feigenbaum</p><p>recalled the words of Gustav Mahler, describing a sensation that he tried to</p><p>capture in the third movement of his Second Symphony. Like the motions of</p><p>dancing figures inabrilliantly litballroom intowhichyou look from thedark</p><p>nightoutsideandfromsuchadistancethatthemusicisinaudible….Lifemay</p><p>appear senseless to you. Feigenbaum was listening to Mahler and reading</p><p>Goethe, immersing himself in their highRomantic attitudes. Inevitably it was</p><p>Goethe’s Faust he most reveled in, soaking up its combination of the most</p><p>passionate ideas about the world with the most intellectual. Without some</p><p>Romantic inclinations, he surely would have dismissed a sensation like his</p><p>confusionatthereservoir.Afterall,whyshouldn’tphenomenalosemeaningas</p><p>they are seen from greater distances? Physical laws provided a trivial</p><p>explanation for their shrinking. On second thought the connection between</p><p>shrinking and loss of meaning was not so obvious.Why should it be that as</p><p>thingsbecomesmalltheyalsobecomeincomprehensible?</p><p>Hetriedquiteseriously toanalyzethisexperiencein termsof the toolsof</p><p>theoreticalphysics,wonderingwhathecould</p><p>sayaboutthebrain’smachineryof</p><p>perception.You see somehuman transactions andyoumakedeductions about</p><p>them.Giventhevastamountofinformationavailabletoyoursenses,howdoes</p><p>yourdecodingapparatussortitout?Clearly—oralmostclearly—thebraindoes</p><p>notownanydirectcopiesofstuffintheworld.Thereisnolibraryofformsand</p><p>ideasagainstwhichtocomparetheimagesofperception.Informationisstored</p><p>in a plastic way, allowing fantastic juxtapositions and leaps of imagination.</p><p>Somechaosexistsout there,and thebrainseems tohavemore flexibility than</p><p>classicalphysicsinfindingtheorderinit.</p><p>Atthesametime,Feigenbaumwasthinkingaboutcolor.Oneoftheminor</p><p>skirmishesofscienceinthefirstyearsofthenineteenthcenturywasadifference</p><p>ofopinionbetweenNewton’sfollowersinEnglandandGoetheinGermanyover</p><p>the nature of color. ToNewtonian physics,Goethe’s ideaswere just somuch</p><p>pseudoscientificmeandering.Goethe refused toviewcolorasastaticquantity,</p><p>tobemeasuredinaspectrometerandpinneddownlikeabutterflytocardboard.</p><p>He argued that color is a matter of perception. “With light poise and</p><p>counterpoise,Natureoscillateswithinherprescribedlimits,”hewrote,“yetthus</p><p>ariseallthevarietiesandconditionsofthephenomenawhicharepresentedtous</p><p>inspaceandtime.”</p><p>The touchstone of Newton’s theory was his famous experiment with a</p><p>prism.A prism breaks a beam ofwhite light into a rainbow of colors, spread</p><p>across thewhole visible spectrum, andNewton realized that those pure colors</p><p>mustbe theelementarycomponents thatadd toproducewhite.Further,witha</p><p>leap of insight, he proposed that the colors corresponded to frequencies. He</p><p>imagined that somevibratingbodies—corpuscleswas the antiqueword—must</p><p>be producing colors in proportion to the speed of the vibrations. Considering</p><p>how little evidence supported this notion, it was as unjustifiable as it was</p><p>brilliant.Whatisred?Toaphysicist,itislightradiatinginwavesbetween620</p><p>to 800 bil-lionths of a meter long. Newton’s optics proved themselves a</p><p>thousand times over, while Goethe’s treatise on color faded into merciful</p><p>obscurity. When Feigenbaum went looking for it, he discovered that the one</p><p>copyinHarvard’slibrarieshadbeenremoved.</p><p>He finally did trackdowna copy, andhe found thatGoethehad actually</p><p>performed an extraordinary set of experiments in his investigation of colors.</p><p>GoethebeganasNewtonhad,withaprism.Newtonhadheldaprismbeforea</p><p>light,castingthedividedbeamontoawhitesurface.Goetheheldtheprismtohis</p><p>eyeand looked through it.Heperceivednocolor at all, neither a rainbownor</p><p>individualhues.Lookingataclearwhitesurfaceoraclearblueskythroughthe</p><p>prismproducedthesameeffect:uniformity.</p><p>Butifaslightspotinterruptedthewhitesurfaceoracloudappearedinthe</p><p>sky, then he would see a burst of color. It is “the interchange of light and</p><p>shadow,”Goetheconcluded, thatcausescolor.Hewenton toexplore theway</p><p>people perceive shadows cast by different sources of colored light. He used</p><p>candlesandpencils,mirrorsandcoloredglass,moonlightandsunlight,crystals,</p><p>liquids,andcolorwheelsinathoroughrangeofexperiments.Forexample,helit</p><p>a candle before a piece of white paper at twilight and held up a pencil. The</p><p>shadow in thecandlelightwasabrilliantblue.Why?Thewhitepaperalone is</p><p>perceivedaswhite,either in thedecliningdaylightor in theadded lightof the</p><p>warmercandle.Howdoesashadowdividethewhiteintoaregionofblueanda</p><p>region of reddish-yellow? Color is “a degree of darkness,” Goethe argued,</p><p>“allied to shadow.”Above all, in amoremodern language, color comes from</p><p>boundaryconditionsandsingularities.</p><p>WhereNewtonwas reductionist,Goethewasholistic.Newtonbroke light</p><p>apart and found themost basic physical explanation for color.Goethewalked</p><p>through flower gardens and studied paintings, looking for a grand, all-</p><p>encompassingexplanation.Newtonmadehistheoryofcolorfitamathematical</p><p>scheme for all of physics. Goethe, fortunately or unfortunately, abhorred</p><p>mathematics.</p><p>Feigenbaum persuaded himself that Goethe had been right about color.</p><p>Goethe’s ideas resemble a facile notion, popular among psychologists, that</p><p>makes a distinction between hard physical reality and the variable subjective</p><p>perceptionofit.Thecolorsweperceivevaryfromtimetotimeandfromperson</p><p>to person—that much is easy to say. But as Feigenbaum understood them,</p><p>Goethe’s ideas hadmore true science in them.Theywere hard and empirical.</p><p>Overandoveragain,Goetheemphasizedtherepeatabilityofhisexperiments.It</p><p>was theperceptionofcolor, toGoethe, thatwasuniversalandobjective.What</p><p>scientific evidence was there for a definable real-world quality of redness</p><p>independentofourperception?</p><p>Feigenbaum found himself asking what sort of mathematical formalisms</p><p>mightcorrespond tohumanperception,particularlyaperception that sifted the</p><p>messymultiplicity of experience and found universal qualities.Redness is not</p><p>necessarilyaparticularbandwidthoflight,astheNewtonianswouldhaveit.Itis</p><p>a territoryofachaoticuniverse,and theboundariesof that territoryarenotso</p><p>easy to describe—yet our minds find redness with regular and verifiable</p><p>consistency. These were the thoughts of a young physicist, far removed, it</p><p>seemed, from such problems as fluid turbulence. Still, to understand how the</p><p>humanmind sorts through the chaos of perception, surely onewould need to</p><p>understandhowdisordercanproduceuniversality.</p><p>WHEN FEIGENBAUM BEGAN to think about nonlinearity at Los Alamos, he</p><p>realizedthathiseducationhadtaughthimnothinguseful.Tosolveasystemof</p><p>nonlinear differential equations was impossible, notwithstanding the special</p><p>examples constructed in textbooks. Perturbative technique, making successive</p><p>corrections to a solvableproblem thatonehopedwould lie somewherenearby</p><p>the real one, seemed foolish. He read through texts on nonlinear flows and</p><p>oscillations and decided that little existed to help a reasonable physicist. His</p><p>computational equipment consisting solely of pencil and paper, Feigenbaum</p><p>decidedtostartwithananalogueofthesimpleequationthatRobertMaystudied</p><p>inthecontextofpopulationbiology.</p><p>It happened to be the equation high school students use in geometry to</p><p>graphaparabola. Itcanbewrittenasy=r(x-x2).Everyvalueofxproducesa</p><p>valueofy,andtheresultingcurveexpressestherelationofthetwonumbersfor</p><p>therangeofvalues.Ifx(thisyear’spopulation)issmall,theny(nextyear’s)is</p><p>small,but larger thanx; thecurveisrisingsteeply. Ifx is in themiddleof the</p><p>range,thenyislarge.Buttheparabolalevelsoffandfalls,sothatifxislarge,</p><p>thenywillbesmallagain.That iswhatproduces theequivalentofpopulation</p><p>crashesinecologicalmodeling,preventingunrealisticunrestrainedgrowth.</p><p>ForMayandthenFeigenbaum,thepointwastousethissimplecalculation</p><p>not once, but repeated endlessly as a feedback loop. The output of one</p><p>calculation was fed back in as input for the next. To see what happened</p><p>graphically, the parabola helped enormously.Pick a startingvalue along thex</p><p>axis.Drawalineuptowhereitmeetstheparabola.Readtheresultingvalueoff</p><p>they axis.And start all overwith thenewvalue.The sequencebounces from</p><p>place toplaceon theparabolaat first,</p><p>and then,perhaps,homes inonastable</p><p>equilibrium,wherexandyareequalandthevaluethusdoesnotchange.</p><p>In spirit, nothing could have been further removed from the complex</p><p>calculationsof standardphysics. Insteadofa labyrinthinescheme tobesolved</p><p>one time, this was a simple calculation performed over and over again. The</p><p>numerical experimenter would watch, like a chemist peering at a reaction</p><p>bubblingawayinsideabeaker.Heretheoutputwasjustastringofnumbers,and</p><p>itdidnotalwaysconvergetoasteadyfinalstate.Itcouldenduposcillatingback</p><p>andforthbetweentwovalues.OrasMayhadexplainedtopopulationbiologists,</p><p>it could keep on changing chaotically as long as anyone cared to watch. The</p><p>choice among these different possible behaviors depended on the value of the</p><p>tuningparameter.</p><p>Feigenbaum carried out numerical work of this faintly experimental sort</p><p>and, at the same time, tried more traditional theoretical ways of analyzing</p><p>nonlinear functions. Even so, he could not see thewhole picture ofwhat this</p><p>equation could do. But he could see that the possibilities were already so</p><p>complicated that they would be viciously hard to analyze. He also knew that</p><p>threeLosAlamosmathematicians—NicholasMetropolis,PaulStein,andMyron</p><p>Stein—hadstudiedsuch“maps” in1971,andnowPaulSteinwarnedhim that</p><p>the complexity was frightening indeed. If this simplest of equations already</p><p>provedintractable,whataboutthefarmorecomplicatedequationsthatascientist</p><p>wouldwritedownforrealsystems?Feigenbaumputthewholeproblemonthe</p><p>shelf.</p><p>In the brief history of chaos, this one innocent-looking equation provides</p><p>the most succinct example of how different sorts of scientists looked at one</p><p>problem inmany differentways. To the biologists, it was an equationwith a</p><p>message:Simplesystemscandocomplicatedthings.ToMetropolis,Stein,and</p><p>Stein,theproblemwastocatalogueacollectionoftopologicalpatternswithout</p><p>referencetoanynumericalvalues.Theywouldbeginthefeedbackprocessata</p><p>particularpointandwatchthesucceedingvaluesbouncefromplacetoplaceon</p><p>the parabola. As the values moved to the right or the left, they wrote down</p><p>sequences ofR’s and L’s. Pattern number one: R. Pattern number two:RLR.</p><p>Pattern number 193: RLLLLLRRLL. These sequences had some interesting</p><p>featurestoamathematician—theyalwaysseemedtorepeatinthesamespecial</p><p>order.Buttoaphysicisttheylookedobscureandtedious.</p><p>No one realized it then, but Lorenz had looked at the same equation in</p><p>1964,asametaphorforadeepquestionaboutclimate.Thequestionwassodeep</p><p>thatalmostnoonehad thought toask itbefore:Doesaclimateexist?That is,</p><p>doestheearth’sweatherhavealongtermaverage?Mostmeteorologists,thenas</p><p>now, took the answer for granted. Surely anymeasurable behavior, nomatter</p><p>howitfluctuates,musthaveanaverage.Yetonreflection,itisfarfromobvious.</p><p>AsLorenzpointedout, theaverageweather for the last12,000yearshasbeen</p><p>notablydifferentthantheaveragefortheprevious12,000,whenmostofNorth</p><p>Americawascoveredbyice.Wasthereoneclimatethatchangedtoanotherfor</p><p>some physical reason? Or is there an even longer-term climate within which</p><p>those periods were just fluctuations? Or is it possible that a system like the</p><p>weathermayneverconvergetoanaverage?</p><p>Lorenzaskedasecondquestion.Supposeyoucouldactuallywritedownthe</p><p>completesetofequationsthatgoverntheweather.Inotherwords,supposeyou</p><p>had God’s own code. Could you then use the equations to calculate average</p><p>statistics for temperature or rainfall? If the equations were linear, the answer</p><p>wouldbeaneasyyes.Buttheyarenonlinear.SinceGodhasnotmadetheactual</p><p>equationsavailable,Lorenzinsteadexaminedthequadraticdifferenceequation.