The Pi ManifestoMicrosoft Word 文档 3.docx
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The Pi ManifestoMicrosoft Word 文档 3.docx
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ThePiManifestoMicrosoftWord文档3
ThePiManifesto
WrittenbyMSC
LastupdatedJuly4th,2011
1\piversus\tau
1.1TheTauMovement
Thisarticleisdedicatedtodefendoneofthemostimportantnumbersinmathematics:
\pi.Quiterecently,aphenomenonknownastheTauMovementhassteadilygrownandisgainingmoreandmorefollowers(calledTauists)bytheday.Thisislargelyduetothreedrivingforces:
1.Theoriginalarticle\piiswrongwrittenbyBobPalais(publishedin2000/2001).
2.TheTauManifestowrittenbyMichaelHartl(launchedonJune28th,2010).
3.ThevideoPiis(still)wrongbyViHart(uploadedonMarch14th,2011).
Weencouragethereadertofirstcheckouttheselinksindetailtoseethepossiblebenefitsofdefiningtheconstant\tau=2\pi\approx6.283185\ldots.
Tauistsclaimthat\piisthewrongcircleconstantandbelievethetruecircleconstantshouldbe\tau=2\pi.TheycelebrateTauDay(June28th),wear\tau-shirtsandspreadpro-taupropoganda.
Butaretauistsdoingmoreharmthangood?
Inthisarticlewewillexplorethisveryquestionandprovideseveralreasonswhy\piwillprevailintheintriguing\piversus\taubattle.
1.2Anypublicityisgoodpublicity
Thebuzzaroundtheblogosphereandonvariousonlinenewssitesisthatthereisabattlehappeninginmathematics,namely\piversus\tau.Headlinesinnewspapersandonblogarticlesoftendeclarethat\piiswrongandtendtomisleadthegeneralpublic:
1.Mathematicianswantpiouttauin(SundayTimes.lk)
2.Downwithuglypi,longliveelegantTau,physicisturges(TheS)
3.Mathematicianswanttosaygoodbyetopi(LiveS)
4.Onnationaltauday,piunderattack(FoxN)
Butpiisfarfrombeinguglyandmathematiciansarecertainlynotgoingtoreplacethecircleconstantanytimesoon.Thefactis,mostmathematicianshaveneverheardoftheTauMovement,andthosewhohave,simplydismisstauistsascranks.
AccordingtoanarticlepublishedbyTheTelegraphonTauday:
"LeadingmathematiciansinIndia,theUKandtheUSappearedoblivioustothiscampaigntodayandassertedthattherehasbeennodebateorevendiscussionoverreplacing2\piwith\tauinseriousmathematicalcircles."
MathematicianAlexandruIonescuatPrincetonUniversitysays:
"Eitheroneisjustfine,itwon'tmakeanydifferencetomathematics."
SiddharthaGadgil,amathematicianattheIISc,says:
"Thewholenotionofreplacing\piby2\piissillysinceweallareverycomfortablewith\piandmultiplicationbytwo."
Infact,onegradstudentinmathematicsgoesontosay:
"Ofcourseithadtobeaphysicistwhowouldwanttogetridoftheusageof\pi...Theconceptof\pihasbeenaroundsincethetimeoftheancientBabylonians(thegreekletterrepresentingthisnumberwaspopularizedbyEulerinthe18thcentury)...sowhychangenowandtrashit?
Thisisn'tthefirstthingthatphysicistshavetriedtochangeinthefieldofmathematics(notationwise,anyways).Iforonebelievethatthemathematicscommunitywillnotbelemmingshereandgowiththisidea;IknowI'mcertainlynotgoingtoaccepttauasareplacementforpi."
Itisdebatablewhetherthemediacoverageof\tauisgoodpublicityorbadpublicityformathematics,butregardless,theTauMovementhasdefinitelysparkedaninterest.Eventhosewithverylittlemathematicalbackgroundarecuriousaboutit!
Ithinkmostmathematicianswouldagreethatanythingthatgeneratesinterestinmathisadefiniteplus.
Asseenfromthequotesabove,alotofmathematicianssimplyshrugofftheTauMovementasbeingsilly.Inthisarticleweattempttogiveaseriousrebuttalto\tauinthedefenceof\pi.Anysuggestionsandreasonswhy\piisbetterthan\tau(or\tauisbetterthan\pi)aremorethanwelcome!
1.3TheTauManifestoiswrong
Tauistsarguethatbyusingtheconstant\tau=2\pialotofformulasbecomesimpler.Unfortunately,theTaoManifestoisfullofselectivebiasinordertoconvincereadersofthebenefitsof\tauover\pi.Theypinpointformulasthatcontain2\piwhileignoringotherformulasthatdonot.Wedemonstratebelowthatwhenmakingthechangeto\tau,therearelotsofformulasthateitherbecomeworseorhavenoclearadvantageofusing\tauover\pi.TauistsalsoclaimthattheirversionofEuler'sformulaisbetterthantheoriginal,butwewillseethatitisinfactweaker.Thebenefitsof\tauonlyappearwhenviewing\pifromanarrowmindedtwodimensionalgeometricalpointofview,butthesebenefitsdisappearwhenlookingatthebiggerpicture.Wewillseehowtheimportanceof\pishinesthroughasitshowsupallovermathematicsandnotjustinelementarygeometry.
2Definitionsof\pi
2.1TheTraditionalDefinition
TheTauManifestoreliesonthetraditionaldefinitionof\pi,namely,theconstantthatisequaltotheratioofacircle'scircumferencetoitsdiameter:
\pi\equiv\frac{C}{D}\approx3.14159\ldots.
