The Pi ManifestoMicrosoft Word 文档 3.docx

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: