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首页 [Robert.J.Lang.-.折纸教程大全].Robert.J.Lang.-.Origami…

[Robert.J.Lang.-.折纸教程大全].Robert.J.Lang.-.Origami.Constructions.pdf

[Robert.J.Lang.-.折纸教程大全].Robert…

remilynn
2009-09-09 0人阅读 举报 0 0 暂无简介

简介:本文档为《[Robert.J.Lang.-.折纸教程大全].Robert.J.Lang.-.Origami.Constructionspdf》,可适用于高等教育领域

Lang,OrigamiandGeometricConstructionsOrigamiandGeometricConstructionsByRobertJLangCopyright©–AllrightsreservedIntroductionPreliminariesandDefinitionsBinaryDivisionsBinaryFoldingAlgorithmBinaryApproximationsRationalFractionsCrossingDiagonalsFujimoto’sConstructionNoma’sMethodHaga’sConstructionIrrationalProportionsContinuedFractionsQuadraticSurdsAngleDivisionsAxiomaticOrigamiPreliminariesFoldingAlignmentsBringingapointtoapointP↔PBringingapointontoaline(P↔L)Bringingonelinetoanotherline(L↔L)AlignmentsbyfoldingMultipleAlignmentsConstructabilityAxiomandCubicCurvesApproximationbyComputerReferencesLang,OrigamiandGeometricConstructionsIntroductionCompassandstraightedgegeometricconstructionsarefamiliartomoststudentsfromhighschoolgeometryNowadays,theyareviewedbymostasaquaintcuriosityofnomorethanacademicinterestTotheancientGreeksandEgyptians,however,geometricconstructionswereusefultools,andforsome,everydaytools,usedforconstructionandsurveying,amongotheractivitiesTheclassicalrulesofcompassandstraightedgeallowasinglecompasstostrikearcsandtransferdistances,andasingleunmarkedstraightedgetodrawstraightlinesthetwomaynotbeusedincombination(forexample,holdingthecompassagainstthestraightedgetoeffectivelymarkthelatter)However,therearemanyvariationsonthegeneralthemeofgeometricconstructionsthatincludeuseofmarkedrulesandtoolsotherthancompassesfortheconstructionofgeometricfiguresOneofthemoreinterestingvariationsistheuseofafoldedsheetofpaperforgeometricconstructionLikecompassandstraightedgeconstructions,foldedpaperconstructionsarebothacademicallyinterestingandpracticallyusefulparticularlywithinorigami,theartoffoldinguncutsheetsofpaperintointerestingandbeautifulshapesModernorigamidesignhasshownthatitispossibletofoldshapesofunbelievablecomplexity,realism,andbeautyfromasingleuncutsquareOrigamifigurespossesanaestheticbeautythatappealstoboththemathematicianandthelaymanPartoftheirappealisthesimplicityoftheconcept:fromthesimplestofbeginningsspringsanobjectofdepth,subtlety,andcomplexitythatoftencanbeconstructedbyapreciselydefinedsequenceoffoldingstepsHowever,manyorigamidesignsevenquitesimpleonesrequirethatonecreatetheinitialfoldsatparticularlocationsonthesquare:dividingitintothirdsortwelfths,forexampleWhileonecouldalwaysmeasureandmarkthesepoints,thereisanaestheticappealtocreatingthesekeypoints,knownasreferencepoints,purelybyfoldingThus,withinorigami,thereisapracticalinterestindevisingfoldingsequencesforparticularproportionsthatoverlapswiththemathematicalfieldofgeometricconstructionsWithinthisarticle,IwillpresentavarietyoftechniquesfororigamigeometricconstructionsThefieldisrichandvaried,withsurprisingconnectionstootherbranchesofmathematicsIwillshoworigamiconstructionsbasedonbinarydivisions,andthenshowhowthesecanbeextendedconstructionofproportionsthatarearbitraryrationalfractionsCertainirrationalproportionsarealsoconstructiblewithorigamiIwillpresentseveralparticularlyinterestingexamplesI’llthenturntothetopicofapproximatefoldingsequences,which,thoughperhapsnotasmathematicallyinteresting,areofconsiderablepracticalutilityAlongtheway,I