Themanifestothengoesontosuggestthatweshouldbemorefocusedontheratioofacircle'scircumferencetoitsradius:
\tau\equiv\frac{C}{r}\approx6.283185\ldots.
Inparticular,sinceacircleisdefinedasthesetofpointsafixeddistance(i.e.,theradius)fromagivenpoint,amorenaturaldefinitionforthecircleconstantusesrinplaceofD.
Sowhydidmathematiciansdefineitusingthediameter?
Likelybecauseitiseasiertomeasurethediameterofacircularobjectthanitistomeasureitsradius.IntheTauManifesto,Hartlsays:
"I’msurprisedthatArchimedes,whofamouslyapproximatedthecircleconstant,didn'trealizethatC/risthemorefundamentalnumber.I’mevenmoresurprisedthatEulerdidn'tcorrecttheproblemwhenhehadthechance."
ButDr.Hartl,thereisnoproblemtocorrect,\piisnotwrong,andwewillsoonseethatwehavebeenusingtherightconstantallalong.
Therearenumerousreasonstodefinethecircleconstantusing\frac{C}{D}.Someofthesereasonsinclude:
1.Thisdefinitionisconsistentwiththeareadefinitiondiscussedinthenextsection.
2.Inpractice,theonlywaytomeasuretheradiusofacircleistofirstmeasurethediameteranddivideby2.
3.WhylookataratiowhereyougoallthewayaroundthecircleyetonlyHALFwayacrossit?
Itjustdoesn'tseemnatural.
4.SomebelievetheBiblesaysweshouldbelookingatcircumferenceanddiameter,nottheradius.(Author'snote:
Thisisn'taseriousreason:
P)
2.2Otherdefinitionsof\pi
Anotherdefinitionfor\piistodefineittobetwicethesmallestpositivexforwhich\cos(x)=0[4],orthesmallestpositivexforwhich\sin(x)=0.Withthisdefinitionneither\pinor\tauissimplerthantheother.Tauistsmayclaimthat\taucanbedefinedastheperiodof\cos(x)or\sin(x)butwhetherthisisbetterisupfordebate(inthesameway,\picanbedefinedastheperiodof\tan(x)).
Anothercommongeometricdefinitionfor\piisintermsofareasratherthanlengths.Takertobetheradiusofacircle.Define\pitobetheratioofthecircle'sareatotheareaofasquarewhosesidelengthisequaltor,thatis,\pi\equiv\frac{A}{r^2}.
Intermsof\tau,thisdefinitionismessyandincludesafactorof2.Inparticular,define\tautobethetwicetheratioofacircle'sareatotheareaofasquarewhosesidelengthisequaltor,thatis,\tau\equiv2\left(\frac{A}{r^2}\right).
Clearly,thisdefinitionfavors\piover\tauandalsoinvolvestheimportantradiusofacircle.Likethetraditionaldefinition,thisdefinitionof\pidependsonresultsofEuclideangeometryandcomesnaturallywhenlookingatareas.
2.3Whystopatredefining\pi?
Mixingthingsupabit,intermsofdiameterwecandefineaconstant(callit\pi/4)asfollows:
\frac{\pi}{4}\equiv\frac{A}{D^2}.
Thissuggeststhatperhapsboth\piand\tauarewrong,and\pi/4isthecorrectcircleconstant.Othershavealsosuggestedsimilarnumbersasthecircleconstant.In1958,Eaglesuggeststhat\pi/2isthecorrectcircleconstant[1].Infact,the\pi/2Manifestoiscomingsoontoawebsitenearyou!
(Justkidding,Ihope).Butwhystopatredefining\pi?
TerryTaosays:
"Itmaybethat2\piiisanevenmorefundamentalconstantthan2\pior\pi.Itis,afterall,thegeneratorof\log
(1).Thefactthatsomanyformulaeinvolving\pi^ndependontheparityofnisanotherclueinthisregard."
Clearly,eachof\pi,2\pi,\pi/2,\pi/4and2\piihavetheirbenefits,butshouldweseriouslyisolate2\piandattempttoredefineitas\tau?
Sure\tauisbetterinafewinstances,butthatisbecauseitisamultipleof\pi.Thisisnoreasontointroduceanewconstantandencouragemathematicstoadoptit.
3Sillyarguments
3.1Asillyargumentfor\tau
Themainargumentfor\tauisitssimplicitytocalculatethenumberofradiansinafractionofacircle.Ithinkweallwouldagreethat\taumakesthistrivialtaskabitmoretrivial.Atauistwouldaskyou:
Quick,howmanyradiansinaneighthofacircle?
Isit\pi/4or\tau/8?
Intermsofturns,\tauhasaslightadvantage.JustlookatthefollowingtwofiguresthatappearedintheTauManifestoandtellmeyouaren'tconvincedbythepowerof\tau!
Figure1:
Somecommonangles.(Source:
)
Butthisisnotareasontoswitchto\tau.Thecontextishighlyrelevantinthisregardandsimilarquestionswhichfavor\picanbeposed.Letmedemonstratewithanexamplebyusingareasratherthanangles.Notethattheareaofaunitcircleis\pi.
Nowquick,whatistheareaofaneigthofaunitcircle?
\pi/8or\tau/16?
Taumayhaveitsbenefitswhenlookingatturns,butwhenlookingatareas\pitakesthecake(orrather,pie).JustliketheTauManifesto,Itoocancreateconvincinglookingpictures:
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