4/16/2015 NOVA|DescribingNatureWithMath

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ByPeterTyson Posted11.10.11 NOVA

"Withafewsymbolsonapage,youcandescribeawealthofphysicalphenomena."

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DescribingNatureWithMathHowdoscientistsusemathematicstodefinereality?Andwhy?

Howisitpossiblethatmathematics,aproductofhumanthoughtthatisindependentofexperience,fitssoexcellentlytheobjectsofreality?AlbertEinstein

Ifyou'relikeme,youunderstandreadilyhowonecandescribenature'swondersusingpoetryormusic,paintingorphotography.Wordsworth's"IWanderedLonelyasaCloud"andVivaldi's"FourSeasons"richlydepicttheirnaturalsubjects,asdoMonet'swaterliliesandAnselAdams'photosofYosemite.Butmathematics?Howcanyoudescribeatreeorcloud,arippledpondorswirlinggalaxyusingnumbersandequations?

Thisphotographdoesaprettygoodjobofdescribingripples.Butamathematiciancoulddoitwithgreaterprecisionandpredictivepower.Enlarge

Photocredit:AlexPotemkin/iStockphoto

Extremelywell,asEinsteinknewbetterthanmost,ofcourse.Infact,mostscientistswouldagreethat,whenitcomestoteasingouttheinherentsecretsoftheuniverse,nothingvisual,verbal,orauralcomesclosetomatchingtheaccuracyandeconomy,thepowerandelegance,andtheinescapabletruthofthemathematical.

Howisthisso?Well,forthemathchallenged,forthatpersonwhohasavoidedanythingbutthemostbasicarithmeticsincehighschool,whofeelsapitinhisstomachwhenheseesanequationthatis,formyselfIwillattempttoexplain,withthehelpofsomewhodomathematicsforaliving.Ifyou'remathphobic,too,Ithinkyou'llgetapainlessfeelforwhyeventhatmasterofdescribingnaturewithwords,Thoreau,wouldholdthat"themostdistinctandbeautifulstatementsofanytruthmusttakeatlastthemathematicalform."

ANCIENTMATHWhilemanyearlycivilizations,includingIslamic,Indian,andChinese,madeimportantcontributionstomathematics,itwastheancientGreekswhoinventedmuchofthemathwe'refamiliarwith.Euclidfatheredthegeometrywenamedafterhimallthoseradiiandhypotenusesandparallellines.Archimedesapproximatedpi.PtolemycreatedaprecisemathematicalmodelthathadalloftheheavenswheelingaroundtheEarth.

TheGreeks'discoveriesaretimeless:Euclid'saxiomsareasunimpeachabletodayaswhenhedevisedthemover2,000yearsago.AndsomeGreekprotophysicistsdidusetheirnewfoundskillstotacklemysteriesofthenaturalworld.Withbasictrigonometry,forexample,theastronomerEratosthenesestimatedthediameteroftheEarthwithover99percentaccuracyin228B.C.

ButwhiletheGreeksbelievedthattheuniversewasmathematicallydesigned,theylargelyappliedmathonlytostaticobjectsmeasuringangles,calculatingvolumesofsolidobjects,andthelikeaswellastophilosophicalpurposes.Platowouldn'tletanyonethroughthefrontdoorofhisacclaimedAcademywhodidn'tknowmathematics."Heisunworthyofthenameofman,"Platosniffed,"whoisignorantofthefactthatthediagonalofasquareisincommensurablewithitsside."Andsoitremainedforamillenniumandahalf.

Galileostrivedtoexplainhowobjectsfallratherthanwhy,amodusoperandithatsetthestagefortheadvancementofscienceasweknowittoday.Enlarge

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4/16/2015 NOVA|DescribingNatureWithMath

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"Mathematicscapturespatternsthattheuniversefindspleasant,ifyoulike."