</p><p>Like May, Lorenz first examined what happened as the equation was</p><p>iterated, given some parameter. With low parameters he saw the equation</p><p>reachingastablefixedpoint.There,certainly,thesystemproduceda“climate”</p><p>in themost trivial sense possible—the “weather” never changed.With higher</p><p>parametershe saw thepossibilityofoscillationbetween twopoints, and there,</p><p>too, the system converged to a simple average. But beyond a certain point,</p><p>Lorenzsawthatchaosensues.Sincehewasthinkingaboutclimate,heaskednot</p><p>onlywhethercontinualfeedbackwouldproduceperiodicbehavior,butalsowhat</p><p>the average outputwould be.And he recognized that the answerwas that the</p><p>average, too, fluctuatedunstably.When theparametervaluewaschangedever</p><p>so slightly, the average might change dramatically. By analogy, the earth’s</p><p>climate might never settle reliably into an equilibriumwith average longterm</p><p>behavior.</p><p>Asamathematicspaper,Lorenz’sclimateworkwouldhavebeenafailure</p><p>—he proved nothing in the axiomatic sense. As a physics paper, too, it was</p><p>seriously flawed,becausehe couldnot justifyusing sucha simple equation to</p><p>drawconclusions about the earth’s climate.Lorenzknewwhat hewas saying,</p><p>though.“Thewriterfeelsthatthisresemblanceisnomereaccident,butthatthe</p><p>differenceequationcapturesmuchofthemathematics,evenifnotthephysics,of</p><p>the transitions from one regime of flow to another, and, indeed, of thewhole</p><p>phenomenon of instability.”Even twenty years later, no one could understand</p><p>what intuition justified such a bold claim, published in Tellus, a Swedish</p><p>meteorology journal. (“Tellus! Nobody reads Tellus,” a physicist exclaimed</p><p>bitterly.) Lorenz was coming to understand ever more deeply the peculiar</p><p>possibilities of chaotic systems—more deeply than he could express in the</p><p>languageofmeteorology.</p><p>As he continued to explore the changing masks of dynamical systems,</p><p>Lorenzrealized thatsystemsslightlymorecomplicated than thequadraticmap</p><p>could produce other kinds of unexpected patterns. Hiding within a particular</p><p>systemcouldbemorethanonestablesolution.Anobservermightseeonekind</p><p>ofbehavioroveravery long time,yet acompletelydifferentkindofbehavior</p><p>couldbejustasnaturalforthesystem.Suchasystemiscalledintransitive.Itcan</p><p>stayinoneequilibriumortheother,butnotboth.Onlyakickfromoutsidecan</p><p>force it to change states. In a trivial way, a standard pendulum clock is an</p><p>intransitivesystem.Asteadyflowofenergycomesinfromawind-upspringor</p><p>abatterythroughanescapementmechanism.Asteadyflowofenergyisdrained</p><p>outbyfriction.Theobviousequilibriumstateisaregularswingingmotion.Ifa</p><p>passerbybumpstheclock,thependulummightspeeduporslowdownfromthe</p><p>momentary jolt butwill quickly return to its equilibrium.But the clock has a</p><p>secondequilibriumaswell—asecondvalidsolutiontoitsequationsofmotion—</p><p>and that is the state inwhich the pendulum is hanging straight down and not</p><p>moving.Alesstrivialintransitivesystem—perhapswithseveraldistinctregions</p><p>ofutterlydifferentbehavior—couldbeclimateitself.</p><p>Climatologistswho use global computermodels to simulate the longterm</p><p>behavioroftheearth’satmosphereandoceanshaveknownforseveralyearsthat</p><p>their models allow at least one dramatically different equilibrium. During the</p><p>entiregeologicalpast,thisalternativeclimatehasneverexisted,butitcouldbe</p><p>an equally valid solution to the system of equations governing the earth. It is</p><p>what some climatologists call the White Earth climate: an earth</p><p>whose</p><p>continents are covered by snow and whose oceans are covered by ice. A</p><p>glaciatedearthwouldreflectseventypercentoftheincomingsolarradiationand</p><p>so would stay extremely cold. The lowest layer of the atmosphere, the</p><p>troposphere, would be much thinner. The storms that would blow across the</p><p>frozensurfacewouldbemuchsmallerthanthestormsweknow.Ingeneral,the</p><p>climatewouldbe lesshospitable to lifeasweknowit.Computermodelshave</p><p>such a strong tendency to fall into the White Earth equilibrium that</p><p>climatologistsfindthemselveswonderingwhyithasnevercomeabout. Itmay</p><p>simplybeamatterofchance.</p><p>To push the earth’s climate into the glaciated statewould require a huge</p><p>kickfromsomeexternalsource.ButLorenzdescribedyetanotherplausiblekind</p><p>ofbehaviorcalled“almost-intransitivity.”Analmost-intransitivesystemdisplays</p><p>one sort of average behavior for a very long time, fluctuating within certain</p><p>bounds. Then, for no reason whatsoever, it shifts into a different sort of</p><p>behavior, still fluctuating but producing a different average. The people who</p><p>designcomputermodelsareawareofLorenz’sdiscovery,buttheytryatallcosts</p><p>to avoid almost-intransitivity. It is too unpredictable. Their natural bias is to</p><p>makemodelswith a strong tendency to return to the equilibriumwemeasure</p><p>everydayontherealplanet.Then,toexplainlargechangesinclimate,theylook</p><p>for external causes—changes in the earth’s orbit around the sun, for example.</p><p>Yet it takes no great imagination for a climatologist to see that almost-</p><p>intransitivitymightwellexplainwhytheearth’sclimatehasdriftedinandoutof</p><p>longIceAgesatmysterious,irregularintervals.Ifso,nophysicalcauseneedbe</p><p>foundforthetiming.TheIceAgesmaysimplybeabyproductofchaos.</p><p>LIKE A GUN COLLECTOR wistfully recalling the Colt .45 in the era of</p><p>automaticweaponry,themodernscientistnursesacertainnostalgiafortheHP–</p><p>65hand-heldcalculator.Inthefewyearsofitssupremacy,thismachinechanged</p><p>many scientists’ working habits forever. For Feigenbaum, it was the bridge</p><p>betweenpencil-and–paper and a style ofworkingwith computers that hadnot</p><p>yetbeenconceived.</p><p>HeknewnothingofLorenz,but in thesummerof1975,atagathering in</p><p>Aspen, Colorado, he heard Steve Smale talk about some of themathematical</p><p>qualitiesof thesamequadraticdifferenceequation.Smaleseemedtothinkthat</p><p>therewere some interestingopenquestions about the exact point atwhich the</p><p>mappingchangesfromperiodictochaotic.Asalways,Smalehadasharpinstinct</p><p>for questionsworth exploring. Feigenbaumdecided to look into it oncemore.</p><p>With his calculator he began to use a combination of analytic algebra and</p><p>numericalexplorationtopiece togetheranunderstandingof thequadraticmap,</p><p>concentratingontheboundaryregionbetweenorderandchaos.</p><p>Metaphorically—but only metaphorically—he knew that this region was</p><p>likethemysteriousboundarybetweensmoothflowandturbulenceinafluid.It</p><p>was the region that Robert May had called to the attention of population</p><p>biologistswhohadpreviouslyfailedtonoticethepossibilityofanybutorderly</p><p>cycles in changing animal populations.En route to chaos in this regionwas a</p><p>cascade of period-doublings, the splitting of two-cycles into four-cycles, four-</p><p>cyclesintoeight-cycles,andsoon.Thesesplittingsmadeaafascinatingpattern.</p><p>Theywerethepointsatwhichaslightchangeinfecundity,forexample,might</p><p>leadapopulationofgypsymothstochangefromafour-yearcycletoaneight-</p><p>year cycle. Feigenbaum decided to begin by calculating the exact parameter</p><p>valuesthatproducedthesplittings.</p><p>Intheend,itwastheslownessofthecalculatorthatledhimtoadiscovery</p><p>that August. It took ages—minutes, in fact—to calculate the exact parameter</p><p>value of each period-doubling.The higher up the chain hewent, the longer it</p><p>took. With a fast computer, and with a printout, Feigenbaum might have</p><p>observednopattern.Buthehadtowritethenumbersdownbyhand,andthenhe</p><p>hadtothinkaboutthemwhilehewaswaiting,andthen,tosavetime,hehadto</p><p>guesswherethenextanswerwouldbe.</p><p>Yet all in an instant he saw that he did not have to guess. Therewas an</p><p>unexpected regularity hidden in this system: the numbers were converging</p><p>geometrically,thewayalineofidenticaltelephonepolesconvergestowardthe</p><p>horizon in a perspective drawing. If you know how big to make any two</p><p>telephonepoles, youknowall the rest; the ratioof the second to the firstwill</p><p>alsobetheratioofthethirdtothesecond,andsoon.Theperiod-doublingswere</p><p>not just coming faster and faster, but theywere coming faster and faster at a</p><p>constantrate.</p><p>Whyshouldthisbeso?Ordinarily,thepresenceofgeometricconvergence</p><p>suggeststhatsomething,somewhere,isrepeatingitselfondifferentscales.Butif</p><p>there was a scaling pattern inside this equation, no one had ever seen it.</p><p>Feigenbaumcalculatedtheratioofconvergencetothefinestprecisionpossible</p><p>onhismachine—threedecimalplaces—andcameupwithanumber,4.669.Did</p><p>thisparticularratiomeananything?Feigenbaumdidwhatanyonewoulddowho</p><p>caredaboutnumbers.Hespenttherestofthedaytryingtofitthenumbertoall</p><p>thestandardconstants—π,e,andsoforth.Itwasavariantofnone.</p><p>Oddly, Robert May realized later that he, too, had seen this geometric</p><p>convergence.Butheforgotitasquicklyashenotedit.FromMay’sperspective</p><p>in ecology, itwas anumerical peculiarity andnothingmore. In the real-world</p><p>systems hewas considering, systems of animal populations or even economic</p><p>models,theinevitablenoisewouldoverwhelmanydetailthatprecise.Thevery</p><p>messiness that had led him so far stopped him at the crucial point.Maywas</p><p>excited by the gross behavior of the equation. He never imagined that the</p><p>numericaldetailswouldproveimportant.</p><p>Feigenbaumknewwhathehad,becausegeometricconvergencemeantthat</p><p>somethinginthisequationwasscaling,andheknewthatscalingwasimportant.</p><p>All of renormalization theory depended on it. In an apparently unruly system,</p><p>scaling meant that some quality was being preserved while everything else</p><p>changed.Someregularitylaybeneaththeturbulentsurfaceoftheequation.But</p><p>where?Itwashardtoseewhattodonext.</p><p>Summer turns rapidly to autumn in the rarefied Los Alamos air, and</p><p>OctoberhadnearlyendedwhenFeigenbaumwasstruckbyanoddthought.He</p><p>knewthatMetropolis,Stein,andSteinhadlookedatotherequationsaswelland</p><p>hadfoundthatcertainpatternscarriedoverfromonesortoffunctiontoanother.</p><p>ThesamecombinationsofR’sandL’sappeared,andtheyappearedinthesame</p><p>order. One function had involved the sine of a number, a twist that made</p><p>Feigenbaum’scarefullyworked-outapproachtotheparabolaequationirrelevant.</p><p>Hewouldhavetostartover.SohetookhisHP–65againandbegantocompute</p><p>theperiod-doublings forxt+1= r sinπxt.Calculatinga trigonometric function</p><p>madetheprocessthatmuchslower,andFeigenbaumwonderedwhether,aswith</p><p>the simpler version of the equation, hewould be able to use a shortcut. Sure</p><p>enough, scanning the numbers, he realized that they were again converging</p><p>geometrically.Itwassimplyamatterofcalculatingtheconvergencerateforthis</p><p>new equation. Again, his precision was limited, but he got a result to three</p><p>decimalplaces:4.669.</p><p>Itwasthesamenumber.Incredibly,thistrigonometric</p><p>functionwasnotjust</p><p>displayingaconsistent,geometricregularity.Itwasdisplayingaregularitythat</p><p>wasnumerically identical to thatofamuchsimplerfunction.Nomathematical</p><p>orphysicaltheoryexistedtoexplainwhytwoequationssodifferentinformand</p><p>meaningshouldleadtothesameresult.</p><p>Feigenbaum called Paul Stein. Stein was not prepared to believe the</p><p>coincidence on such scanty evidence. The precision was low, after all.</p><p>Nevertheless,FeigenbaumalsocalledhisparentsinNewJerseytotellthemhe</p><p>had stumbled across something profound. He told hismother it was going to</p><p>make him famous. Then he started trying other functions, anything he could</p><p>think of thatwent through a sequence of bifurcations on theway to disorder.</p><p>Everyoneproducedthesamenumber.