Mathematics and culture II PDF
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Mathematics and Culture II MicheleEmmer Mathematics and Culture II Visual Perfection: Mathematics and Creativity Editor MicheleEmmer DipartimentodiMatematica“G.Castelnuovo” UniversitàdegliStudi“LaSapienza”,Roma,Italy PiazzaleAldoMoro 2 00185Roma,Italy e-mail:[email protected] Library ofCongress ControlNumber: 2003064905 Acatalog recordfor this bookis availablefrom theLibrary ofCongress. Bibliographicinformationpublishedby DieDeutscheBibliothek DieDeutscheBibliotheklists this publicationin theDeutscheNationalbibliografie; detailedbibliographicdatais availablein theInternet at <http://dnb.ddb.de>. ThearticlesbyApostolosDoxiadisEuclid’sPoetics:Anexaminationofthesimilaritybetweennar- rativeandproof,SimonSinghTheRiseofNarrativeNon-FictionandRobertOssermanMathe- maticsTakesCenterStagewereoriginallypublishedinItalianinMatematicaeCultura2002, 88-470-0154-4bySpringerItalia,Milano2002. The Articles Mathematics and Culture: Ever New Ideas; Mathematics, Art and Architecture; Mathland: From Topology to Virtual Architecture; Mathematics, Literature and Cinema and MathematicsandRaymondQueneaubyMicheleEmmerweretranslatedfromtheItalianLan- guagebyGianfrancoMarletta. e-mail:[email protected] Mathematics Subject Classification(2000): 00Axx, 00B10, 01-XX ISBN 3-540-21368-6 Springer BerlinHeidelbergNew York Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthemate- rialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recita- tion,broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks. Duplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsofthe GermanCopyrightLawofSeptember9,1965,initscurrentversion,andpermissionforusemust always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is apart ofSpringer Science+Business Media springeronline.com ©Springer-VerlagBerlinHeidelberg 2005 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromthe relevant protectivelaws and regulations and thereforefreefor general use. EngravingoncoverandpartbeginningsbyMatteoEmmer, fromthebook:M.Emmer “La Veneziaperfetta”,CentroInternazionaledellaGrafica,Venezia, 2003; by kindpermission. Typeset:performelectronicpublishingGmbH,Heidelberg Cover design:ErichKirchner,Heidelberg Production:LE-TEXJelonek,Schmidt &Vöckler GbR,Leipzig Printedonacid-freepaper 46/3142/YL-54 3 2 1 0 Mathematics andCulture:EverNew Ideas MicheleEmmer Oneoftheinterestingquestionsinthestudyoflinksbetweenmathematics,art andcreativityiswhetheramathematician’screativityleadshimtoinventanew world,orrathermakeshimdiscoveronethatalreadyexistsinitsownright.It could seem like a superfluous question of very little interest. Though it might seemsotononmathematicians,itcertainlyisn’tsoformanymathematicianslike RogerPenrose,whohasdedicatedapartofthebookTheEmperor’sNewMind[1] tothissubject.“Inmathematics,shouldonetalkofinventionorofdiscovery?” asks Roger Penrose. There are two possible answers to the question: when the mathematicianobtainsnewresults,hecreatesonlysomeelaboratementalcon- structionswhich,althoughhavenorelationtophysicalrealitywhatsoever,never- thelesspossesssuchpowerandelegancethattheyareabletomaketheresearcher believethatthese“merementalconstructions”havetheirveryownreality.Alter- V natively,domathematiciansdiscoverthat“thesemerementalconstructions”are alreadythere,atruthwhoseexistenceiscompletelyindependentoftheirwork- ings out? Penrose is inclined toward the second point of view, even though he addsthattheproblemisnotassimpleasitseems.Hisopinionisthatinmathe- maticsonedeterminessituationsforwhichthetermdiscoveryiscertainlymore appropriatethantheterminvention.Therearecasesinwhichtheresultsessen- tially derive from the structure itself, more than from the input of mathemati- cians.Penrosecitestheexampleofcomplexnumbers:“Laterwefindmanyother magicalpropertiesthatthesecomplexnumberspossess,propertiesthatwehad noinklingaboutatfirst.Thesepropertiesarejustthere.Theywerenotputthere byCardano,norbyBombelli,norWallis,norCoates,norEuler,norWessel,nor Gauss,despitetheundoubtedfarsightednessofthese,andother,greatmathema- ticians;suchmagicwasinherentintheverystructurethattheygraduallyuncov- ered.” When mathematicians discover a structure of this kind, it means that they stumbleduponthatwhichPenrosecalls“worksofGod”.So,aremathematicians mereexplorers?Fortunatelynotallmathematicalstructuresaresostrictlyprede- termined.Therearecasesinwhich“theresultsareobtainedequallybymeritof thestructureandofthemathematicians’calculations”;inthiscase,Penrosesays, itismoreappropriatetousethewordinventionthantheworddiscovery.Hence thereisroomforwhathecallsworksofman,thoughhenotesthatthediscoveries of structures that are works of God are of vastly greater importance than the ‘mere’ inventions that are the works ofman. Mathematics andCultureII One can make analogous distinctions in the arts and in engineering. “Great works ofart areindeed‘closer toGod’ thanarelesser ones.” Amongartists,Penroseclaims,theideathattheirmostimportantworksreveal eternaltruthsisnotuncommon,thattheyhave“somekindofprioretherealexis- tence”,whilethelesserimportantworkshaveamorepersonalcharacter,andare arbitrary,mortalconstructions.Inmathematics,thisneedtobelieveinanimma- terialandeternalexistence,atleastofthemostprofoundmathematicalconcepts (theworksofGod),isfeltevenmorestrongly.Penroseobservesthat“thereisa compellinguniquenessanduniversalityinsuchmathematicalideaswhichseems tobeofquiteadifferentorderfromthatwhichonecouldexpectintheartsoror engineering.”Aworkofartcanbeappreciatedorbroughtintoquestionindiffer- entepochs,butno-onecanputindoubtacorrectproofofamathematicalresult. Penroseexplainsveryexplicitlythatmathematiciansthinkoftheirdiscipline asahighlycreativeactivitythathasnothingtoenvyofthecreativityofartistsand indeed,becauseoftheuniquenessanduniversalityofmathematicalcreation,one that should be regarded as superior to the artistic discipline; Penrose doesn’t writeitexplicitly,butmanymathematiciansthinkthatmathematicsisthetrue Art; a difficult, laborious art, with it’s very own language and symbolism, that produces universally accepted results. Penrosetakesupthequestionoftheexistenceinitsownrightoftheworldof mathematicalideas,aquestionthathasdoggedthescienceofmathematicssince itsinception.ItcomesfromthebookMatièreàpensée,co-authoredbyamathe- matician,AlainConnes,winneroftheFieldsmedal,andaneurobiologist,Jean- VI PierreChangeux [2]. In one of the chapters in the book, entitled Invention ou découverte the neurobiologist,speakingofthenatureofmathematicalobjects,recallsthatthere isarealistattitudedirectlyinspiredbyPlato,anattitudethatcanbesummarised inthephrase:theworldispopulatedbyideasthathavearealityseparatefrom physicalreality.Supportinghisobservation,ChangeuxquotesaclaimofDieu- donné, according to whom “mathematicians accept that mathematical objects possessarealitydistinctfromphysicalreality”,arealitythatcanbecomparedto thatwhichPlatoaccordstohisIdeas.Fromthispointofviewitisofsecondaryim- portancewhetherornotthemathematicalworldisadivinecreation,asthemath- ematicianCantor(1874–1918)believed:“ThehighestperfectionofGodisthepos- sibility ofcreatinganinfinite set,andhis immensebounty leads him todo so.” Weareincompletedivinemathesis,incompletemetaphysics.Thisissurpris- ingtoseriousscientists,commentsChangeux.Connes,themathematician,notat all bothered by the biologist’s arguments, replies very clearly that he identifies strongly with the realist point of view. After emphasising that the sequence of primenumbershasamorestablerealitythanthematerialrealitysurroundingus, he notes that “one can comparethe workof a mathematicianto that of an ex- plorerdiscoveringthe world.” If practicalexperience leads usto discoverpure andsimplefactssuchas,forexample,thatthesequenceofprimenumbersappear tohavenoend,theworkofamathematicianconsistsofprovingthatthereexists aninfinitenumberofprimes.Oncethispropertyhasbeenproved,no-onecan everclaimtohavingfoundthelargestprimeofall.Itwouldbeeasytoshowthem Preface thattheyarewrong.Connesconcludes“soweclashwitharealityjustasincon- testableasthephysicalone”.Indeed,itisarguablymorerealthananyphysicalre- ality. Stillintheaforementionedbook,wereadthatmathematicsisauniversallan- guage,butthatisnotall,forthegenerativenatureofmathematicshasacrucial role.Otherwiseonecouldnotexplaintheincredibleandunforeseeableusefulness ofmathematics,frommeteorologytothestructureofDNA,tothesimulationof boats for theAmerica’s Cup. Imaginationisnotenoughtodomathematics;youneedknowledgeofthelan- guageandtechniques;imagination,creativity,theabilitytomakeproblemsex- plicit, toformaliseand solve them, whenever possible. Inrecentyears,thankstotheadventofcomputergraphics,theroleofvisuali- sationinsomesectorsofmathematicshasincreasedsignificantly.Itwasinevita- blethatthissortofnewVisualMathematicswouldopennewwaysforlinksbe- tween creativity in the arts and in mathematics. We wish to deal with some of theseaspectsinthisnewbookoftheseriescalledsignificantly“Mathematicsand Culture”,whichwasbornofanideaofValeriaMarchiafavaandMicheleEmmer inSeptember 1997,inTurin. References VII [1] Penrose,R.:TheEmperor’s New Mind.OxfordUniversity Press,New York(1989). [2] Changeux,Je.-P.,Connes,A.:MatièreàPenseée.ed.OdileJacob,Paris (1989). TableofContents Mathematics,Art andArchitecture Mathematics,Art andArchitecture MicheleEmmer .............................................................................................. 3 From Tiling the Plane to Paving Town Square JudithFlaggMoranandKimWilliams......................................................... 5 Mathematics inContemporary Arts –FiniteandInfinity DietmarGuderian.......................................................................................... 23 DeInsanaGeometria AchillePerilli.................................................................................................. 