THEMEASUREOFALLTHINGSGalileochangedallthatintheearly17thcentury.EschewingtheGreeks'attemptstoexplainwhyapebblefallswhenyoudropit,Galileosetouttodeterminehow.The"greatbook"oftheuniverseiswritteninthelanguageofmathematics,hefamouslydeclared,andunlessweunderstandthetriangles,circles,andothergeometricalfiguresthatformitscharacters,hewrote,"itishumanlyimpossibletocomprehendasinglewordofit[and]onewandersinvainthroughadarklabyrinth."(WordsworthorMonetmighttakeissuewiththatstatement,butwait.)

Galileosoughtcharacteristicsofourworldthathecouldmeasurevariableaspectslikeforceandweight,timeandspace,velocityandacceleration.Withsuchmeasurements,Galileowasabletoconstructthosegemsofscientificshorthandmathematicalformulaswhichdefinedphenomenamoreconciselyandmorepowerfullythanhadeverbeenpossiblebefore.(Hiscontemporary,theGermanmathematicianJohannesKepler,didthesamefortheheavens,craftingmathematicallawsthataccuratelydescribetheorbitsofplanetsaroundthesunandledtothescrappingofPtolemy'sEarthcentricmodel.)

ATIDYSUMAclassicexampleistheformulacommonlyshownasd=16t .(Hanginthere,mathphobes.Yourqueasiness,whichIshare,shouldgoawaywhenyouseehowstraightforwardthisis.)WhatGalileodiscoveredandensconcedinthissimpleequation,oneofthemostconsequentialinscientifichistory,isthat,whenairresistanceisleftout,thedistanceinfeet,d,thatanobjectfallsisequalto16timesthesquareofthetimeinseconds,t.Thus,ifyoudropapebbleoffacliff,inoneseconditwillfall16feet,intwoseconds64feet,inthreeseconds144feet,andsoon.

Galileo'ssuccinctformulaneatlyexpressesthenotionofaccelerationofobjectsnearthesurfaceoftheEarth,butthatisjustthestartofitsusefulness.First,justaswithanyvalueoftyoucancalculated,foranyvalueofdyoucanfiguret.Togettot,simplydividebothsidesoftheformulad=16t by16,thentakethesquarerootofbothsides.Thisleavesanewformula:

t= d16Thiscompactequationtellsyouthetimeneededforyourpebbletofallagivendistanceanydistance.Sayyourcliffis150high.Howlongwouldthepebbletaketoreachthebottom?Aquickcalculationrevealsjustoverthreeseconds.Athousandfeethigh?Justshyofeightseconds.

Boulder,pebble,pea:Despitetheirgreatdifferencesinmass,allthreeobjects,ifdroppedfromourhypotheticalcliffinavacuum,wouldreachthegroundbelowinthesameamountoftime.ThisiswhatGalileossimpleformulareveals.Enlarge

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BROADSTROKESWhatelsecanyoudowithapithyformulaliked=16t ?Well,ashintedabove,youcanmakecalculationsforaninfinitenumberofdifferentvaluesforeitherdort.Inessence,thismeansthatd=16t containsaninfiniteamountofinformation.Youcanalsosubstituteanyobjectforyourpebbleapea,say,oraboulderandtheformulastillholdsupperfectly(undertheconditionspreviouslymentioned).Couldasinglepoemorpaintingdoasmuch?

Andbecausethesamemathematicallawmaygovernmultiplephenomena,acuriousscientistcandiscoverrelationshipsbetweenthosephenomenathatmighthaveotherwisegoneundetected.Trigonometricfunctions,forinstance,applytoallwavemotionslight,sound,andradiowavesaswellaswavesinwater,wavesingas,andmanyothertypesofwavemotions.Thepersonwho"gets"thesetrigfunctionsandtheirpropertieswillipsofacto"get"allthephenomenathatthesefunctionsgovern.