</p><p>Feigenbaumhadplayedwithnumbersallhislife.Whenhewasateen-ager</p><p>heknewhowtocalculatelogarithmsandsinesthatmostpeoplewouldlookup</p><p>in tables.But he had never learned to use any computer bigger than his hand</p><p>calculator—and in this he was typical of physicists and mathematicians, who</p><p>tended to disdain themechanistic thinking that computerwork implied.Now,</p><p>though,itwastime.HeaskedacolleaguetoteachhimFortran,and,bytheend</p><p>of the day, for a variety of functions, he had calculated his constant to five</p><p>decimal places, 4.66920. That night he read about double precision in the</p><p>manual, and thenextdayhegotas faras4.6692016090—enoughprecision to</p><p>convinceStein.Feigenbaumwasn’tquitesurehehadconvincedhimself,though.</p><p>Hehadsetouttolookforregularity—thatwaswhatunderstandingmathematics</p><p>meant—buthehadalsosetoutknowingthatparticularkindsofequations, just</p><p>like particular physical systems, behave in special, characteristic ways. These</p><p>equationsweresimple,afterall.Feigenbaumunderstoodthequadraticequation,</p><p>heunderstoodthesineequation—themathematicswastrivial.Yetsomethingin</p><p>theheartoftheseverydifferentequations,repeatingoverandoveragain,created</p><p>a single number. He had stumbled upon something: perhaps just a curiosity;</p><p>perhapsanewlawofnature.</p><p>Imagine that a prehistoric zoologist decides that some things are heavier</p><p>than other things—they have some abstract quality he calls weight—and he</p><p>wants to investigate this idea scientifically. He has never actually measured</p><p>weight, but he thinks he has some understanding of the idea.He looks at big</p><p>snakesandlittlesnakes,bigbearsandlittlebears,andheguessesthattheweight</p><p>oftheseanimalsmighthavesomerelationshiptotheirsize.Hebuildsascaleand</p><p>startsweighing snakes.Tohisastonishment, every snakeweighs the same.To</p><p>his consternation, every bear weighs the same, too. And to his further</p><p>amazement, bears weigh the same as snakes. They all weigh 4.6692016090.</p><p>Clearlyweightisnotwhathesupposed.Thewholeconceptrequiresrethinking.</p><p>Rolling streams, swinging pendulums, electronic oscillators—many</p><p>physical systems went through a transition on the way to chaos, and those</p><p>transitions had remained too complicated for analysis. Thesewere all systems</p><p>whose mechanics seemed perfectly well understood. Physicists knew all the</p><p>right equations; yetmoving from the equations to an understanding of global,</p><p>longtermbehaviorseemedimpossible.Unfortunately,equationsforfluids,even</p><p>pendulums,werefarmorechallengingthanthesimpleone-dimensionallogistic</p><p>map.ButFeigenbaum’sdiscoveryimpliedthatthoseequationswerebesidethe</p><p>point.Theywere irrelevant.Whenorder emerged, it suddenly seemed tohave</p><p>forgottenwhat theoriginalequationwas.Quadraticor trigonometric, theresult</p><p>wasthesame.“Thewholetraditionofphysicsisthatyouisolatethemechanisms</p><p>andthenalltherestflows,”hesaid.“That’scompletelyfallingapart.Hereyou</p><p>know the right equations but they’re just not helpful. You add up all the</p><p>microscopicpiecesandyoufindthatyoucannotextendthemtothelongterm.</p><p>They’re not what’s important in the problem. It completely changes what it</p><p>meanstoknowsomething.”</p><p>Although the connection between numerics and physics was faint,</p><p>Feigenbaum had found evidence that he needed to work out a new way of</p><p>calculating complex nonlinear problems. So far, all available techniques had</p><p>depended on the details of the functions. If the function was a sine function,</p><p>Feigenbaum’s carefully worked-out calculations were sine calculations. His</p><p>discovery of universality meant that all those techniques would have to be</p><p>thrown out. The regularity had nothing to dowith sines. It had nothing to do</p><p>with parabolas. It hadnothing to dowith anyparticular function.Butwhy? It</p><p>was frustrating.Nature had pulled back a curtain for an instant and offered a</p><p>glimpseofunexpectedorder.Whatelsewasbehindthatcurtain?</p><p>WHENINSPIRATIONCAME,itwasintheformofapicture,amentalimageof</p><p>two small wavy forms and one big one. That was all—a bright, sharp image</p><p>etched inhismind,nomore,perhaps, than thevisible topofavast icebergof</p><p>mentalprocessingthathadtakenplacebelowthewaterlineofconsciousness.It</p><p>hadtodowithscaling,anditgaveFeigenbaumthepathheneeded.</p><p>He was studying attractors. The steady equilibrium reached by his</p><p>mappings is a fixed point that attracts all others—nomatter what the starting</p><p>“population,”itwillbouncesteadilyintowardtheattractor.Then,withthefirst</p><p>period-doubling, the attractor splits in two, like a dividing cell.At first, these</p><p>twopointsarepracticallytogether;then,astheparameterrises,theyfloatapart.</p><p>Then another period-doubling: each point of the attractor divides again, at the</p><p>samemoment.Feigenbaum’snumberlethimpredictwhentheperiod-doublings</p><p>wouldoccur.Nowhediscoveredthathecouldalsopredicttheprecisevaluesof</p><p>each point on this ever-more–complicated attractor—two points, four points,</p><p>eight points…He could predict the actual populations reached in the year-to–</p><p>yearoscillations.Therewasyetanothergeometricconvergence.Thesenumbers,</p><p>too,obeyedalawofscaling.</p><p>Feigenbaum was exploring a forgotten middle ground be tween</p><p>mathematicsandphysics.Hisworkwashardtoclassify.Itwasnotmathematics;</p><p>hewasnotprovinganything.Hewasstudyingnumbers,yes,butnumbersareto</p><p>amathematicianwhatbagsofcoinsaretoaninvestmentbanker:nominallythe</p><p>stuff of his profession, but actually too gritty and particular towaste time on.</p><p>Ideas are the real currencyofmathematicians.Feigenbaumwascarryingout a</p><p>program in physics, and, strange as it seemed, it was almost a kind of</p><p>experimentalphysics.</p><p>ZEROINGINONCHAOS.Asimpleequation,repeatedmanytimesover:MitchellFeigenbaumfocused</p><p>on straightforward functions, taking one number as input and producing another as output. For animal</p><p>populations,afunctionmightexpresstherelationshipbetweenthisyear’spopulationandnextyear’s.</p><p>Oneway to visualize such functions is tomake a graph, plotting input on the horizontal axis and</p><p>outputon theverticalaxis.Foreachpossible input,x, there is justoneoutput,y,andtheseformashape</p><p>representedbytheheavyline.</p><p>Then, to represent the longtermbehaviorof the system,Feigenbaumdrewa trajectory that started</p><p>withsomearbitraryx.Becauseeachywasthenfedbackintothesamefunctionasnewinput,hecouldusea</p><p>sortofschematicshortcut:Thetrajectorywouldbounceoffthe45–degreeline,thelinewherexequalsy.</p><p>Foran</p><p>ecologist,themostobvioussortoffunctionforpopulationgrowthislinear—theMalthusian</p><p>scenarioofsteady,limitlessgrowthbyafixedpercentageeachyear(left).Morerealisticfunctionsformed</p><p>anarch,sendingthepopulationbackdownwardwhenitbecametoohigh.Illustratedisthe“logisticmap,”a</p><p>perfect parabola, definedby the function y= rx(1–x),where the value of r, from0 to 4, determines the</p><p>parabola’ssteepness.ButFeigenbaumdiscoveredthatitdidnotmatterpreciselywhatsortofarchheused;</p><p>the details of the equation were beside the point. What mattered was that the function should have a</p><p>“hump.”</p><p>The behavior depended sensitively, though, on the steepness—the degree of nonlinearity, orwhat</p><p>RobertMaycalled“boom-and–bustiness.”Tooshallowafunctionwouldproduceextinction:Anystarting</p><p>populationwouldleadeventuallytozero.Increasingthesteepnessproducedthesteadyequilibriumthata</p><p>traditionalecologistwouldexpect;thatpoint,drawinginalltrajectories,wasaone-dimensional“attractor.”</p><p>Beyondacertainpoint,abifurcationproducedanoscillatingpopulationwithperiodtwo.Thenmore</p><p>period-doublingswouldoccur,andfinally(bottomright)thetrajectorywouldrefusetosettledownatall.</p><p>Such imageswere a starting point for Feigenbaumwhen he tried to construct a theory.He began</p><p>thinking in termsofrecursion:functionsoffunctions,andfunctionsoffunctionsoffunctions,andsoon;</p><p>mapswithtwohumps,andthenfour….</p><p>Numbers and functions were his object of study, instead of mesons and</p><p>quarks. They had trajectories and orbits. He needed to inquire into their</p><p>behavior.Heneeded—inaphrasethatlaterbecameaclichéofthenewscience</p><p>—tocreateintuition.Hisacceleratorandhiscloudchamberwerethecomputer.</p><p>Alongwithhis theory, hewasbuilding amethodology.Ordinarily a computer</p><p>userwouldconstructaproblem,feeditin,andwaitforthemachinetocalculate</p><p>itssolution—oneproblem,onesolution.Feigenbaumandthechaosresearchers</p><p>whofollowedneededmore.TheyneededtodowhatLorenzhaddone,tocreate</p><p>miniature universes and observe their evolution. Then they could change this</p><p>feature or that and observe the changed paths that would result. They were</p><p>armedwith the new conviction, after all, that tiny changes in certain features</p><p>couldleadtoremarkablechangesinoverallbehavior.</p><p>Feigenbaum quickly discovered how ill-suited the computer facilities of</p><p>Los Alamos were for the style of computing he wanted to develop. Despite</p><p>enormousresources, fargreater thanatmostuniversities,LosAlamoshadfew</p><p>terminalscapableofdisplayinggraphsandpictures,and those fewwere in the</p><p>WeaponsDivision.Feigenbaumwantedtotakenumbersandplotthemaspoints</p><p>onamap.Hehadtoresorttothemostprimitivemethodconceivable:longrolls</p><p>of printout paper with lines made by printing rows of spaces followed by an</p><p>asterisk or a plus sign. The official policy at Los Alamos held that one big</p><p>computerwasworth farmore thanmany little computers—a policy thatwent</p><p>withtheoneproblem,onesolutiontradition.Littlecomputerswerediscouraged.</p><p>Furthermore, any division’s purchase of a computer would have to meet</p><p>stringent government guidelines and a formal review. Only later, with the</p><p>budgetary complicity of theTheoreticalDivision, didFeigenbaumbecome the</p><p>recipientofa$20,000“desktopcalculator.”Thenhecouldchangehisequations</p><p>andpictureson therun, tweaking themand tuning them,playing thecomputer</p><p>like a musical instrument. For now, the only terminals capable of serious</p><p>graphics were in high-security areas—behind the fence, in local parlance.</p><p>Feigenbaum had to use a terminal hooked up by telephone lines to a central</p><p>computer. The reality of working in such an arrangement made it hard to</p><p>appreciatetherawpowerofthecomputerattheotherendoftheline.Eventhe</p><p>simplesttaskstookminutes.ToeditalineofaprogrammeantpressingReturn</p><p>and waiting while the terminal hummed incessantly and the central computer</p><p>playeditselectronicroundrobinwithotherusersacrossthelaboratory.</p><p>While hewas computing, hewas thinking.What newmathematics could</p><p>producethemultiplescalingpatternshewasobserving?Somethingaboutthese</p><p>functions must be recursive, he realized, self-referential, the behavior of one</p><p>guided by the behavior of another hidden inside it. Thewavy image that had</p><p>cometohiminamomentofinspirationexpressedsomethingaboutthewayone</p><p>function could be scaled to match another. He applied the mathematics of</p><p>renormalization group theory,with its use of scaling to collapse infinities into</p><p>manageable quantities. In the spring of 1976 he entered a mode of existence</p><p>more intense than any he had lived through. Hewould concentrate as if in a</p><p>trance, programming furiously, scribblingwith his pencil, programming again.</p><p>HecouldnotcallCdivisionforhelp,becausethatwouldmeansigningoffthe</p><p>computertousethetelephone,andreconnectionwaschancy.Hecouldnotstop</p><p>formorethanfiveminutes’thought,becausethecomputerwouldautomatically</p><p>disconnect his line. Every so often the computer would go down anyway,</p><p>leavinghimshakingwithadrenalin.Heworkedfor twomonthswithoutpause.</p><p>Hisfunctionaldaywastwenty-twohours.Hewouldtrytogotosleepinakind</p><p>ofbuzz,andawakentwohourslaterwithhisthoughtsexactlywherehehadleft</p><p>them.Hisdietwasstrictlycoffee.(Evenwhenhealthyandatpeace,Feigenbaum</p><p>subsisted exclusively on the reddest possible meat, coffee, and red wine. His</p><p>friendsspeculatedthathemustbegettinghisvitaminsfromcigarettes.)</p><p>Intheend,adoctorcalleditoff.HeprescribedamodestregimenofValium</p><p>and an enforced vacation. But by then Feigenbaum had created a universal</p><p>theory.</p><p>UNIVERSALITY MADE THE DIFFERENCE between beautiful and useful.