39 Ovals inBorromini’s Geometry MicheaSimona............................................................................................... 45 Fractals:AResonancebetweenArt andNature IX RichardTaylor,BenNewell,BrankaSpeharandColinClifford.................. 53 Mathland–FromTopology toVirtualArchitecture MicheleEmmer .............................................................................................. 65 VisualMathematics andComputerGraphics Introduction:VisualMathematics andComputer Graphics MicheleEmmer............................................................................................... 81 MathAwareness Month 2000:AnInteractiveExperience ThomasF.BanchoffandDavideP.Cervone................................................. 83 VisualTopology andVariationalProblems onTwo-DimensionalSurfaces AnatolyT.Fomenko,AlexandrO.IvanovandAlexeyA.Tuzhilin ............. 99 MetarealisticRenderingofReal-timeInteractive Computer Animations GeorgeK.Francis............................................................................................ 125 VirtualSculptures HerbertW.Franke.......................................................................................... 145 PublicationofElectronicGeometry Models MichaelJoswigandKonradPolthier ............................................................ 151 Trees,Roots andaBrain: AMethaphoricalFoundationfor MathematicalArt JohnSims ........................................................................................................ 163 TableofContents Mathematics,LiteratureandCinema Mathematics,LiteratureandCinema MicheleEmmer .............................................................................................. 173 Euclid’sPoetics:Anexaminationofthesimilarity betweennarrativeandproof ApostolosDoxiadis......................................................................................... 175 TheRiseofNarrativeNon-Fiction SimonSingh.................................................................................................... 183 MathematicsTakesCenterStage RobertOsserman............................................................................................ 187 Mathematics andRaymondQueneau MicheleEmmer .............................................................................................. 195 Authors ............................................................................................................... 201 X Mathematics, Art and Architecture Mathematics,Art andArchitecture MicheleEmmer Thecarethatmathematicianshavefortheaestheticqualityoftheirdisciplineis noteworthy:thisgivesrisetomanymathematicians’ideathatmathematicalac- tivity andartisticactivity are,in some sense, very similar andcomparable. MorrisKline,amathematician,(andheisbutoneofmanyexamplesthatwe couldcite)hasdedicatedseveralpagesofhisbookMathematicsInWesternCul- ture[1]tothissubject.Afterrecallingthatmathematicianshaveforhundredsof yearsacceptedwhattheGreeksmaintained,thatis,thefactthatmathematicsis an art and that mathematical work must satisfy aesthetic requirements, Kline asksthefundamentalquestionofwhymanypeoplemaintainthattheinclusionof mathematics in thearts is unjustified. Oneofthemostcommonobjectionsisthatmathematicsdoesnotevokeany emotion.Kline,ontheotherhand,observesthatmathematicsprovokesundeni- 3 ablefeelingsofaversionandreaction,andmoreovergeneratesgreatjoyinthere- searcherswhentheymanagetomakeapreciseformulationoftheirideasandon obtainingcleverandinspiredproofs.Theproblemliesinthefactthatitisonlyre- searcherswhocanexperiencetheseemotionsandno-oneelse.“Justlikeinthe arts,eachparticularofthefinalworkisnotdiscovered,butcomposed.Ofcourse thecreativeprocessmustproduceaworkthathasdesign,harmonyandbeauty. Thesequalities tooarepresent inmathematicalcreations.” CommentingonthesewordsofKline,Ihaveobservedthat[2]“ifitisnotinter- esting,inthisrealm,todiscussmathematicians’ideasonart,itishoweverworth- whilehighlightinghowthisartisticaspirationiswidespreadinthemathematical community.Complementarytothisrequirementistheneed forrecognitionof theartisticcreativityofthemathematicianbythemathematicallaity;arecogni- tionthatisnotgenerallygiven,inparticularbythosewhotakeaprofessionalin- terestinart.Especiallysincethiswouldentailhavingtounderstandsomethingof contemporarymathematics.Everyonecanlookataworkofart,listentoasym- phony,butonecannotlookatorlistentomathematics.Klineclearlyrecognises thiswhen he saysthat the definitive test ofaworkofartisit’scontributionto aestheticpleasureor tobeauty. Fortunately,orunfortunately,thetestinquestionissubjective,onethatdepends onthelevelofcultureinaspecificsector.Thequestionastowhetherornotmathe- maticspossessesitsownkindofbeautycanthusbegivenareplyonlybythosewho haveacultureinthisdiscipline...Unfortunately,tomastermathematicalideasre- quiresstudyandthereexistsnodirectroutethatsubstantiallyaccomplishesthis”.
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