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Science Networks Historical Studies 59 Michael Friedman A History of Folding in Mathematics Mathematizing the Margins Science Networks. Historical Studies Science Networks. Historical Studies Founded by Erwin Hiebert and Hans Wußing Volume 59 Edited by Eberhard Knobloch and Olivier Darrigol Editorial Board: J. Barrow-Green, Milton Keynes R. Krömer, Wuppertal U. Bottazzini, Milano J. Peiffer, Paris K. Chemla, Paris W. Purkert, Bonn A. Cogliati, Milano D. Rowe, Mainz S.S. Demidov, Moskva Ch. Sasaki, Kasugai C. Eckes, Nancy T. Sauer, Mainz J. Hughes, Manchester A. Simões, Lisboa V.P. Vizgin, Moskva Moreinformationaboutthisseriesathttp://www.springer.com/series/4883 Michael Friedman A History of Folding in Mathematics Mathematizing the Margins MichaelFriedman Humboldt-UniversitätzuBerlin Berlin,Germany ThispublicationwasmadepossiblebytheImageKnowledgeGestaltung.AnInterdisciplinary LaboratoryClusterofExcellenceattheHumboldt-UniversitätzuBerlinwithfinancialsupport fromtheGermanResearchFoundationasapartoftheExcellenceInitiative. ISSN1421-6329 ISSN2296-6080 (electronic) ScienceNetworks.HistoricalStudies ISBN978-3-319-72486-7 ISBN978-3-319-72487-4 (eBook) https://doi.org/10.1007/978-3-319-72487-4 LibraryofCongressControlNumber:2018932403 MathematicsSubjectClassification(2010):01A05;01A40;01A45;01A55;01A60;03F99 ©SpringerInternationalPublishingAG,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this bookarebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Coverillustration:FromWallerMsde-00215,AugustBeer:ÜberdieCorrectiondesCosinusgesetzesbei derAnwendungdesNicol’schenPrismasinderPhotometrie,after1850.WithfriendlypermissionbyThe WallerManuscriptCollection(partoftheUppsalaUniversityLibraryCollections). Printedonacid-freepaper ThisbookispublishedunderthetradenameBirkhäuser,www.birkhauser-science.combytheregistered companySpringerInternationalPublishingAGpartofSpringerNature. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Acknowledgments I wish to thank and acknowledge the many scholars who have helped me in the writing of this book. In particular, I want to express my gratitude to Vincenzo de Risi,whoconstantlyencouragedmetocontinueworkingontheproject;withouthis support, the book would not have been published. I also want to emphasise how gratefulIamtoWolfgangSchäffner:theprojectwouldnothaveunfoldedintheway thatitdidwithouthisconstantencouragementandsupport. During myresearch into thework ofFriedrich Fröbel and theway inwhichhis mathematicalconceptionoffoldingwasfurtherdeveloped,TimAllender,Kornelia Cepok,AvrilPowell,JaneReadandMargittaRocksteinhelpedmegreatly.Ithank alsoTiemenCocquyt,GerdinevandenDool,UrsulaKlein,PeterRambergandAlan Rocke,whoprovidedhelpandimportantinsightsconcerningJacobusHenricusvan ’tHoff’sfoldedmodelsandtheroleoffoldinginchemicalmodels.JaapTop,Irene Polo Blanco, Rolf ter Sluis and Gerda Huisman helped me during the research on Alicia Boole Stott, while Cyrille Foasso assisted me greatly during my visit to the ConservatoirenationaldesartsetmétiersinParisandafterwards.Theresearchon Margherita Beloch Piazzolla and Humiaki Huzita would not have been possible withoutthehelpofClaudiaSerra,MariaTeresaBorgato,EmmaFrigerio,Benedetto Scimemi, Luisa Canovi, Antonio Coiana, Francesco Decio and Consolato Pellegrino; for the valuable information and insights they shared, I am also extremelygrateful.SpecialthanksmustalsogotoHorstBredekampforhisgenerous conversations on Leibniz and his conception of the fold, conversations which, I hope,willbearadditionalfruitsinthenearfuture.IwouldlikealsotothankAngela Axworthy,RenzoBaldasso,MichaelBarany,ClaudiaBlümle,NathanielBoyd,June Barrow-Green, Maurizio Cornalba, Elizabeth Evenden-Kenyon, Ulf Hashagen, Jacques Justin, Suzanne Karr Schmidt, Friedhelm Kürpig, Erik Kwakkel, José Muñoz Alvis, Nicola Oswald, Colin Rittberg, Lisa Rougetet, Anja