</p><p>Mathematicians,beyondacertainpoint,carelittlewhethertheyareprovidinga</p><p>technique for calculation. Physicists, beyond a certain point, need numbers.</p><p>Universalityoffered thehope thatbysolvinganeasyproblemphysicistscould</p><p>solve much harder problems. The answers would be the same. Further, by</p><p>placinghis theory in the frameworkof the renormalizationgroup,Feigenbaum</p><p>gave it a clothing that physicists would recognize as a tool for calculating,</p><p>almostsomethingstandard.</p><p>But what made universality useful also made it hard for physicists to</p><p>believe.Universalitymeantthatdifferentsystemswouldbehaveidentically.Of</p><p>course, Feigenbaum was only studying simple numerical functions. But he</p><p>believed that his theory expressed a natural law about systems at the point of</p><p>transitionbetweenorderlyandturbulent.Everyoneknewthatturbulencemeanta</p><p>continuousspectrumofdifferentfrequencies,andeveryonehadwonderedwhere</p><p>the different frequencies came from. Suddenly you could see the frequencies</p><p>coming in sequentially. The physical implication was that real-world systems</p><p>wouldbehavein thesame,recognizableway,andthatfurthermore itwouldbe</p><p>measurablythesame.Feigenbaum’suniversalitywasnotjustqualitative,itwas</p><p>quantitative;notjuststructural,butmetrical.Itextendednotjusttopatterns,but</p><p>toprecisenumbers.Toaphysicist,thatstrainedcredulity.</p><p>Years laterFeigenbaumstillkept inadeskdrawer,wherehecouldgetat</p><p>themquickly,hisrejectionletters.Bythenhehadalltherecognitionheneeded.</p><p>HisLosAlamosworkhadwonhimprizesandawardsthatbroughtprestigeand</p><p>money.But</p><p>itstillrankledthateditorsofthetopacademicjournalshaddeemed</p><p>hiswork unfit for publication for two years after he began submitting it. The</p><p>notionofascientificbreakthroughsooriginalandunexpectedthatitcannotbe</p><p>publishedseemsaslightlytarnishedmyth.Modernscience,withitsvastflowof</p><p>information and its impartial system of peer review, is not supposed to be a</p><p>matteroftaste.OneeditorwhosentbackaFeigenbaummanuscriptrecognized</p><p>yearslaterthathehadrejectedapaperthatwasaturningpointforthefield;yet</p><p>he still argued that the paper had been unsuited to his journal’s audience of</p><p>applied mathematicians. In the meantime, even without publication,</p><p>Feigenbaum’s breakthrough became a superheated piece of news in certain</p><p>circlesofmathematicsandphysics.Thekernelof theorywasdisseminated the</p><p>way most science is now disseminated—through lectures and preprints.</p><p>Feigenbaumdescribedhisworkatconferences,andrequestsforphotocopiesof</p><p>hispaperscameinbythescoreandthenbythehundred.</p><p>MODERN ECONOMICS RELIES HEAVILY on the efficient market theory.</p><p>Knowledge is assumed to flow freely fromplace toplace.Thepeoplemaking</p><p>importantdecisionsaresupposedtohaveaccesstomoreorlessthesamebody</p><p>of information. Of course, pockets of ignorance or inside information remain</p><p>hereandthere,butonthewhole,onceknowledgeispublic,economistsassume</p><p>that it is known everywhere. Historians of science often take for granted an</p><p>efficientmarkettheoryoftheirown.Whenadiscoveryismade,whenanideais</p><p>expressed,itisassumedtobecomethecommonpropertyofthescientificworld.</p><p>Each discovery and each new insight builds on the last. Science rises like a</p><p>building,brickbybrick.Intellectualchroniclescanbe,forallpracticalpurposes,</p><p>linear.</p><p>Thatviewofscienceworksbestwhenawell-defineddisciplineawaitsthe</p><p>resolutionofawell-definedproblem.Noonemisunderstoodthediscoveryofthe</p><p>molecularstructureofDNA,forexample.Butthehistoryofideasisnotalways</p><p>soneat.Asnonlinear science arose in odd corners of different disciplines, the</p><p>flowofideasfailedtofollowthestandardlogicofhistorians.Theemergenceof</p><p>chaos as an entity unto itself was a story not only of new theories and new</p><p>discoveries,butalsoof thebelatedunderstandingofold ideas.Manypiecesof</p><p>the puzzle had been seen long before—by Poincaré, by Maxwell, even by</p><p>Einstein—andthenforgotten.Manynewpieceswereunderstoodatfirstonlyby</p><p>afewinsiders.Amathematicaldiscoverywasunderstoodbymathematicians,a</p><p>physicsdiscoverybyphysicists,ameteorologicaldiscoverybynoone.Theway</p><p>ideasspreadbecameasimportantasthewaytheyoriginated.</p><p>Eachscientisthadaprivateconstellationof intellectualparents.Eachhad</p><p>hisownpictureofthelandscapeofideas,andeachpicturewaslimitedinitsown</p><p>way.Knowledgewas imperfect.Scientistswerebiasedby thecustomsof their</p><p>disciplines or by the accidental paths of their own educations. The scientific</p><p>worldcanbesurprisinglyfinite.Nocommitteeofscientistspushedhistoryintoa</p><p>new channel—a handful of individuals did it, with individual perceptions and</p><p>individualgoals.</p><p>Afterwards,aconsensusbegantotakeshapeaboutwhichinnovationsand</p><p>which contributions had been most influential. But the consensus involved a</p><p>certainelementofrevisionism.Intheheatofdiscovery,particularlyduringthe</p><p>late 1970s, no two physicists, no two mathematicians understood chaos in</p><p>exactly the same way. A scientist accustomed to classical systems without</p><p>friction or dissipation would place himself in a lineage descending from</p><p>RussianslikeA.N.KolmogorovandV.I.Arnold.Amathematicianaccustomed</p><p>toclassicaldynamicalsystemswouldenvisionalinefromPoincarétoBirkhoff</p><p>to Levinson to Smale. Later, a mathematician’s constellation might center on</p><p>Smale, Guckenheimer, and Ruelle. Or it might emphasize a computationally</p><p>inclinedsetofforebearsassociatedwithLosAlamos:Ulam,Metropolis,Stein.</p><p>A theoretical physicist might think of Ruelle, Lorenz, Rössler, and Yorke. A</p><p>biologistwould thinkofSmale,Guckenheimer,May,andYorke.Thepossible</p><p>combinationswereendless.Ascientistworkingwithmaterials—ageologistora</p><p>seismologist—would credit the direct influence of Mandelbrot; a theoretical</p><p>physicistwouldbarelyacknowledgeknowingthename.</p><p>Feigenbaum’s role would become a special source of contention. Much</p><p>later,whenhewasridingacrestofsemicelebrity,somephysicistswentoutof</p><p>theirwaytociteotherpeoplewhohadbeenworkingonthesameproblematthe</p><p>sametime,giveortakeafewyears.Someaccusedhimoffocusingtoonarrowly</p><p>onasmallpieceofthebroadspectrumofchaoticbehavior.“Feigenbaumology”</p><p>wasoverrated,aphysicistmightsay—abeautifulpieceofwork,tobesure,but</p><p>not asbroadly influential asYorke’swork, for example. In1984,Feigenbaum</p><p>was invited to address the Nobel Symposium in Sweden, and there the</p><p>controversy swirled. Benoit Mandelbrot gave a wickedly pointed talk that</p><p>listeners laterdescribedashis“antifeigenbaumlecture.”SomehowMandelbrot</p><p>had exhumed a twenty-year–old paper on period-doubling by a Finnish</p><p>mathematician named Myrberg, and he kept describing the Feigenbaum</p><p>sequencesas“Myrbergsequences.”</p><p>ButFeigenbaumhaddiscovereduniversalityandcreatedatheorytoexplain</p><p>it.Thatwasthepivotonwhichthenewscienceswung.Unabletopublishsuch</p><p>an astonishing and counterintuitive result, he spread the word in a series of</p><p>lectures at a New Hampshire conference in August 1976, an international</p><p>mathematics meeting at Los Alamos in September, a set of talks at Brown</p><p>University inNovember.Thediscovery and the theorymet surprise, disbelief,</p><p>andexcitement.Themoreascientisthadthoughtaboutnonlinearity,themorehe</p><p>felt the force of Feigenbaum’s universality.One put it simply: “Itwas a very</p><p>happy and shocking discovery that therewere structures in nonlinear systems</p><p>thatarealwaysthesameifyoulookedatthemtherightway.”Somephysicists</p><p>pickedupnotjusttheideasbutalsothetechniques.Playingwiththesemaps—</p><p>just playing—gave them chills. With their own calculators, they could</p><p>experiencethesurpriseandsatisfactionthathadkeptFeigenbaumgoingatLos</p><p>Alamos. And they refined the theory. Hearing his talk at the Institute for</p><p>Advanced Study in Princeton, PredragCvitanović, a particle physicist, helped</p><p>Feigenbaum simplify his theory and extend its universality. But all thewhile,</p><p>Cvitanovićpretendeditwasjustapastime;hecouldnotbringhimselftoadmit</p><p>tohiscolleagueswhathewasdoing.</p><p>Amongmathematicians, too,areservedattitudeprevailed, largelybecause</p><p>Feigenbaumdidnotprovidea rigorousproof. Indeed,notuntil1979didproof</p><p>comeonmathematicians’ terms, inworkbyOscarE.LanfordIII.Feigenbaum</p><p>often recalled presenting his theory to a distinguished audience at the Los</p><p>AlamosmeetinginSeptember.Hehadbarelybeguntodescribetheworkwhen</p><p>the eminentmathematicianMarkKac rose to ask: “Sir, do youmean to offer</p><p>numericsoraproof?”</p><p>Morethantheoneandlessthantheother,Feigenbaumreplied.</p><p>“Isitwhatanyreasonablemanwouldcallaproof?”</p><p>Feigenbaum said that the listeners would have to judge for themselves.</p><p>Afterhewasdonespeaking,hepolledKac,whoresponded,withasardonically</p><p>trilledr:“Yes,that’sindeedareasonableman’sproof.Thedetailscanbeleftto</p><p>ther-r–rigorousmathematicians.”</p><p>A movement had begun,</p><p>and the discovery of universality spurred it</p><p>forward.Inthesummerof1977,twophysicists,JosephFordandGiulioCasati,</p><p>organizedthefirstconferenceonasciencecalledchaos.Itwasheldinagracious</p><p>villainComo,Italy,atinycityatthesouthernfootofthelakeofthesamename,</p><p>a stunningly deep blue catchbasin for themelting snow from the ItalianAlps.</p><p>Onehundred people came—mostly physicists, but also curious scientists from</p><p>other fields. “Mitch had seen universality and found out how it scaled and</p><p>workedoutawayofgettingtochaosthatwasintuitivelyappealing,”Fordsaid.</p><p>“Itwasthefirsttimewehadaclearmodelthateverybodycouldunderstand.</p><p>“Anditwasoneofthosethingswhosetimehadcome.Indisciplinesfrom</p><p>astronomy to zoology, peoplewere doing the same things, publishing in their</p><p>narrow disciplinary journals, just totally unaware that the other people were</p><p>around.Theythoughttheywerebythemselves,andtheywereregardedasabit</p><p>eccentricintheirownareas.Theyhadexhaustedthesimplequestionsyoucould</p><p>askandbeguntoworryaboutphenomenathatwereabitmorecomplicated.And</p><p>these peoplewere justweepingly grateful to find out that everybody elsewas</p><p>there,too.”</p><p>LATER,FEIGENBAUMLIVEDinabarespace,abedinoneroom,acomputerin</p><p>another, and, in the third, three black electronic towers for playing his solidly</p><p>Germanic record collection. His one experiment in home furnishing, the</p><p>purchaseofanexpensivemarblecoffeetablewhilehewasinItaly,hadendedin</p><p>failure;hereceivedaparcelofmarblechips.Pilesofpapersandbookslinedthe</p><p>walls.He talked rapidly,his longhair,graynowmixedwithbrown, sweeping</p><p>backfromhisforehead.“Somethingdramatichappenedinthetwenties.Forno</p><p>good reason physicists stumbled upon an essentially correct description of the</p><p>worldaroundthem—becausethetheoryofquantummechanicsisinsomesense</p><p>essentiallycorrect.Ittellsyouhowyoucantakedirtandmakecomputersfrom</p><p>it.It’sthewaywe’velearnedtomanipulateouruniverse.It’sthewaychemicals</p><p>aremadeandplasticsandwhatnot.Oneknowshowtocomputewithit.It’san</p><p>extravagantlygoodtheory—exceptatsomelevelitdoesn’tmakegoodsense.</p><p>“Somepartoftheimageryismissing.Ifyouaskwhattheequationsreally</p><p>meanandwhatisthedescriptionoftheworldaccordingtothistheory,it’snota</p><p>descriptionthatentailsyourintuitionoftheworld.Youcan’tthinkofaparticle</p><p>movingasthoughithasatrajectory.You’renotallowedtovisualizeitthatway.</p><p>Ifyoustartaskingmoreandmoresubtlequestions—whatdoes this theory tell</p><p>you the world looks like?—in the end it’s so far out of your normal way of</p><p>picturing things that you run into all sorts of conflicts.Nowmaybe that’s the</p><p>waytheworldreallyis.Butyoudon’treallyknowthatthereisn’tanotherwayof</p><p>assemblingallthisinformationthatdoesn’tdemandsoradicaladeparturefrom</p><p>thewayinwhichyouintuitthings.