’llpresenttheaxiomatictheoryoforigamiconstructions,whichnotonlystipulateswhatclassesofproportionsarefoldable,butalsoprovidesthebasisforfindingextremelyefficientapproximatefoldingsequencesbycomputersolutionatechniquethathasfoundapplicationinanumberofpublishedorigamibooksofdesignsPreliminariesandDefinitionsOrigami,likegeometricconstructions,hasmanyvariationsInthemostcommonversion,onestartswithanunmarkedsquaresheetofpaperOnlyfoldingisallowed:nocuttingThegoaloforigamiconstructionistopreciselylocateoneormorepointsonthepaper,oftenaroundtheedgesofthesheet,butalsopossiblyintheinteriorThesepoints,knownasreferencepoints,arethenusedtodefinetheremainingfoldsthatshapethefinalobjectTheprocessoffoldingthemodelcreatesnewreferencepointsalongtheway,whicharegeneratedasintersectionsofcreasesorpointswhereacreasehitsafoldededgeInanidealorigamifoldingsequenceastepLang,OrigamiandGeometricConstructionsbystepseriesoforigamiinstructionseachfoldactionispreciselydefinedbyaligningcombinationsoffeaturesofthepaper,wherethosefeaturesmightbepoints,edges,creaselines,orintersectionsofsameTwoexamplesofcreatingsuchalignmentsareshowninFiguresandFigureillustratesfoldingasheetofpaperinhalfalongitsdiagonalThefoldisdefinedbybringingonecornertotheoppositecornerandflatteningthepaperWhenthepaperisflattened,acreaseisformedthat(ifthepaperwastrulysquare)connectstheothertwocornersFoldthebottomrightcorneruptothetopleftUnfold“Foldandunfold”isindicatedbyadoubleheadedarrowFigureThesequenceforfoldingasquareinhalfdiagonallyAsashorthandnotation,thetwostepsoffoldingandunfoldingarecommonlyindicatedbyasingledoubleheadedarrowasinthethirdstepofFigureFigureillustratesanotherwayoffoldingthepaperinhalf(“bookwise”)Thisfoldcanbedefinedindistinct,butequivalentways:()Foldthebottomleftcorneruptothetopleftcorner()Foldthebottomrightcorneruptothetoprightcorner()FoldthebottomedgeuptobealignedwiththetopedgeForasquare,thesethreemethodsareequivalentHowever,ifyoustartwithslightlyskewpaper(aparallelogramratherthanasquare),youwillgetslightlydifferentresultsfromthethreeLang,OrigamiandGeometricConstructionsFoldthebottomedgeuptothetopedgeUnfoldThenewcreasedefinestwonewpointsFigureThesequenceforfoldingasquareinhalfbookwiseInbothcases,ifyouunfoldthepaperbacktotheoriginalsquare,youwillfindyouhavecreatedanewcreaseonthepaperForthesequenceoffigure,youwillalsohavenowdefinedtwonewpoints:themidpointsofthetwosidesEachpointispreciselydefinedbytheintersectionofthecreasewitharawedgeofthepaperThesetwosequencesalsoillustratetheruleswewilladoptfororigamigeometricconstructionsThegoaloforigamigeometricconstructionsistodefineoneormorepointsorlineswithinasquarethathaveageometricspecification(eg,linesthatbisectortrisectangles)orthathaveaquantitativedefinition(eg,apointofthewayalonganedge)Weassumethefollowingrules:()Alllinesaredefinedbyeithertheedgeofthesquareoracreaseonthepaper()Allpointsaredefinedbytheintersectionoftwolines()Allfoldsmustbeuniquelydefinedbyaligningcombinationsofpointsandlines()Acreaseisformedbymakingasinglefold,flatteningtheresult,and(optionally)unfoldingRule(),inparticular,isfairlyrestrictiveitsaysthatfoldsmustbemadeoneatatimeBycontrast,allbutthesimplestorigamifiguresincludestepsinwhichmultiplefoldsoccursimultaneouslyLaterinthisarticle,IwilldiscusswhathappenswhenwerelaxthisconstraintBinaryDivisionsOneofthemostcommonorigamiconstructionsthatturnsupinpracticalfoldingistheproblemofdividingoneorbothsidesofthesquareintoNequaldivisions,whereNissomeintegerFigureillustratedthesimplestcasedividingtheedgeofasquareintotwopartsanditssolutionOfcourse,thismethodisnotrestrictedtoasquareitworksequallywellonanylinesegmentinasquareThus,thetwohalvesofthesquaremaybeindividuallydividedintotwoparts,andsoonByrepeatedlydividingthesegmentsinhalf,itispossibletodividetheedgeofasquare(orrectangle)intoths,ths,andsoforth,asshowninFigureLang,OrigamiandGeometricConstructionsDivisionintothsDivisionintothsDivisionintothsFigureDivisionofasquareintoths,ths,andthsThismethodallowsustodivideasquareintoproportionsof,,,…andingeneral,nforintegernEachdivisionisnofthesideofthesquareByscalingallnumberstothesizeofthesquare,wecansaywehaveconstructedthefractionn,wherethefractionisgivenintermsofthesideofthesquareItisalsopossibletoconstructafractionoftheformmnforanyintegerm<n(Inallthediscussionthatfollows,wewillconsideronlyfractionsbetweenand)Themostdirectmethodistosubdividetheedgeofthesquarecompletelyintonths,thencountupmdivisionsfromthebottomThismethodclearlyrequiresn−creases,andisnotveryefficient,becausecompletelysubdividingthesquareresultsinthecreationofmanyunnecessarycreasesThereisanelegantmethodforconstructinganyfractionofthistypethatusestheminimalnumberoffoldsArationalfractionwhosedenominatorisaperfectpoweroftwoiscalledabinaryfractionthefoldingmethodiscalledthebinaryfoldingalgorithmBinaryFoldingAlgorithmThebinaryfoldingalgorithmwasdescribedbyBruntonandexpandeduponbyLangItproducesanefficientfoldingsequencetoconstructanyproportionthatisabinaryfractionandisbasedonbinarynotationInbinarynotation,thereareonlytwodigits,andallnumbersarewrittenasstringsofonesandzerosAnynumbercanbewritteninbinarynotationasastringofonesandzerosForexample,thenumbersthroughcanbewritteninbinaryasshowninTableLang,OrigamiandGeometricConstructionsDecimalBinaryTableBinaryequivalentsfordecimalnumbers–Anybinaryfractionoftheformmncanbefoldedinexactlyncreases,andtherequiredfoldingsequenceisencodedinthebinaryexpressionofthefractionBinarynotationforfractionsisbestunderstoodinanalogywithordinarydecimalnotationIndecimalnotation,eachdigittotheleftofthedecimalpointisunderstoodtomultiplyapowerofforexample,=××××=()Thesamethinghappensinbinarynotation,exceptyouusepowersofratherthanpowersofandthereareonlytwopossibledigits:andTherefore,thebinarynumberis=××××==eleven()Bythismeans,anyintegermaybewritteninbinarynotationwithauniquecombinationofonesandzerosWhileitislesscommonlydone,itisalsopossibletowritefractionalquantitiesinabinarynotationthatisanalogoustoourdecimalnotation,inwhichfractionalquantitiesappearasdigitstotherightofthedecimalpoint(althoughperhapsitshouldbecalleda“binarypoint”ratherthana“decimalpoint”)Forexample,justasthedecimalmeans=×−×−×−=,()thebinaryfractionmaybeinterpretedas=×−×−×−=()Otherexamples:thefractionisgivenbyinbinarythefractionisinbinary,whileisThefractionis,and,writteninbinary,isAnyfractionwhoseLang,OrigamiandGeometricConstructionsdenominatorisaperfectpoweroftwohasabinaryrepresentationwithafinitenumberofdigitstotherightofthedecimalpointYoucanconstructthebinaryfractionforanynumberbyfollowingthisalgorithm:()Writedownadecimalpoint()Multiplythefractionby()Subtractofftheintegerpart(eitheror)andwriteitdowntotherightofthelastthingyouwrote()Repeatsteps()and()asmanytimesasnecessary,eachtimeaddingdigitstotheright,untilyougetaremainderofEquivalently,thefractionmniswrittenasadecimalpointplusthebinaryexpansionoftheintegerm,paddedwithenoughzerostotheimmediaterightofthedecimaltogetatotalofndigitsWhataboutfractionswhosedenominatorisnotaperfectpowerof(whichincludesmostnumbers)Ifyouwriteanumbersuchasinbinaryusingthealgorithmdescribedabove,youwillnevergetaremainderofzeroInstead,itformsaninfinitestringofdigitsforexample,=…IfthenumberisarationalnumbertheratiooftwointegersthenthefractionwilleventuallystarttorepeatitselfThebinaryexpressionforafractiongivesaprecisedescriptionofthefoldingsequenceneededtomakeamarkatagivendistanceupthesideofthepaperFirst,here’sthefoldingalgorithm:Tomarkoffadistanceequaltoabinaryfractionbyfolding,writedownitsbinaryformThen,beginningfromtherightsideofthefraction(theleastsignificantdigit):forthefirstdigit(whichisalwaysabecauseyoudropanytrailingzeros)foldthetopdowntothebottomandunfoldForeachremainingdigit,ifitisa,foldthetopofthepapertothepreviouscrease,pinch,andunfoldifitisa,foldthebottomofthepapertothepreviouscrease,pinch,andunfoldBycomparingthisalgorithmwiththeexpansionformulaforabinaryfraction,youcanseehowthefoldingalgorithmworksLet’stakethenumber()asanexampleTheconventionalwayofexpandingthisistoexpandthenumberinpowersof,asshowninequation()=×−×−×−×−×−()Anotherwayofwritingthisbinaryexpansionistoexpanditasanestedseries,asinequation()Lang,OrigamiandGeometricConstructions=×(×(×(×(×()))))()Toevaluatethisform,youstartattheinnermostnumberintheexpression(theterminal“”)andworkyourwaybacktotheleft,slowlyworkingyourwayoutofthenestedparenthesesIfwewritethefractionthisway,itbecomesaseriesofnestedoperationswhereeachoperationiseither:(a)Addandmultiplyby,or(b)AddandmultiplybyNowlet’slookattheorigamifoldingsequenceintherecipeaboveIfwehaveasquarewithacreasemarklocatedadistancerfromthebottomandfoldthebottomofthesquareupandunfold,thenewcreaseismadeadistance()rfromthebottomIfinstead,wefoldthetopofthesquaredowntothemarkandunfold,thenewcreaseismadeadistance()(r)fromthebottomThus,foldingthebottomuportopdownisequivalenttoperformingoperations(a)or(b),respectivelyr×(r)r×(r)rrFigure(Top)Foldingthebottomedgeuptoacreasergivesanewcrease(r)fromthebottom(Bottom)Foldingthetopedgedowntoacreasergivesanewcrease((r))fromthebottomSinceanybinaryfractioncanbewrittenasanestedsequenceofthetwooperations(a)and(b)andthetwofoldingstepsshowninfigureimplementthesetwooperations,itfollowsthatanyproportioncanbefoldedfromitsbinaryexpansionLang,OrigamiandGeometricConstructionsThedifferenceinefficiencybetweenfoldingalldivisionsandcountingupward,versusthebinarymethod,issubstantialForafractionmn,theformermethodrequiresn−foldsthelatter,onlynBinaryApproximationsOnlyfractionswhosedenominatorisaperfectpowerofpossessabinaryexpansionwithafinitenumberofdigitsFormostfractions,thebinaryexpansionofthefractionisinfiniteButifwetruncatethebinaryexpansionatsomepoint,wegetabinaryfractionthatprovidesacloseapproximationofthenumberThisworksinanynumberbaseForexample,indecimalnotation,=…(alsoaninfinitedecimal)Ifwetruncateatonedigit(),wegetthefraction,whichisonlyroughlyequaltoIfwetaketwodigits(),weget,whichisveryclosetoandifwetakedigits(),weget,whichisverycloseindeedThesamethinghappensinbinarynotationIfwetruncatethebinaryexpansionofatdigits,weget=arathercrudeapproximationofButis,whichiscloserto,andis,whichdiffersfrombylessthanThus,anynumbercanbeapproximatedbyabinaryfractiontoarbitraryaccuracy,whichleadstoaneasywaytofindanapproxi

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