AWEALTHOFDATAThepowerofapotentequationextendsstillfurther.TakeIsaacNewton'suniversallawofgravitation,whichbrilliantlycombinesGalileo'slawsoffallingbodieswithKepler'slawsofplanetarymotion.Manyofusknowgravityvaguelyasthatunseenforcethatkeepsthepebbleinyourpalmoryourfeetontheground.Newtondescribeditthisway:

F=Gm mr

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4/16/2015 NOVA|DescribingNatureWithMath

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"Einsteinusedmathematicstoseeapieceoftheuniversethatnoonehadeverseenbefore."

Iwon'tgointothisformula,butjustknowthatfromityoucancalculatethegravitationaltugbetweenjustaboutanytwoobjectsyoucanthinkof,fromthatbetweenyourcoffeecupandthetableitrestson,tothatbetweenonegalaxyandanother.Or,dependingonwhichvariablesyouknow,youcannaildowneverythingfromtheaccelerationofanyfreelyfallingobjectneartheEarth'ssurface(32feetpersecondduringeverysecondofitsfall)tothemassofourplanet(about6,000,000,000,000,000,000,000tons).

IfallothervariablesareknownandtheyaretodayonecanevencalculatethemassofourplanetusingNewtonsterseformulaongravitationalattraction.Enlarge

Photocredit:CourtesyNASA

"Withafewsymbolsonapage,youcandescribeawealthofphysicalphenomena,"saysastrophysicistBrianGreene,hostofNOVA'sseries(/wgbh/nova/physics/fabricofcosmos.html)basedonhisbookTheFabricoftheCosmos."Andthatis,insomesense,whatwemeanbyelegancethatthemessy,complexworldaroundusemanatesfromthisverysimpleequationthatyouhavewrittenonapieceofpaper."

AndlikeGalileo'sd=16t ,Newton'sformulaisamazinglyaccurate.In1997,UniversityofWashingtonresearchersdeterminedthatNewton'sinversesquarelawholdsdowntoadistanceof56,000thsofamillimeter.Itmayholdfurther,butthat'saspreciseasresearchershavegottenatthemoment.

EXACTSCIENCEWhatamazesmemostaboutGalileoandNewton'sformulasistheirexactitude.InGalileo's,thedistanceequalsexactlythesquareofthetimemultipliedby16inNewton's,theforceofattractionbetweenanytwoobjectsisexactlythesquareofthedistancebetweenthem.(That'sther inhisequation.)Suchexactnesscropsupregularlyinmathematicaldescriptionsofreality.Einsteinfound,forinstance,thattheenergyboundupin,say,apebbleequalsthepebble'smasstimesthesquareofthespeedoflight,orE=mc .

Eventhingswecanseeandtouchinnatureflirtwithmathematicalproportionsandpatterns.ConsidertheFibonaccisequence:1,1,2,3,5,8,13,21,34,55,89,144Noticeapattern?Afterthefirst,everynumberisthesumoftheprevioustwo.TheFibonaccisequencehasmanyinterestingproperties.OneisthatfractionsformedbysuccessiveFibonaccinumberse.g.,3/2and5/3and8/5getcloserandclosertoaparticularvalue,whichmathematiciansknowasthegoldennumber.Butwhataboutthis:ManyplantsadheretoFibonaccinumbers.Theblackeyedsusanhas13petals.Astershave21.Manydaisieshave34,55,or89petals,whilesunflowersusuallyhave55,89,or144.

Whydosunflowersoftenhaveprecisely55,89,or144petals,numbersthatfigureinthefamousFibonaccisequence?Nature,itseems,hascertainmathematicalunderpinnings.Enlarge

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ISGODAMATHEMATICIAN?TheapparentmathematicalnatureofNature,fromforcestoflowers,hasleftmanysincethetimeoftheGreekswondering,asthemathematicianMarioLiviodoesinhisbookofthesametitle,"IsGodamathematician?"Doestheuniverse,thatis,haveanunderlyingmathematicalstructure?Manybelieveitdoes."Justasmusicisauditorypatternsthatthehumanmindfindspleasant,"saysStanfordmathematicianKeithDevlin,"mathematicscapturespatternsthattheuniversefindspleasant,ifyoulikepatternsthatareimplicitinthewaytheuniverseworks."