Sattelmacher, DainaTaiminaandMatthiasWiebelformorethansimplyhelpfulconversations,but alsothoughtfulinsightsandadvice.Aspecialthankstomymother,AlisaPadovano- Friedman,fortheinvaluablehelpwiththeItaliantexts. v vi Acknowledgments The following universities and museums, which allowed me to visit and to investigate their archives, must also be mentioned: Friedrich-Fröbel-Museum, Bad Blankenburg; Froebel Archive University of Roehampton, London; Conservatoire nationaldesartsetmétiers,Paris;MuseumBoerhaave,Leiden. Lastbutcertainlynotleast,Iwouldliketothankmydearcolleaguesandfriends inIsraelandinGermany.Withouttheirsupport,myownconvolutedreflectionsand ideaswouldnothavematerializedintotheclearandtransparent formofthisbook: Renen Amir, Karin Krauthausen, Thomas Nitschke, Angelika Seppi and Samo Tomšič. Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 SettingtheScene:WhichInstrumentIsStronger?. . . . . . . . . . . . . 1 1.2 MarginalizationandItsEpistemologicalConsequences. . . . . . . . . 5 1.3 MarginalizationandtheMedium:Or—WhyDidMarginalization Occur?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 TheEconomyofExcessandLack. . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 HistoriographicalPerspectivesandanOverview. . . . . . . . . . . . . . 19 1.5.1 MarginalizedTraditions. . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5.2 TheHistoricalResearchtoDateandOverview. . . . . . . . . 22 1.5.3 ArgumentandStructure. . . . . . . . . . . . . . . . . . . . . . . . . . 24 2 FromtheSixteenthCenturyOnwards:FoldingPolyhedra—New EpistemologicalHorizons?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1 Dürer’sNets. . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . . .. . . . .. . 30 2.1.1 UnderweysungderMessungandtheUnfoldedNets. . . . . . 32 2.1.2 FoldedTilesandFoldsofDrapery. . . . . . . . . . . . . . . . . . 39 2.1.3 Dürer’sFolding:AnEpistemologicalOffer?. . . . . . . . . . . 44 2.2 Dürer’sUnfoldedPolyhedra:ContextandRamifications. . . . . . . . 48 2.2.1 PacioliandBovelles,PaperInstrumentsandFoldedBooks: EncountersofFoldingandGeometry. . . . . . . . . . . . . . . . 49 2.2.1.1 PaperInstruments:FoldingforScience. . . . . . . . 53 2.2.1.2 AHistoricalDetour:BatBooksandImpositionof theBook—TheStandardizationofFolding. . . . . 59 2.2.2 Dürer’sFollowersFoldaNet. . . . . . . . . . . . . . . . . . . . . . 66 2.2.2.1 Stevin’sandCowley’sImpossibleNets. . . . . . . . 76 2.2.2.2 NetsofPolyhedra:AMathematicalStagnation?. . 80 2.3 IgnoringFoldingasaMethodofProofinMathematics. . . . . . . . . 83 vii viii Contents 2.3.1 FoldingandGeometry:AForgottenBeginning—Pacioli FoldsaGnomon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.3.2 FoldingandGeometry:AProblematicBeginning. . . . . . . 86 3 PrologtotheNineteenthCentury:AcceptingFoldingasaMethod ofInference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.1 FoldingandtheParallelPostulate. . . . . . . . . . . . . . . . . . . . . . . . 94 3.1.1 FoldingandParallelLine:AnImplicitEncounterDuring theArabicMiddleAges. . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.1.2 FoldingandParallelLine:AnExplicitEncounterDuring theEighteenthCentury. . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.2 FoldinginProofs:SuzanneandFrancœur. . . . . . . . . . . . . . . . . . 98 3.2.1 SymmetryandFoldingDiderotandSymmetryin Francœur’sCoursComplet. . . . . . . . . . . . . . . . . . . . . . . . 100 3.3 Lardner,Wright,Henrici:SymmetrywithFoldingin GreatBritain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4 TheNineteenthCentury:WhatCanandCannotBe (Re)presented—OnModelsandKindergartens. . . . . . . . . . . . . . . . . 113 4.1 OnModelsinGeneralandFoldedModelsinParticular. . . . . . . . . 114 4.1.1 MathematicalModelsDuringtheEighteenthandNineteenth Centuries. . . .. . . .. . . . .. . . .. . . . .. . . . .. . . .. . . . .. 115 4.1.2 FoldedModelsinMathematics:Dupin,Schlegel,Beltrami, SchwarzandtheTwoWieners. . . . . . . . . . . . . . . . . . . . . 126 4.1.2.1 LouisDupinandVictorSchlegel:HowtoFold NetsintheNineteenthCentury. . . . . . . . . . . . . . 126 4.1.2.2 EugenioBeltramiandModelsinItaly. . . . . . . . . 141 4.1.2.3 Schwarz,PeanoandChristianWiener. . . . . . . . . 152 4.1.2.4 HermannWiener. . . . . . . . . . . . . . . . . . . . . . . . 165 4.1.3 ADetourintotheRealmofChemistry:TheFoldedModels ofVan’tHoffandSachse. . . . . . . . . . . . . . . . . . . . . . . . 180 4.1.3.1 Van’tHoffFoldsaLetter. . . . . . . . . . . . . . . . . 181 4.1.3.2 HermannSachse’sThreeEquations. . . . . . . . . . . 194 4.1.3.3 FoldedModelsinChemistryandMathematics: AFailedEncounter. . . . . . . . . . . . . . . . . . . . . . 200 4.1.4 ModelingwiththeFold:AMinorityInsideaVanished Tradition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 4.2 FoldinginKindergarten:HowChildren’sPlayEnteredthe MathematicalScene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 4.2.1 Fröbel’sFolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 4.2.1.1 FröbelandMathematics. . . . . . . . . . . . . . . . . . . 209 4.2.1.2 FröbelFolds. . . . . . . . . . . . . . . . . . . . . . . . . . . 216 4.2.1.3 Fröbel’sInfluenceandtheVanishingofFolding- BasedMathematicsfromKindergarten. . . . . . . . 227 4.2.2 FromGreatBritaintoIndia. . . . . . . . . . . . . . . . . . . . . . . 247 Contents ix 4.2.2.1 FirstLessonsinGeometry:Bhimanakunte HanumanthaRao’sBook. . . . . . . . . . . . . . . . . . 250 4.2.2.2 TheBooksofTandalamSundaraRow. . . . . . . . . 254 4.2.3 FoldinginKindergartens:ASuccessfulMarginalization. . . 268 5 TheTwentiethCentury:TowardstheAxiomatization, OperationalizationandAlgebraizationoftheFold. . . . . . . . . . . . . . 271 5.1 TheInfluenceofRow’sBook. . . . . . . . . . . . . . . . . . . . . . . . . . . 272 5.1.1 FirstStepsTowardsOperativeAxiomatization:Ahrens, Hurwitz,Rupp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 5.1.1.1 Anhrens’sFundamentalFoldingConstructions. . . 274 5.1.1.2 TheBasicOperationsofAdolfHurwitz. . . . . . . . 278 5.1.1.3 LotkaandRupp:CreasesasEnvelopes. . . . . . . . 282 5.1.2 TheDistinctionBetweenAxiomsandOperations:ABook byYoungandYoung. . . . . .. . . . . . . . . . . . . . . . . . . .. . 285 5.1.2.1 TheYoungs’sTheFirstBookofGeometry. . . . . 286 5.1.2.2 TranslationsandAcceptance. . . . . . . . . . . . . . . . 293 5.1.3 ADetour:HowDoesOneFoldaPentagon?. . . . . . . . . . . 295 5.1.3.1 TheConstructionofEuclid. . . . . . . . . . . . . . . . . 296 5.1.3.2 HowDoesOneFoldaRegularPentagon?. . . . . . 297 5.1.3.3 HowDoesOneKnotaRegularPentagon?. . . . . . 305 5.2 AnAlgebraicEntwinementofTheoryandPraxis:Beloch’s Fold. . . . . . .. . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . . . . .. . 318 5.2.1 Vacca’s1930Manuscript. . . . . . . . . . . . . . . . . . . . . . . . . 319 5.2.2 Beloch’s1934Discoveries. . . . . . . . . . . . . . . . . . . . . . . . 323 5.2.3 After1934:FurtherDevelopmentandReception. . . . . . . . 327 5.2.3.1 Lill’sMethodofSolvingAnyEquation. . . . . . . . 330 5.2.3.2 AFallTowardsOblivion?. . . . . . . . . . . . . . . . . 336 5.3 EpilogfortheTwentiethCentury:TheFoldingofAlgebraic Symbols. . . .. . . . . .. . . . . .. . . . . .. . . . . . .. . . . . .. . . . . .. . 340 5.3.1 TheFaltungofBilinearForms. . . . . . . . . . . . . . . . . . . . . 341 5.3.2 ConvolutionasFaltung. . . . . . . . . . . . . . . . . . . . . . . . . . 350 6 Coda:1989—TheAxiomatization(s)oftheFold. . . . . . . . . . . . . . . . 355 6.1 TheOperationsofHumiakiHuzita. . . .. . . . . .. . . . . . .. . . . . .. 358 6.2 TheOperationsofJacquesJustin. . . . . . . . . . . . . . . . . . . . . . . . . 363 6.3 Conclusion:Too-Much,Too-Little—UnfoldinganEpistemological Non-equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 AppendixA:MargheritaBelochPiazzolla:“Alcuneapplicazioni delmetododelripiegamentodellacartadiSundaraRow”. . . . . . . . . . . 377 AppendixB:Deleuze,LeibnizandtheUnmathematicalFold. . . . . . . . . 381 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415