Chaos Making a New Science ( PDFDrive ) - Inglês (2025)

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Pedro Pereira 20/09/2024

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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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  • Exercício avaliativo - Módulo 4_ Revisão da tentativa
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  • Assinale a alternativa correta: a. V, V, V, V. b. F, V, V, F. c. F, V, F, V. d. F, F, V, F. e. F, F, V, V.a. V, V, V, V.b. F, V, V, F.c. F, V, ...
  • Assinale a alternativa correta: a. Apenas I, II e III estão corretas. b. Apenas I e IV estão corretas. c. Todas as alternativas estão corretas. d. ...
  • Assinale a alternativa correta: a. Todas as alternativas estão corretas. b. Apenas II e III estão corretas. c. Apenas I, II e III estão corretas. d...
  • Assinale a alternativa correta: a. Apenas I, II e III estão corretas. b. Apenas II e III estão corretas. c. Apenas II, III e IV estão corretas. d. ...
  • Tal afirmação se refere a que tipo de Marketing? Assinale a alternativa correta:a. Marketing de Atraçãob. Marketing Digitalc. Marketing Multiní...
  • Com base nisto, analise as afirmativas abaixo e classifique-as com “V” para verdadeiro e “F” para falso:( ) A compra de termos relacionados ao ...
  • Sendo, a respeito das transformações ocasionadas pelas redes sociais, analise as afirmativas abaixo:I. O acesso à internet e o aumento significa...
  • Tal afirmação diz respeito a que ramificação do Marketing? Assinale a alternativa correta:a. Marketing de Escalab. Marketing Multinívelc. Marke...
  • Sendo assim, analise as afirmativas abaixo:I. O Marketing Multinível é uma estratégia de mercado não intrusiva e muito mais segmentada. Por isso...
  • Dentro deste contexto, analise as afirmativas abaixo:I. Estamos hoje passando por este processo de mutação, onde organizações buscam as maneiras...
  • Diante disto, analise as afirmativas abaixo: I. ROI: é a sigla em inglês para Retorno Sobre o Investimento. Esta é uma métrica usada para saber qua...
  • Assim sendo, quanto aos princípios da valorização de dados, analise as afirmativas abaixo: I. Foco nas métricas erradas; II. Desvalorização dos cli...
  • O Marketing de Conteúdo é um dos pilares do Inbound Marketing. De acordo com Lima et al. (2016) o Marketing de Conteúdo busca entregar o conteúdo c...
  • Lista de Exercícios 7 - Funções
  • Lista de Exercícios 7.1 - funções do 1º grau

Perguntas dessa disciplina

Grátis

4. O trecho “By making room for reports of single experiments or minor technical advances, journals increased the chaos of science” pode ser reescr...

Grátis

Questão – 04 Alternativa: B O trecho “By making room for reports of single experiments or minor technical advances, journals increased the ch...
Kuhn (1998, p. 122) says that the transition to a new paradigm is a revolution when a new paradigm completely or partially replaces an older one. T...
Ainda sobre a seçãoData science, engineering, and data-driven decision making, no exemplo mencionado sobre o WalMart e a Target, que problema étic...

UNICAP

Na seçãoData science, engineering, and data-driven decision making, os autores afirmam que, atualmente, estamos vendo uma revolução de propagandas...

UNICAP

Chaos  Making a New Science ( PDFDrive ) - Inglês (2025)

FAQs

What is chaos making a new science by James Gleick about? ›

Overview. Chaos: Making a New Science was the first popular book about chaos theory. It describes the Mandelbrot set, Julia sets, and Lorenz attractors without using complicated mathematics. It portrays the efforts of dozens of scientists whose separate work contributed to the developing field.

How many pages is chaos making a new science? ›

Product Details
ISBN-13:9780143113454
Pages:384
Sales rank:98,439
Product dimensions:5.50(w) x 8.40(h) x 1.30(d)
Age Range:18 Years
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What does chaos theory teach us? ›

Chaos theory ultimately teaches that us that uncertainty and unpredictably will always be a constant in life. And while three months may be less than a blink of an eye in the history of the universe, it's a significant amount of time in our lives.

What is the chaos theory easily explained? ›

While most traditional science deals with supposedly predictable phenomena like gravity, electricity, or chemical reactions, Chaos Theory deals with nonlinear things that are effectively impossible to predict or control, like turbulence, weather, the stock market, our brain states, and so on.

What is the chaos book about? ›

The book presents O'Neill's research into the background and motives for the Tate–LaBianca murders committed by the Manson Family in 1969. O'Neill questions the Helter Skelter scenario argued by lead prosecutor Vincent Bugliosi in the trials and in his book Helter Skelter (1974).

How many chapters are in total chaos? ›

Someone wants to be found. Survive in 6 chapters, against over 8 horrifying fiends with a large assortment of weapons.

How many pages is 12 Rules for Life an antidote to chaos? ›

12 Rules for Life
First edition cover
AuthorJordan Peterson
Media typePrint, digital, audible
Pages448 (hardcover) 320 (ebook)
ISBN978-0-345-81602-3 (Canada) 978-0-241-35163-5 (UK)
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What is the scientific explanation of chaos? ›

Chaos theory has been developed from the recognition that apparently simple physical systems which obey deterministic laws may nevertheless behave unpredictably. Nonlinear systems can converge to an equilibrium (steady state) or there can be a stable oscillation (periodic behavior) or there can be chaotic change.

What is the book chaos about? ›

The book presents O'Neill's research into the background and motives for the Tate–LaBianca murders committed by the Manson Family in 1969. O'Neill questions the Helter Skelter scenario argued by lead prosecutor Vincent Bugliosi in the trials and in his book Helter Skelter (1974).

What is James Gleick referring to? ›

James Gleick's statement about "the gloom that has fallen over the book publishing industry" refers to: The practice of digitizing books that is putting publishers of only printed matter out of business.

What is the movie chaos theory about? ›

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