Sodidwehumansinventmathematics,orwasitalreadyoutthere,limningthecosmos,awaitingthelikesofEuclidtorevealit?InhisbookMathematicsinWesternCulture,themathematicianMorrisKlinechosetosidestepthephilosophicalandfocusonthescientific:"Theplanthatmathematicseitherimposesonnatureorrevealsinnaturereplacesdisorderwithharmoniousorder.ThisistheessentialcontributionofPtolemy,Copernicus,Newton,andEinstein."

SEEINGTHEINVISIBLEFormulaslikeGalileo'sandNetwon'smaketheinvisiblevisible.Withd=16t ,wecan"see"themotionoffallingobjects.WithNewton'sequationongravity,wecan"see"theforcethatholdsthemooninorbitaroundtheEarth.WithEinstein'sequations,wecan"see"atoms."Einsteinisfamousforalotofthings,butonethingthatisoftenoverlookedishe'sthefirstpersonwhoactuallysaidhowbiganatomis,"saysJimGates,aphysicistattheUniversityofMaryland."Einsteinusedmathematicstoseeapieceoftheuniversethatnoonehadeverseenbefore."

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"DonotworryaboutyourdifficultiesinmathematicsIcanassureyouthatminearestillgreater."

Today,withadvancedtechnology,wecanobserveindividualatoms,butsomenaturalphenomenadefyanydescriptionbutamathematicalone."Theonlythingyoucansayabouttherealityofanelectronistociteitsmathematicalproperties,"notedthelatemathematicswriterMartinGardner."Sothere'sasenseinwhichmatterhascompletelydissolvedandwhatisleftisjustamathematicalstructure."CharlesDarwin,whoadmittedtohavingfoundmathematics"repugnant"asastudent,mayhaveputitbestwhenhewrote,"Mathematicsseemstoendowonewithsomethinglikeanewsense."

Mathematicspredictedwhatnaturehaslongknownthatthestripesonthemarineangelfishactuallymigrateacrossitsbodyovertime.Enlarge

Photocredit:IliutaGoean/iStockphoto

FORTUNETELLINGMathematicsalsoendowsonewithanabilitytopredict,asGalileo'sandNewton'sformulasmakeclear.Suchpredictivecapabilityoftenleadstonewdiscoveries.Inthemid1990s,KyotoUniversityresearchersrealizedtotheirsurprisethatequationsoriginallydevisedbythemathematicalgeniusAlanTuringpredictedthattheparallelyellowandpurplestripesofthemarineangelfishhavetomoveovertime.Stable,unmovingpatternsdidn'tjivewiththemathematics.Tofindoutifthiswastrue,theresearchersphotographedangelfishinanaquariumoverseveralmonths.Sureenough,anangelfish'sstripesdomigrateacrossitsbodyovertime,andinjustthewaytheequationshadindicated.Mathhadrevealedthesecret.

"Therereallyisafacingthemusicthatmathforces,andthat'swhyit'sawonderfullanguagefordescribingnature,"Greenesays."Itdoesmakepredictionsforwhatshouldhappen,and,whenthemathisaccuratelydescribingreality,thosepredictionsareborneoutbyobservation."

AMATHFORALLSEASONSSomuchmathematicsexistsnowonescholarestimatesthatamillionpagesofnewmathematicalideasarepublishedeachyearthatwhenscientistsfaceproblemsnotsolvablewithmaththeyknow,theycanoftenturntoanothervarietyforhelp.WhenEinsteinbeganworkonhistheoryofgeneralrelativity,heneededamathematicsthatcoulddescribewhathewasproposingthatspaceiscurved.HefounditinthenonEuclideangeometryof19thcenturymathematicianGeorgF.B.Riemann,whichprovidedjustthetoolherequired:ageometryofcurvedspacesinanynumberofdimensions.