</p><p>“There’safundamentalpresumptioninphysicsthatthewayyouunderstand</p><p>theworldisthatyoukeepisolatingitsingredientsuntilyouunderstandthestuff</p><p>thatyouthinkistrulyfundamental.Thenyoupresumethattheotherthingsyou</p><p>don’tunderstandaredetails.Theassumptionisthatthereareasmallnumberof</p><p>principlesthatyoucandiscernbylookingatthingsintheirpurestate—thisisthe</p><p>true analytic notion—and then somehow you put these together in more</p><p>complicatedwayswhenyouwanttosolvemoredirtyproblems.Ifyoucan.</p><p>“In the end, to understand you have to change gears. You have to</p><p>reassemble how you conceive of the important things that are going on. You</p><p>could have tried to simulate a model fluid system on a computer. It’s just</p><p>beginningtobepossible.Butitwouldhavebeenawasteofeffort,becausewhat</p><p>really happens has nothing to do with a fluid or a particular equation. It’s a</p><p>general descriptionofwhathappens in a largevarietyof systemswhen things</p><p>work on themselves again and again. It requires a different way of thinking</p><p>abouttheproblem.</p><p>“Whenyoulookatthisroom—youseejunksittingoverthereandaperson</p><p>sittingoverhereanddoorsoverthere—you’resupposedtotaketheelementary</p><p>principlesofmatterandwritedownthewavefunctionstodescribethem.Well,</p><p>this is not a feasible thought.MaybeGod coulddo it, but no analytic thought</p><p>existsforunderstandingsuchaproblem.</p><p>“It’snotanacademicquestionanymoretoaskwhat’sgoingtohappentoa</p><p>cloud.Peopleverymuchwanttoknow—andthatmeansthere’smoneyavailable</p><p>forit.Thatproblemisverymuchwithintherealmofphysicsandit’saproblem</p><p>verymuchof the samecaliber.You’re looking at something complicated, and</p><p>the present way of solving it is to try to look at as many points as you can,</p><p>enoughstufftosaywherethecloudis,wherethewarmairis,whatitsvelocity</p><p>is,andsoforth.Thenyoustick it into thebiggestmachineyoucanaffordand</p><p>you try to get an estimate of what it’s going to do next. But this is not very</p><p>realistic.”</p><p>Hestubbedoutonecigaretteandlitanother.“Onehastolookfordifferent</p><p>ways.Onehastolookforscalingstructures—howdobigdetailsrelatetolittle</p><p>details. You look at fluid disturbances, complicated structures in which the</p><p>complexity has come about by a persistent process. At some level they don’t</p><p>careverymuchwhatthesizeoftheprocessis—itcouldbethesizeofapeaor</p><p>the size of a basketball.Theprocess doesn’t carewhere it is, andmoreover it</p><p>doesn’t care how long it’s been going. The only things that can ever be</p><p>universal,inasense,arescalingthings.</p><p>“Inaway,artisatheoryaboutthewaytheworldlookstohumanbeings.</p><p>It’s abundantly obvious that one doesn’t know the world around us in detail.</p><p>Whatartistshaveaccomplishedisrealizingthatthere’sonlyasmallamountof</p><p>stuffthat’simportant,andthenseeingwhatitwas.Sotheycandosomeofmy</p><p>researchforme.WhenyoulookatearlystuffofVanGoghtherearezillionsof</p><p>detailsthatareputintoit,there’salwaysanimmenseamountofinformationin</p><p>hispaintings.Itobviouslyoccurredtohim,whatistheirreducibleamountofthis</p><p>stuff that you have to put in. Or you can study the horizons in Dutch ink</p><p>drawingsfromaround1600,withtinytreesandcowsthatlookveryreal.Ifyou</p><p>lookclosely,thetreeshavesortofleafyboundaries,butitdoesn’tworkifthat’s</p><p>all it is—there are also, sticking in it, little pieces of twiglike stuff. There’s a</p><p>definite interplaybetween thesofter texturesand the thingswithmoredefinite</p><p>lines. Somehow the combination gives the correct perception.With Ruysdael</p><p>andTurner,ifyoulookatthewaytheyconstructcomplicatedwater,itisclearly</p><p>doneinaniterativeway.There’ssomelevelofstuff,andthenstuffpaintedon</p><p>top of that, and then corrections to that. Turbulent fluids for those painters is</p><p>alwayssomethingwithascaleideainit.</p><p>“Itrulydowanttoknowhowtodescribeclouds.Buttosaythere’sapiece</p><p>overherewiththatmuchdensity,andnexttoitapiecewiththismuchdensity—</p><p>toaccumulatethatmuchdetailedinformation,Ithinkiswrong.It’scertainlynot</p><p>howahumanbeingperceivesthosethings,andit’snothowanartistperceives</p><p>them.Somewhere thebusinessofwritingdownpartialdifferentialequations is</p><p>nottohavedonetheworkontheproblem.</p><p>“Somehow the wondrous promise of the earth is that there are things</p><p>beautiful in it, thingswondrous and alluring, and by virtue of your trade you</p><p>want to understand them.” He put the</p><p>swingingnorthand</p><p>south.Hediscoveredthatwhenalinewentfromhightolowwithoutabump,a</p><p>doublebumpwouldcomenext,andhesaid,“That’sthekindofruleaforecaster</p><p>coulduse.”But therepetitionswereneverquiteexact.Therewaspattern,with</p><p>disturbances.Anorderlydisorder.</p><p>To make the patterns plain to see, Lorenz created a primitive kind of</p><p>graphics.Insteadofjustprintingouttheusuallinesofdigits,hewouldhavethe</p><p>machine print a certain number of blank spaces followed by the letter a. He</p><p>wouldpickonevariable—perhaps thedirectionof theairstream.Gradually the</p><p>a’smarched down the roll of paper, swinging back and forth in a wavy line,</p><p>makingalongseriesofhillsandvalleysthatrepresentedthewaythewestwind</p><p>would swing north and south across the continent. The orderliness of it, the</p><p>recognizable cycles coming around again and again but never twice the same</p><p>way,hadahypnotic fascination.Thesystemseemedslowly toberevealing its</p><p>secretstotheforecaster’seye.</p><p>Onedayinthewinterof1961,wantingtoexamineonesequenceatgreater</p><p>length,Lorenztookashortcut.Insteadofstartingthewholerunover,hestarted</p><p>midway through. To give the machine its initial conditions, he typed the</p><p>numbersstraightfromtheearlierprintout.Thenhewalkeddownthehalltoget</p><p>awayfromthenoiseanddrinkacupofcoffee.Whenhereturnedanhourlater,</p><p>hesawsomethingunexpected,somethingthatplantedaseedforanewscience.</p><p>THISNEWRUNshouldhaveexactlyduplicatedtheold.Lorenzhadcopiedthe</p><p>numbers into the machine himself. The program had not changed. Yet as he</p><p>staredatthenewprintout,Lorenzsawhisweatherdivergingsorapidlyfromthe</p><p>pattern of the last run that, within just a few months, all resemblance had</p><p>disappeared.Helookedatonesetofnumbers,thenbackattheother.Hemight</p><p>aswellhavechosentworandomweathersoutofahat.Hisfirstthoughtwasthat</p><p>anothervacuumtubehadgonebad.</p><p>Suddenly he realized the truth. There had been no malfunction. The</p><p>problem lay in the numbers he had typed. In the computer’s memory, six</p><p>decimalplaceswere stored: .506127.On theprintout, to save space, just three</p><p>appeared:.506.Lorenzhadenteredtheshorter,rounded-offnumbers,assuming</p><p>thatthedifference—onepartinathousand—wasinconsequential.</p><p>Itwasareasonableassumption.Ifaweathersatellitecanreadocean-surface</p><p>temperature towithinonepart ina thousand, itsoperatorsconsider themselves</p><p>lucky.Lorenz’sRoyalMcBeewasimplementingtheclassicalprogram.Ituseda</p><p>purely deterministic systemof equations.Given a particular starting point, the</p><p>weatherwouldunfoldexactlythesamewayeachtime.Givenaslightlydifferent</p><p>starting point, the weather should unfold in a slightly different way. A small</p><p>numericalerrorwas likeasmallpuffofwind—surely thesmallpuffsfadedor</p><p>canceledeachotheroutbeforetheycouldchangeimportant,large-scalefeatures</p><p>of the weather. Yet in Lorenz’s particular system of equations, small errors</p><p>provedcatastrophic.</p><p>HOWTWOWEATHERPATTERNSDIVERGE.Fromnearlythesamestartingpoint,EdwardLorenzsaw</p><p>hiscomputerweatherproducepatternsthatgrewfartherandfartherapartuntilallresemblancedisappeared.</p><p>(FromLorenz’s1961printouts.)</p><p>He decided to lookmore closely at theway two nearly identical runs of</p><p>weather flowed apart. He copied one of the wavy lines of output onto a</p><p>transparencyandlaiditovertheother,toinspectthewayitdiverged.First,two</p><p>humps matched detail for detail. Then one line began to lag a hairsbreadth</p><p>behind.Bythetimethetworunsreachedthenexthump,theyweredistinctlyout</p><p>ofphase.Bythethirdorfourthhump,allsimilarityhadvanished.</p><p>Itwasonlyawobblefromaclumsycomputer.Lorenzcouldhaveassumed</p><p>something was wrong with his particular machine or his particular model—</p><p>probablyshouldhaveassumed.Itwasnotasthoughhehadmixedsodiumand</p><p>chlorine and got gold. But for reasons of mathematical intuition that his</p><p>colleagueswould begin to understand only later, Lorenz felt a jolt: something</p><p>was philosophically out of joint. The practical import could be staggering.</p><p>Althoughhisequationsweregrossparodiesoftheearth’sweather,hehadafaith</p><p>thattheycapturedtheessenceoftherealatmosphere.Thatfirstday,hedecided</p><p>thatlong-rangeweatherforecastingmustbedoomed.</p><p>“Wecertainlyhadn’tbeensuccessfulindoingthatanywayandnowwehad</p><p>an excuse,” he said. “I think one of the reasons people thought it would be</p><p>possible to forecast so far ahead is that there are real physical phenomena for</p><p>which one can do an excellent job of forecasting, such as eclipses,where the</p><p>dynamicsofthesun,moon,andeartharefairlycomplicated,andsuchasoceanic</p><p>tides.Ineverusedtothinkoftideforecastsaspredictionatall—Iusedtothink</p><p>of them as statements of fact—but of course, you are predicting. Tides are</p><p>actuallyjustascomplicatedastheatmosphere.Bothhaveperiodiccomponents</p><p>—youcanpredict thatnext summerwillbewarmer than thiswinter.Butwith</p><p>weather we take the attitude that we knew that already. With tides, it’s the</p><p>predictable part that we’re interested in, and the unpredictable part is small,</p><p>unlessthere’sastorm.</p><p>“The average person, seeing that we can predict tides pretty well a few</p><p>monthsaheadwouldsay,whycan’twedothesamethingwiththeatmosphere,</p><p>it’sjustadifferentfluidsystem,thelawsareaboutascomplicated.ButIrealized</p><p>thatanyphysicalsystemthatbehavednonperiodicallywouldbeunpredictable.”</p><p>THE FIFTIES AND SIXTIES were years of unreal optimism about weather</p><p>forecasting. Newspapers and magazines were filled with hope for weather</p><p>science, not just for prediction but for modification and control. Two</p><p>technologieswerematuringtogether,thedigitalcomputerandthespacesatellite.</p><p>An international program was being prepared to take advantage of them, the</p><p>GlobalAtmosphereResearch Program. Therewas an idea that human society</p><p>would free itself fromweather’s turmoil and become itsmaster instead of its</p><p>victim. Geodesic domes would cover cornfields. Airplanes would seed the</p><p>clouds.Scientistswouldlearnhowtomakerainandhowtostopit.</p><p>TheintellectualfatherofthispopularnotionwasVonNeumann,whobuilt</p><p>hisfirstcomputerwiththepreciseintention,amongotherthings,ofcontrolling</p><p>theweather.Hesurroundedhimselfwithmeteorologistsandgavebreathtaking</p><p>talks about his plans to the general physics community. He had a specific</p><p>mathematical reason for his optimism. He recognized that a complicated</p><p>dynamicalsystemcouldhavepointsofinstability—criticalpointswhereasmall</p><p>pushcanhave largeconsequences, aswithaballbalancedat the topofahill.</p><p>With the computer up and running, Von Neumann imagined that scientists</p><p>would calculate the equations of fluid motion for the next few days. Then a</p><p>centralcommitteeofmeteorologistswouldsendupairplanestolaydownsmoke</p><p>screens or seed clouds to push the weather into the desired mode. But Von</p><p>Neumannhadoverlookedthepossibilityofchaos,withinstabilityateverypoint.</p><p>By the1980s avast andexpensivebureaucracydevoted itself to carrying</p><p>out Von Neumann’s mission, or at least the prediction part of it. America’s</p><p>premierforecastersoperatedoutofanunadornedcubeofabuildinginsuburban</p><p>Maryland, near theWashington beltway,with a spy’s nest of radar and radio</p><p>antennasontheroof.Their</p><p>cigarette down. Smoke rose from the</p><p>ashtray, first in a thin column and then (with a nod to universality) in broken</p><p>tendrilsthatswirledupwardtotheceiling.</p><p>TheExperimenter</p><p>It’sanexperiencelikenootherexperienceIcandescribe,thebestthingthatcan</p><p>happen to a scientist, realizing that something that’s happened in his or her</p><p>mind exactly corresponds to something that happens in nature. It’s startling</p><p>every time it occurs.One is surprised that a construct of one’sownmind can</p><p>actuallybe realized in thehonest-to–goodnessworldout there.Agreat shock,</p><p>andagreat,greatjoy.</p><p>—LEOKADANOFF</p><p>“ALBERT IS GETTINGMATURE.” So they said at ÉcoleNormale Supérieure,</p><p>theacademywhich,withÉcolePolytechnique,sitsatoptheFrencheducational</p><p>hierarchy.TheywonderedwhetheragewastakingitstollonAlbertLibchaber,</p><p>whohadmadeadistinguishednameforhimselfasalow-temperaturephysicist,</p><p>studying the quantum behavior of superfluid helium at temperatures a breath</p><p>awayfromabsolutezero.Hehadprestigeandasecureplaceonthefaculty.And</p><p>now in 1977 he was wasting his time and the university’s resources on an</p><p>experiment that seemed trivial. Libchaber himself worried that he would be</p><p>jeopardizingthecareerofanygraduatestudentheemployedonsuchaproject,</p><p>sohegottheassistanceofaprofessionalengineerinstead.</p><p>FiveyearsbeforetheGermansinvadedParis,Libchaberwasbornthere,the</p><p>sonofPolishJews,thegrandsonofarabbi.Hesurvivedthewarthesameway</p><p>BenoitMandelbrotdid,byhidinginthecountryside,separatedfromhisparents</p><p>because theiraccentswere toodangerous.Hisparentsmanaged tosurvive; the</p><p>restofthefamilywaslosttotheNazis.Inaquirkofpoliticalfate,Libchaber’s</p><p>ownlifewassavedbytheprotectionofalocalchiefofthePétainsecretpolice,a</p><p>man whose fervent right-wing beliefs were matched only by his fervent</p><p>antiracism.Afterthewar, theten-year–oldboyreturnedthefavor.Hetestified,</p><p>only half-comprehending, before awar crimes commission, and his testimony</p><p>savedtheman.</p><p>Moving through theworldofFrenchacademicscience,Libchaber rose in</p><p>his profession, his brilliance never questioned. His colleagues did sometimes</p><p>think he was a little crazy—a Jewish mystic amid the rationalists, a Gaullist</p><p>wheremostscientistswereCommunists.TheyjokedabouthisGreatMantheory</p><p>of history, his fixation on Goethe, his obsession with old books. He had</p><p>hundreds of original editions of works by scientists, some dating back to the</p><p>1600s.He read themnotashistoricalcuriositiesbutasa sourceof fresh ideas</p><p>about thenatureofreality, thesamerealityhewasprobingwithhis lasersand</p><p>his high-technology refrigeration coils. In his engineer, Jean Maurer, he had</p><p>founda compatible spirit, aFrenchmanwhoworkedonlywhenhe felt like it.</p><p>LibchaberthoughtMaurerwouldfindhisnewprojectamusing—hisunderstated</p><p>Galliceuphemismforintriguingorexcitingorprofound.Thetwosetoutin1977</p><p>tobuildanexperimentthatwouldrevealtheonsetofturbulence.</p><p>Asan experimenter,Libchaberwasknown for anineteenth-century style:</p><p>clever mind, nimble hands, always preferring ingenuity to brute force. He</p><p>dislikedgianttechnologyandheavycomputation.Hisideaofagoodexperiment</p><p>waslikeamathematician’sideaofagoodproof.Elegancecountedasmuchas</p><p>results.Evenso,somecolleaguesthoughthewascarryingthingstoofarwithhis</p><p>onset-of–turbulence experiment. It was small enough to carry around in a</p><p>matchbox—and sometimes Libchaber did carry it around, like some piece of</p><p>conceptual art. He called it “Helium in a Small Box.” The heart of the</p><p>experimentwas even smaller, a cell about the size of a lemon seed, carved in</p><p>stainlesssteelwith thesharpestpossibleedgesandwalls. Into thecellwasfed</p><p>liquidheliumchilledtoaboutfourdegreesaboveabsolutezero,warmcompared</p><p>toLibchaber’soldsuperfluidexperiments.</p><p>The laboratoryoccupied thesecondfloorof theÉcolephysicsbuilding in</p><p>Paris,justafewhundredfeetfromLouisPasteur’soldlaboratory.Likeallgood</p><p>general-purposephysics laboratories,Libchaber’sexisted ina stateof constant</p><p>mess, paint cans and hand tools strewn about on floors and tables, odd-sized</p><p>pieces ofmetal and plastic everywhere. Amid the disarray, the apparatus that</p><p>heldLibchaber’sminusculefluidcellwasastrikingbitofpurposefulness.Below</p><p>thestainlesssteelcellsatabottomplateofhigh-puritycopper.Abovesatatop</p><p>plate of sapphire crystal. The materials were chosen according to how they</p><p>conductedheat.Therewere tinyelectricheatingcoils andTeflongaskets.The</p><p>liquid helium flowed down from a reservoir, itself just a half-inch cube. The</p><p>whole system sat inside a container thatmaintained an extreme vacuum.And</p><p>that container, in turn, sat in a bath of liquid nitrogen, to help stabilize the</p><p>temperature.</p><p>Vibration always worried Libchaber. Experiments, like real nonlinear</p><p>systems, existed against a constant background of noise. Noise hampered</p><p>measurementandcorrupteddata.Insensitiveflows—andLibchaber’swouldbe</p><p>assensitiveashecouldmakeit—noisemightsharplyperturbanonlinearflow,</p><p>knocking it from one kind of behavior into another. But nonlinearity can</p><p>stabilizeasystemaswellasdestabilizeit.Nonlinearfeedbackregulatesmotion,</p><p>makingitmorerobust.Inalinearsystem,aperturbationhasaconstanteffect.In</p><p>thepresenceofnonlinearity,aperturbationcanfeedonitselfuntilitdiesaway</p><p>and the system returns automatically to a stable state. Libchaber believed that</p><p>biological systems used their nonlinearity as a defense against noise. The</p><p>transfer of energy by proteins, the wavemotion of the heart’s electricity, the</p><p>nervous system—all these kept their versatility in a noisy world. Libchaber</p><p>hopedthatwhateverstructureunderlayfluidflowwouldproverobustenoughfor</p><p>hisexperimenttodetect.</p><p>“HELIUM IN A SMALL BOX.” Albert Libchaber’s delicate experiment: Its heart was a carefully</p><p>machined rectangular cell containing liquid helium; tiny sapphire “bolometers” measured the fluid’s</p><p>temperature.Thetinycellwasembeddedinacasingdesignedtoshielditfromthenoiseandvibrationand</p><p>toallowprecisecontroloftheheating.</p><p>His plan was to create convection in the liquid helium by making the</p><p>bottom plate warmer than the top plate. It was exactly the convection model</p><p>described by Edward Lorenz, the classic system known as Rayleigh-Bénard</p><p>convection.LibchaberwasnotawareofLorenz—notyet.Norhadheanyideaof</p><p>MitchellFeigenbaum’stheory.In1977Feigenbaumwasbeginningtotravelthe</p><p>scientific lecture circuit, and his discoveries were making their mark where</p><p>scientistsknewhow to interpret them.Butas farasmostphysicistscould tell,</p><p>thepatternsandregularitiesofFeigenbaumologyborenoobviousconnectionto</p><p>real systems.Thosepatternscameoutof adigital calculator.Physical systems</p><p>were infinitely more complicated. Without more evidence, the most anyone</p><p>could say was that Feigenbaum had discovered a mathematical analogy that</p><p>lookedlikethebeginningofturbulence.</p><p>LibchaberknewthatAmericanandFrenchexperimentshadweakenedthe</p><p>Landauideafortheonsetofturbulencebyshowingthatturbulencearrivedina</p><p>sudden transition, instead of a continuous piling-up of different frequencies.</p><p>ExperimenterslikeJerryGollubandHarrySwinney,withtheirflowinarotating</p><p>cylinder,haddemonstratedthatanew</p><p>supercomputerranamodelthatresembledLorenz’s</p><p>only in its fundamental spirit.Where the RoyalMcBee could carry out sixty</p><p>multiplications each second, the speed of a Control Data Cyber 205 was</p><p>measuredinmegaflops,millionsoffloating-pointoperationspersecond.Where</p><p>Lorenz had been happy with twelve equations, the modern global model</p><p>calculated systems of 500,000 equations. The model understood the way</p><p>moisturemoved heat in and out of the airwhen it condensed and evaporated.</p><p>Thedigitalwindswereshapedbydigitalmountainranges.Datapouredinhourly</p><p>from every nation on the globe, from airplanes, satellites, and ships. The</p><p>NationalMeteorologicalCenterproducedtheworld’ssecondbestforecasts.</p><p>The best came out of Reading, England, a small college town an hour’s</p><p>drivefromLondon.TheEuropeanCentreforMediumRangeWeatherForecasts</p><p>occupied a modest tree-shaded building in a generic United Nations style,</p><p>modern brick-and–glass architecture, decoratedwith gifts frommany lands. It</p><p>wasbuiltintheheydayoftheall-EuropeanCommonMarketspirit,whenmost</p><p>ofthenationsofwesternEuropedecidedtopooltheirtalentandresourcesinthe</p><p>cause of weather prediction. The Europeans attributed their success to their</p><p>young, rotating staff—no civil service—and their Cray supercomputer, which</p><p>alwaysseemedtobeonemodelaheadoftheAmericancounterpart.</p><p>Weatherforecastingwasthebeginningbuthardlytheendofthebusinessof</p><p>usingcomputerstomodelcomplexsystems.Thesametechniquesservedmany</p><p>kinds of physical scientists and social scientists hoping to make predictions</p><p>about everything from the small-scale fluid flows that concerned propeller</p><p>designers to thevast financial flows thatconcernedeconomists. Indeed,by the</p><p>seventies and eighties, economic forecasting by computer bore a real</p><p>resemblance to global weather forecasting. The models would churn through</p><p>complicated,somewhatarbitrarywebsofequations,meanttoturnmeasurements</p><p>ofinitialconditions—atmosphericpressureormoneysupply—intoasimulation</p><p>of future trends. The programmers hoped the results were not too grossly</p><p>distorted by the many unavoidable simplifying assumptions. If a model did</p><p>anythingtooobviouslybizarre—floodedtheSaharaortripledinterestrates—the</p><p>programmerswould revise the equations to bring the output back in linewith</p><p>expectation.Inpractice,econometricmodelsproveddismallyblindtowhatthe</p><p>futurewould bring, butmany peoplewho should have known better acted as</p><p>though they believed in the results. Forecasts of economic growth or</p><p>unemployment were put forward with an implied precision of two or three</p><p>decimalplaces.Governmentsandfinancialinstitutionspaidforsuchpredictions</p><p>and acted on them, perhaps out of necessity or for want of anything better.</p><p>Presumablytheyknewthatsuchvariablesas“consumeroptimism”werenotas</p><p>nicelymeasurableas“humidity”and that theperfectdifferentialequationshad</p><p>notyetbeenwrittenforthemovementofpoliticsandfashion.Butfewrealized</p><p>howfragilewas theveryprocessofmodeling flowsoncomputers, evenwhen</p><p>the data was reasonably trustworthy and the lawswere purely physical, as in</p><p>weatherforecasting.</p><p>Computermodelinghadindeedsucceededinchangingtheweatherbusiness</p><p>fromanarttoascience.TheEuropeanCentre’sassessmentssuggestedthatthe</p><p>worldsavedbillionsofdollarseachyearfrompredictionsthatwerestatistically</p><p>betterthannothing.Butbeyondtwoorthreedaystheworld’sbestforecastswere</p><p>speculative,andbeyondsixorseventheywereworthless.</p><p>TheButterflyEffectwasthereason.Forsmallpiecesofweather—andtoa</p><p>global forecaster, smallcanmean thunderstormsandblizzards—anyprediction</p><p>deterioratesrapidly.Errorsanduncertaintiesmultiply,cascadingupwardthrough</p><p>achainof turbulent features, fromdustdevils and squallsup to continent-size</p><p>eddiesthatonlysatellitescansee.</p><p>Themodernweathermodelsworkwithagridofpointsontheorderofsixty</p><p>miles apart, and even so, some starting data has to be guessed, since ground</p><p>stations and satellites cannot see everywhere. But suppose the earth could be</p><p>coveredwith sensors spacedone foot apart, rising at one-foot intervals all the</p><p>waytothetopoftheatmosphere.Supposeeverysensorgivesperfectlyaccurate</p><p>readings of temperature, pressure, humidity, and any other quantity a</p><p>meteorologist would want. Precisely at noon an infinitely powerful computer</p><p>takes all thedata and calculateswhatwill happen at eachpoint at 12:01, then</p><p>12:02,then12:03…</p><p>ThecomputerwillstillbeunabletopredictwhetherPrinceton,NewJersey,</p><p>willhavesunorrainonadayonemonthaway.Atnoonthespacesbetweenthe</p><p>sensors will hide fluctuations that the computer will not know about, tiny</p><p>deviations from the average. By 12:01, those fluctuations will already have</p><p>createdsmallerrorsonefootaway.Soon theerrorswillhavemultiplied to the</p><p>ten-footscale,andsoonuptothesizeoftheglobe.</p><p>Evenforexperiencedmeteorologists,allthisrunsagainstintuition.Oneof</p><p>Lorenz’soldestfriendswasRobertWhite,afellowmeteorologistatM.I.T.who</p><p>later became head of the National Oceanic and Atmospheric Administration.</p><p>Lorenz toldhimabout theButterflyEffect andwhathe felt itmeant for long-</p><p>rangeprediction.WhitegaveVonNeumann’sanswer.“Prediction,nothing,”he</p><p>said.