Withfractalgeometry,youcanwritedownformulasthatdescriberoughshapesliketrees,incontrasttosmoothshapeslikeripples.Enlarge

Photocredit:ChrisHepburn/iStockphoto

Or,ifnecessary,theyinventnewmath.WhenthelatemathematicianBenoitMandelbrotconcludedthatstandardEuclideangeometry,whichisallaboutsmoothshapes,fellshortwhenhetriedtomathematicallyportray"rough"shapeslikebushytreesorjaggedcoastlines,heinventedanewmathematicscalledfractalgeometry."Mathisouroneandonlystrategyforunderstandingthecomplexityofnature,"saysRalphAbraham,amathematicianattheUniversityofCaliforniaSantaCruz,inNOVA'sHuntingtheHiddenDimension(/wgbh/nova/physics/huntinghiddendimension.html)."Fractalgeometryhasgivenusamuchlargervocabulary,andwithalargervocabularywecanreadmoreofthebookofnature."Galileowouldbesoproud.

TECHNOLOGICALWONDERSGalileowouldalsobeproudofjusthowmuchhissuccessorshaveachievedwithhisscientificmethod.FormulasfromhisownonfallingbodiestoWernerHeisenberg'sonquantummechanicshaveprovidedusthemeanstocollectandinterpretthemostvaluableknowledgewehaveeverattainedabouttheworkingsofnature.Altogether,themostgroundbreakingadvancesofmodernscienceandtechnology,boththeoreticalandpractical,havecomeaboutthroughthekindofdescriptive,quantitativeknowledgegatheringthatGalileopioneeredandNewtonrefined.

Newton'slawofgravity,forinstance,hasbeencriticalinallourmissionsintospace."Byunderstandingthemathematicsorforceofgravitybetweenlotsofdifferentbodies,yougetcompletecontrolandunderstanding,withveryhighprecision,ofexactlythebestwaytosendaspaceprobetoMarsorJupiterortoputsatellitesinorbitallofthosethings,"saysIanStewart,anemeritusprofessorofmathematicsattheUniversityofWarwickinEngland."Withoutthemath,youwouldnotbeabletodoit.Wecan'tsendathousandsatellitesupandhopeoneofthemgetsintotherightplace."

WithoutNewtonsformulaongravitationalattraction,wewouldneverhavebeenabletosendsatellitesandothercraftintospacesosuccessfully.Here,theInternationalSpaceStationasseenin2007.Enlarge

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Mathematicsunderliesvirtuallyallofourtechnologytoday.JamesMaxwell'sfourequationssummarizingelectromagnetismleddirectlytoradioandallotherformsoftelecommunication.E=mc leddirectlytonuclearpowerandnuclearweapons.Theequationsofquantummechanicsmadepossibleeverythingfromtransistorsandsemiconductorstoelectronmicroscopyandmagneticresonanceimaging.

Indeed,manyofthetechnologiesyouandIenjoyeverydaysimplywouldnotworkwithoutmathematics.WhenyoudoaGooglesearch,you'rerelyingon19thcenturyalgebra,onwhichthesearchengine'salgorithmsarebased.Whenyouwatchamovie,youmaywellbeseeingmountainsandothernaturalfeaturesthat,whileappearingasrealasrock,ariseentirelyfrommathematicalmodels.WhenyouplayyouriPod,you'rehearingamathematicalrecreationofmusicthatisstoreddigitallyyourcellphonedoesthesameinrealtime.

"Whenyoulistentoamobilephone,you'renotactuallyhearingthevoiceofthepersonspeaking,"Devlintoldme."You'rehearingamathematicalrecreationofthatvoice.Thatvoiceisreducedtomathematics."

AFTERMATHAndI'mreducedtoconcedingthatmathdoesn'tscaremesomuchanymore.Howaboutyou?Ifyoustillfeelqueasy,perhapsyoucantakesolacefromEinsteinhimself,whooncereassuredajuniorhighschoolstudent,"DonotworryaboutyourdifficultiesinmathematicsIcanassureyouthatminearestillgreater."

PeterTysoniseditorinchiefofNOVAOnline.

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