“This isweathercontrol.”His thoughtwas that smallmodifications,well</p><p>withinhumancapability,couldcausedesiredlarge-scalechanges.</p><p>Lorenz saw it differently.Yes, you could change theweather.You could</p><p>makeitdosomethingdifferentfromwhatitwouldotherwisehavedone.Butif</p><p>you did, then you would never know what it would otherwise have done. It</p><p>wouldbelikegivinganextrashuffletoanalreadywell-shuffledpackofcards.</p><p>Youknow itwill changeyour luck, but youdon’t knowwhether for better or</p><p>worse.</p><p>LORENZ’SDISCOVERYWASANACCIDENT,onemoreinalinestretchingbackto</p><p>Archimedes and his bathtub. Lorenz never was the type to shout Eureka.</p><p>Serendipitymerely ledhim to aplacehehadbeenall along.Hewas ready to</p><p>exploretheconsequencesofhisdiscoverybyworkingoutwhatitmustmeanfor</p><p>thewayscienceunderstoodflowsinallkindsoffluids.</p><p>HadhestoppedwiththeButterflyEffect,animageofpredictabilitygiving</p><p>waytopurerandomness,thenLorenzwouldhaveproducednomorethanapiece</p><p>of very bad news. But Lorenz saw more than randomness embedded in his</p><p>weather model. He saw a fine geometrical structure, order masquerading as</p><p>randomness.Hewasamathematicianinmeteorologist’sclothing,afterall,and</p><p>now he began to lead a double life. He would write papers that were pure</p><p>meteorology.Buthewouldalsowritepapersthatwerepuremathematics,witha</p><p>slightly misleading dose of weather talk as preface. Eventually the prefaces</p><p>woulddisappearaltogether.</p><p>Heturnedhisattentionmoreandmoretothemathematicsofsystemsthat</p><p>never found a steady state, systems that almost repeated themselves but never</p><p>quitesucceeded.Everyoneknewthattheweatherwassuchasystem—aperiodic.</p><p>Nature is full of others: animal populations that rise and fall almost regularly,</p><p>epidemics that come and go on tantalizingly near-regular schedules. If the</p><p>weathereverdidreachastateexactlylikeoneithadreachedbefore,everygust</p><p>andcloudthesame,thenpresumablyitwouldrepeatitselfforeverafterandthe</p><p>problemofforecastingwouldbecometrivial.</p><p>Lorenz saw that there must be a link between the unwillingness of the</p><p>weather to repeat itself and the inability of forecasters to predict it—a link</p><p>between aperiodicity and unpredictability. It was not easy to find simple</p><p>equations that would produce the aperiodicity he was seeking. At first his</p><p>computertendedtolockintorepetitivecycles.ButLorenztrieddifferentsortsof</p><p>minorcomplications,andhefinallysucceededwhenheput inanequationthat</p><p>variedtheamountofheatingfromeasttowest,correspondingtothereal-world</p><p>variationbetween theway thesunwarms theeastcoastofNorthAmerica, for</p><p>example,andthewayitwarmstheAtlanticOcean.Therepetitiondisappeared.</p><p>The Butterfly Effect was no accident; it was necessary. Suppose small</p><p>perturbationsremainedsmall,hereasoned,insteadofcascadingupwardthrough</p><p>thesystem.Thenwhentheweathercamearbitrarilyclosetoastateithadpassed</p><p>throughbefore,itwouldstayarbitrarilyclosetothepatternsthatfollowed.For</p><p>practical purposes, the cycles would be predictable—and eventually</p><p>uninteresting. To produce the rich repertoire of real earthly weather, the</p><p>beautiful multiplicity of it, you could hardly wish for anything better than a</p><p>ButterflyEffect.</p><p>The Butterfly Effect acquired a technical name: sensitive dependence on</p><p>initial conditions. And sensitive dependence on initial conditions was not an</p><p>altogethernewnotion.Ithadaplaceinfolklore:</p><p>“Forwantofanail,theshoewaslost;</p><p>Forwantofashoe,thehorsewaslost;</p><p>Forwantofahorse,theriderwaslost;</p><p>Forwantofarider,thebattlewaslost;</p><p>Forwantofabattle,thekingdomwaslost!”</p><p>Inscienceasinlife,itiswellknownthatachainofeventscanhaveapoint</p><p>of crisis that couldmagnify small changes. But chaosmeant that such points</p><p>were everywhere. Theywere pervasive. In systems like theweather, sensitive</p><p>dependence on initial conditions was an inescapable consequence of the way</p><p>smallscalesintertwinedwithlarge.</p><p>HiscolleagueswereastonishedthatLorenzhadmimickedbothaperiodicity</p><p>andsensitivedependenceoninitialconditionsinhistoyversionoftheweather:</p><p>twelve equations, calculated over and over again with ruthless mechanical</p><p>efficiency.Howcould such richness, suchunpredictability—suchchaos—arise</p><p>fromasimpledeterministicsystem?</p><p>LORENZ PUT THE WEATHER ASIDE and looked for even simpler ways to</p><p>producethiscomplexbehavior.Hefoundoneinasystemofjustthreeequations.</p><p>Theywere nonlinear,meaning that they expressed relationships that were not</p><p>strictlyproportional.Linearrelationshipscanbecapturedwithastraightlineon</p><p>a graph. Linear relationships are easy to think about: the more the merrier.</p><p>Linearequationsaresolvable,whichmakesthemsuitablefortextbooks.Linear</p><p>systems have an important modular virtue: you can take them apart, and put</p><p>themtogetheragain—thepiecesaddup.</p><p>Nonlinear systems generally cannot be solved and cannot be added</p><p>together.Influidsystemsandmechanicalsystems,thenonlineartermstendtobe</p><p>the features thatpeoplewant to leaveoutwhen they try toget agood, simple</p><p>understanding.Friction, for example.Without friction a simple linear equation</p><p>expresses the amount of energy you need to accelerate a hockey puck.With</p><p>frictiontherelationshipgetscomplicated,becausetheamountofenergychanges</p><p>dependingonhowfastthepuckisalreadymoving.Nonlinearitymeansthatthe</p><p>actofplaying thegamehasawayofchanging the rules.Youcannotassigna</p><p>constantimportancetofriction,becauseitsimportancedependsonspeed.Speed,</p><p>inturn,dependsonfriction.Thattwistedchangeabilitymakesnonlinearityhard</p><p>tocalculate,butitalsocreatesrichkindsofbehaviorthatneveroccurinlinear</p><p>systems.Influiddynamics,everythingboilsdowntoonecanonicalequation,the</p><p>Navier-Stokes equation. It is a miracle of brevity, relating a fluid’s velocity,</p><p>pressure,density,andviscosity,butithappenstobenonlinear.Sothenatureof</p><p>those relationships often becomes impossible to pin down. Analyzing the</p><p>behaviorofanonlinearequationliketheNavier-Stokesequationislikewalking</p><p>throughamazewhosewalls rearrange themselveswitheachstepyou take.As</p><p>Von Neumann himself put it: “The character of the equation…changes</p><p>simultaneously in all relevant respects:Both order and degree change.Hence,</p><p>badmathematicaldifficultiesmustbeexpected.”Theworldwouldbeadifferent</p><p>place—andsciencewouldnotneedchaos—ifonly theNavier-Stokesequation</p><p>didnotcontainthedemonofnonlinearity.</p><p>A particular kind of fluid motion inspired Lorenz’s three equations: the</p><p>risingofhotgasorliquid,knownasconvection.Intheatmosphere,convection</p><p>stirs air heatedby the sun-baked earth, and shimmering convectivewaves rise</p><p>ghost-like abovehot tar and radiators.Lorenzwas just as happy talking about</p><p>convection in a cup of hot coffee. As he put it, this was just one of the</p><p>innumerable hydrodynamical processes in our universewhose future behavior</p><p>wemightwishtopredict.Howcanwecalculatehowquicklyacupofcoffeewill</p><p>cool?Ifthecoffeeisjustwarm,itsheatwilldissipatewithoutanyhydrodynamic</p><p>motion at all. The coffee remains in a steady state. But if it is hot enough, a</p><p>convectiveoverturningwillbringhotcoffeefromthebottomofthecupuptothe</p><p>coolersurface.Convectionincoffeebecomesplainlyvisiblewhenalittlecream</p><p>isdribbledintothecup.Theswirlscanbecomplicated.Butthelongtermdestiny</p><p>of such a system is obvious.Because the heat dissipates, and because friction</p><p>slowsamovingfluid,themotionmustcometoaninevitablestop.Lorenzdrily</p><p>told a gathering of scientists, “We might have trouble forecasting the</p><p>temperature of the coffee one minute in advance, but we should have little</p><p>difficultyinforecastingitanhourahead.”Theequationsofmotionthatgoverna</p><p>cooling cup of coffee must reflect the system’s destiny. They must be</p><p>dissipative.Temperaturemustheadforthetemperatureoftheroom,andvelocity</p><p>mustheadforzero.</p><p>Lorenz tooka setofequations forconvectionandstripped it to thebone,</p><p>throwing out everything that could possibly be extraneous, making it</p><p>unrealisticallysimple.Almostnothingremainedoftheoriginalmodel,buthedid</p><p>leavethenonlinearity.Totheeyeofaphysicist,theequationslookedeasy.You</p><p>wouldglanceatthem—manyscientistsdid,inyearstocome—andsay,Icould</p><p>solvethat.</p><p>“Yes,”Lorenzsaidquietly,“thereisatendencytothinkthatwhenyousee</p><p>them.There are some nonlinear terms in them, but you think theremust be a</p><p>waytogetaroundthem.Butyoujustcan’t.”</p><p>AROLLINGFLUID.When a liquid or gas is heated frombelow, the fluid tends to organize itself into</p><p>cylindricalrolls(left).Hotfluidrisesononeside,losesheat,anddescendsontheotherside—theprocessof</p><p>convection.Whentheheatisturnedupfurther(right),aninstabilitysetsin,andtherollsdevelopawobble</p><p>thatmovesbackandforthalongthelengthofthecylinders.Atevenhighertemperatures,theflowbecomes</p><p>wildandturbulent.</p><p>The simplest kindof textbook convection takes place in a cell of fluid, a</p><p>box with a smooth bottom that can be heated and a smooth top that can be</p><p>cooled. The temperature difference between the hot bottom and the cool top</p><p>controlstheflow.Ifthedifferenceissmall,thesystemremainsstill.Heatmoves</p><p>towardthetopbyconduction,asifthroughabarofmetal,withoutovercoming</p><p>the natural tendency of the</p><p>fluid to remain at rest. Furthermore, the system is</p><p>stable. Any random motions that might occur when, say, a graduate student</p><p>knocksintotheapparatuswilltendtodieout,returningthesystemtoitssteady</p><p>state.</p><p>Turnuptheheat,though,andanewkindofbehaviordevelops.Asthefluid</p><p>underneathbecomeshot,itexpands.Asitexpands,itbecomeslessdense.Asit</p><p>becomes less dense, it becomes lighter, enough to overcome friction, and it</p><p>pushes up toward the surface. In a carefully designed box, a cylindrical roll</p><p>develops,withthehotfluidrisingaroundonesideandcoolfluidsinkingdown</p><p>aroundtheother.Viewedfromtheside, themotionmakesacontinuouscircle.</p><p>Outof the laboratory, too,natureoftenmakes its ownconvection cells.When</p><p>the sun heats a desert floor, for example, the rolling air can shape shadowy</p><p>patternsinthecloudsaboveorthesandbelow.</p><p>Turn up the heat evenmore, and the behavior growsmore complex.The</p><p>rolls begin to wobble. Lorenz’s pared-down equations were far too simple to</p><p>model that sort of complexity. They abstracted just one feature of real-world</p><p>convection: the circularmotion of hot fluid rising up and around like a Ferris</p><p>wheel. The equations took into account the velocity of that motion and the</p><p>transfer of heat. Those physical processes interacted. As any given bit of hot</p><p>fluidrosearoundthecircle,itwouldcomeintocontactwithcoolerfluidandso</p><p>begintoloseheat.Ifthecirclewasmovingfastenough,theballoffluidwould</p><p>not lose all its extra heat by the time it reached the top and started swinging</p><p>downtheothersideoftheroll,soitwouldactuallybegintopushbackagainst</p><p>themomentumoftheotherhotfluidcomingupbehindit.</p><p>AlthoughtheLorenzsystemdidnotfullymodelconvection,itdidturnout</p><p>to have exact analogues in real systems. For example, his equations precisely</p><p>describeanold-fashionedelectricaldynamo,theancestorofmoderngenerators,</p><p>wherecurrentflowsthroughadiscthatrotatesthroughamagneticfield.Under</p><p>certain conditions the dynamo can reverse itself. And some scientists, after</p><p>Lorenz’sequationsbecamebetterknown,suggestedthatthebehaviorofsucha</p><p>dynamo might provide an explanation for another peculiar reversing</p><p>phenomenon: the earth’s magnetic field. The “geodynamo” is known to have</p><p>flippedmanytimesduringtheearth’shistory,atintervalsthatseemerraticand</p><p>inexplicable. Faced with such irregularity, theorists typically look for</p><p>explanationsoutsidethesystem,proposingsuchcausesasmeteoritestrikes.Yet</p><p>perhapsthegeodynamocontainsitsownchaos.</p><p>THE LORENZIAN WATERWHEEL. The first, famous chaotic system discovered by Edward Lorenz</p><p>corresponds exactly to a mechanical device: a waterwheel. This simple device proves capable of</p><p>surprisinglycomplicatedbehavior.</p><p>Therotationofthewaterwheelsharessomeofthepropertiesoftherotatingcylindersoffluidinthe</p><p>processofconvection.Thewaterwheelislikeaslicethroughthecylinder.Bothsystemsaredrivensteadily</p><p>—bywater or by heat—and both dissipate energy. The fluid loses heat; the buckets losewater. In both</p><p>systems,thelongtermbehaviordependsonhowhardthedrivingenergyis.</p><p>Waterpoursinfromthetopatasteadyrate.Iftheflowofwaterinthewaterwheelisslow,thetop</p><p>bucketneverfillsupenoughtoovercomefriction,andthewheelneverstartsturning.(Similarly,inafluid,</p><p>iftheheatistoolowtoovercomeviscosity,itwillnotsetthefluidinmotion.)</p><p>Iftheflowisfaster,theweightofthetopbucketsetsthewheelinmotion(left).Thewaterwheelcan</p><p>settleintoarotationthatcontinuesatasteadyrate(center).</p><p>Butiftheflowisfasterstill(right),thespincanbecomechaotic,becauseofnonlineareffectsbuilt</p><p>intothesystem.Asbucketspassundertheflowingwater,howmuchtheyfilldependsonthespeedofspin.</p><p>If thewheel is spinning rapidly, the buckets have little time to fill up. (Similarly, fluid in a fast-turning</p><p>convectionrollhaslittletimetoabsorbheat.)Also,ifthewheelisspinningrapidly,bucketscanstartupthe</p><p>othersidebeforetheyhavetimetoempty.Asaresult,heavybucketsonthesidemovingupwardcancause</p><p>thespintoslowdownandthenreverse.</p><p>Infact,Lorenzdiscovered,overlongperiods,thespincanreverseitselfmanytimes,neversettling</p><p>downtoasteadyrateandneverrepeatingitselfinanypredictablepattern.</p><p>THE LORENZ ATTRACTOR (on facing page). This magical image, resembling an owl’s mask or</p><p>butterfly’swings,becameanemblemfortheearlyexplorersofchaos.Itrevealedthefinestructurehidden</p><p>withinadisorderlystreamofdata.Traditionally,thechangingvaluesofanyonevariablecouldbedisplayed</p><p>in a so-called time series (top). To show the changing relationships among three variables required a</p><p>different technique. At any instant in time, the three variables fix the location of a point in three-</p><p>dimensional space; as the system changes, themotion of the point represents the continuously changing</p><p>variables.</p><p>Becausethesystemneverexactlyrepeatsitself,thetrajectoryneverintersectsitself.Insteaditloops</p><p>aroundandaroundforever.Motionontheattractorisabstract,butitconveystheflavorofthemotionofthe</p><p>realsystem.Forexample,thecrossoverfromonewingoftheattractortotheothercorrespondstoareversal</p><p>inthedirectionofspinofthewaterwheelorconvectingfluid.</p><p>Another system precisely described by the Lorenz equations is a certain</p><p>kindofwaterwheel,amechanicalanalogueoftherotatingcircleofconvection.</p><p>Atthetop,waterdripssteadilyintocontainershangingonthewheel’srim.Each</p><p>containerleakssteadilyfromasmallhole.Ifthestreamofwaterisslow,thetop</p><p>containersneverfillfastenoughtoovercomefriction,butifthestreamisfaster,</p><p>theweightstartstoturnthewheel.Therotationmightbecomecontinuous.Orif</p><p>the stream is so fast that the heavy containers swing all the way around the</p><p>bottomandstartuptheotherside,thewheelmightthenslow,stop,andreverse</p><p>itsrotation,turningfirstonewayandthentheother.</p><p>A physicist’s intuition about such a simple mechanical system—his pre-</p><p>chaos intuition—tellshimthatover the longterm, if thestreamofwaternever</p><p>varied,asteadystatewouldevolve.Eitherthewheelwouldrotatesteadilyorit</p><p>wouldoscillatesteadilybackandforth,turningfirstinonedirectionandthenthe</p><p>otheratconstantintervals.Lorenzfoundotherwise.</p><p>Threeequations,with threevariables, completelydescribed themotionof</p><p>this system. Lorenz’s computer printed out the changing values of the three</p><p>variables: 0–10–0; 4–12–0; 9–20–0; 16–36–2; 30–66–7; 54–115–24; 93–192–</p><p>74.Thethreenumbersroseandthenfellasimaginarytimeintervalstickedby,</p><p>fivetimesteps,ahundredtimesteps,athousand.</p><p>Tomakeapicturefromthedata,Lorenzusedeachsetofthreenumbersas</p><p>coordinates to specify the locationofapoint in three-dimensional space.Thus</p><p>the sequence of numbers produced a sequence of points tracing a continuous</p><p>path,arecordofthesystem’sbehavior.Suchapathmightleadtooneplaceand</p><p>stop, meaning that the system had settled down to a steady state, where the</p><p>variablesforspeedandtemperaturewerenolongerchanging.Orthepathmight</p><p>formaloop,goingaroundandaround,meaningthatthesystemhadsettledintoa</p><p>patternofbehaviorthatwouldrepeatitselfperiodically.</p><p>Lorenz’s systemdidneither. Instead, themapdisplayedakindof infinite</p><p>complexity. Italwaysstayedwithincertainbounds,never</p><p>runningoff thepage</p><p>butneverrepeatingitself,either.Ittracedastrange,distinctiveshape,akindof</p><p>doublespiralinthreedimensions,likeabutterflywithitstwowings.Theshape</p><p>signaledpuredisorder, sincenopointorpatternofpointsever recurred.Yet it</p><p>alsosignaledanewkindoforder.</p><p>YEARSLATER,PHYSICISTSwouldgivewistful lookswhenthey talkedabout</p><p>Lorenz’spaperonthoseequations—“thatbeautifulmarvelofapaper.”Bythen</p><p>itwastalkedaboutasifitwereanancientscroll,preservingsecretsofeternity.</p><p>In the thousandsof articles thatmadeup the technical literatureof chaos, few</p><p>were cited more often than “Deterministic Nonperiodic Flow.” For years, no</p><p>single object would inspire more illustrations, even motion pictures, than the</p><p>mysteriouscurvedepictedattheend,thedoublespiralthatbecameknownasthe</p><p>Lorenzattractor.Forthefirsttime,Lorenz’spictureshadshownwhatitmeantto</p><p>say,“Thisiscomplicated.”Alltherichnessofchaoswasthere.</p><p>At the time, though, few could see it. Lorenz described it to Willem</p><p>Malkus, a professor of appliedmathematics atM.I.T., a gentlemanly scientist</p><p>withagrandcapacityforappreciating theworkofcolleagues.Malkus laughed</p><p>andsaid,“Ed,weknow—weknowverywell—thatfluidconvectiondoesn’tdo</p><p>thatatall.”Thecomplexitywouldsurelybedampedout,Malkustoldhim,and</p><p>thesystemwouldsettledowntosteady,regularmotion.</p><p>“Of course, we completely missed the point,” Malkus said a generation</p><p>later—years after he had built a real Lorenzian waterwheel in his basement</p><p>laboratorytoshownonbelievers.“Edwasn’tthinkingintermsofourphysicsat</p><p>all.Hewas thinking in terms of some sort of generalized or abstractedmodel</p><p>which exhibited behavior that he intuitively felt was characteristic of some</p><p>aspectsoftheexternalworld.Hecouldn’tquitesaythattous,though.It’sonly</p><p>afterthefactthatweperceivedthathemusthaveheldthoseviews.”</p><p>Few laymen realized how tightly compartmentalized the scientific</p><p>community had become, a battleship with bulkheads sealed against leaks.</p><p>Biologists had enough to read without keeping up with the mathematics</p><p>literature—for that matter, molecular biologists had enough to read without</p><p>keeping upwith population biology. Physicists had betterways to spend their</p><p>timethansiftingthroughthemeteorologyjournals.Somemathematicianswould</p><p>have been excited to see Lorenz’s discovery; within a decade, physicists,</p><p>astronomers,andbiologistswereseekingsomethingjust likeit,andsometimes</p><p>rediscovering it for themselves. But Lorenz was a meteorologist, and no one</p><p>thought to look for chaos on page 130 of volume 20 of the Journal of the</p><p>AtmosphericSciences.</p><p>Revolution</p><p>Ofcourse,theentireeffortistoputoneself</p><p>Outsidetheordinaryrange</p><p>Ofwhatarecalledstatistics.</p><p>—STEPHENSPENDER</p><p>THE HISTORIAN OF SCIENCE Thomas S. Kuhn describes a disturbing</p><p>experiment conducted by a pair of psychologists in the 1940s. Subjects were</p><p>givenglimpsesofplayingcards,oneatatime,andaskedtonamethem.There</p><p>wasatrick,ofcourse.Afewofthecardswerefreakish:forexample,aredsixof</p><p>spadesorablackqueenofdiamonds.</p><p>Athighspeedthesubjectssailedsmoothlyalong.Nothingcouldhavebeen</p><p>simpler.Theydidn’t see the anomalies at all.Showna red sixof spades, they</p><p>wouldsingouteither“sixofhearts”or“sixofspades.”Butwhenthecardswere</p><p>displayed for longer intervals, the subjects started to hesitate. They became</p><p>awareofaproblembutwerenotsurequitewhatitwas.Asubjectmightsaythat</p><p>hehadseensomethingodd,likearedborderaroundablackheart.</p><p>Eventually,asthepacewasslowedevenmore,mostsubjectswouldcatch</p><p>on.Theywouldseethewrongcardsandmakethementalshiftnecessarytoplay</p><p>the game without error. Not everyone, though. A few suffered a sense of</p><p>disorientationthatbroughtrealpain.“Ican’tmakethatsuitout,whateveritis,”</p><p>saidone.“Itdidn’tevenlooklikeacardthattime.Idon’tknowwhatcoloritis</p><p>noworwhetherit’saspadeoraheart.I’mnotevensurewhataspadelookslike.</p><p>MyGod!”</p><p>Professional scientists, given brief, uncertain glimpses of nature’s</p><p>workings,arenolessvulnerabletoanguishandconfusionwhentheycomeface</p><p>to facewith incongruity.And incongruity,when itchanges thewaya scientist</p><p>sees,makespossible themost importantadvances.SoKuhnargues,andsothe</p><p>storyofchaossuggests.</p><p>Kuhn’snotionsofhowscientistsworkandhowrevolutionsoccurdrewas</p><p>much hostility as admiration when he first published them, in 1962, and the</p><p>controversyhasneverended.Hepushedasharpneedleintothetraditionalview</p><p>thatscienceprogressesbytheaccretionofknowledge,eachdiscoveryaddingto</p><p>the last, and that new theories emerge when new experimental facts require</p><p>them.Hedeflatedtheviewofscienceasanorderlyprocessofaskingquestions</p><p>and finding theiranswers.Heemphasizedacontrastbetween thebulkofwhat</p><p>scientists do, working on legitimate, well-understood problems within their</p><p>disciplines, and theexceptional,unorthodoxwork that creates revolutions.Not</p><p>byaccident,hemadescientistsseemlessthanperfectrationalists.</p><p>In Kuhn’s scheme, normal science consists largely of mopping up</p><p>operations. Experimentalists carry out modified versions of experiments that</p><p>havebeencarriedoutmanytimesbefore.Theoristsaddabrickhere,reshapea</p><p>cornice there, inawallof theory. Itcouldhardlybeotherwise. Ifall scientists</p><p>had to begin from the beginning, questioning fundamental assumptions, they</p><p>wouldbehardpressedtoreachtheleveloftechnicalsophisticationnecessaryto</p><p>dousefulwork.InBenjaminFranklin’stime,thehandfulofscientiststryingto</p><p>understand electricity could choose their own first principles—indeed, had to.</p><p>One researcher might consider attraction to be the most important electrical</p><p>effect, thinking of electricity as a sort of “effluvium” emanating from</p><p>substances.Anothermightthinkofelectricityasafluid,conveyedbyconducting</p><p>material. These scientists could speak almost as easily to laymen as to each</p><p>other,becausetheyhadnotyetreachedastagewheretheycouldtakeforgranted</p><p>a common, specialized language for the phenomena they were studying. By</p><p>contrast, a twentieth-century fluid dynamicist could hardly expect to advance</p><p>knowledge in his field without first adopting a body of terminology and</p><p>mathematical technique. In return, unconsciously, he would give up much</p><p>freedomtoquestionthefoundationsofhisscience.</p><p>CentraltoKuhn’sideasisthevisionofnormalscienceassolvingproblems,</p><p>thekindsofproblemsthatstudentslearnthefirsttimetheyopentheirtextbooks.</p><p>Such problems define an accepted style of achievement that carries most</p><p>scientists through graduate school, through their thesis work, and through the</p><p>writingof journalarticles thatmakesup thebodyofacademiccareers.“Under</p><p>normal conditions the research scientist is not an innovator but a solver of</p><p>puzzles, and the puzzles upon which he concentrates are just those which he</p><p>believes canbeboth statedand solvedwithin the existing scientific tradition,”</p><p>Kuhnwrote.</p><p>Thentherearerevolutions.Anewsciencearisesoutofonethathasreached</p><p>a dead end. Often a revolution has an interdisciplinary character—its central</p><p>discoveriesoftencomefrompeoplestrayingoutsidethenormalboundsoftheir</p><p>specialties. The problems that obsess these theorists are not recognized as</p><p>legitimate</p>
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- Lista de Exercícios 7 - Funções
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