3.POLYHEDRA,GRAPHSANDSURFACES3.2.PlatonicSolidsandBeyond - PDF document

3.POLYHEDRA,GRAPHSANDSURFACES3.2.PlatonicSolidsandBeyond Sincethreefacesmeetateveryvertex,weknowthateveryvertexinthepolyhedronmusthavedegreethree.Thehandshakinglemmaassertsthatthesumofthedegreesisequaltotwicethenumberofedges,sowehavetheequation3V=2Eorequivalently,V=2 3E.ThismeansthatifweknowthevalueofE,thenwealsoknowthevalueofV. Sinceeveryfaceisapentagon,weknowthateveryvertexinthedualmusthavedegreeve.Thehandshakinglemmaonthedualassertsthatthesumofthenumbersofedgesaroundeachfaceisequaltotwicethenumberofedges,sowehavetheequation5F=2Eorequivalently,F=2 5E.ThismeansthatifweknowthevalueofE,thenwealsoknowthevalueofF. NowwecanuseEuler'sformulaV�E+F=2andsubstituteforVandF.Ifyoudothisproperly,youshouldobtain2 3E�E+2 5E=2)E=30.Fromthis,wecaneasilydeducethatV=2 3E=20andF=2 5E=12.Youcanandshouldusethismethodtodeterminethenumberofvertices,edgesandfacesforeachofthePlatonicsolidsand,ifyoudoso,you'llendupwithatablelikethefollowing. polyhedronndVEF tetrahedron33464cube438126octahedron346128dodecahedron53203012icosahedron35123020 ThefollowingdiagramshowsthevePlatonicsolids—theyarecalledthetetrahedron,thehexahedron,theoctahedron,thedodecahedronandtheicosahedron.1Therstthreeareeasiertoimagine,becausetheyaresimplythetriangularpyramid,thecube,andtheshapeobtainedfromgluingtwosquarepyramidstogether. 1ThesenamescomefromtheancientGreekandsimplymeanfourfaces,sixfaces,eightfaces,twelvefacesandtwentyfaces,respectively.Ofcourse,wealmostalwaysrefertothehexahedronmoreaffectionatelyasthecube.2 3.POLYHEDRA,GRAPHSANDSURFACES3.2.PlatonicSolidsandBeyond arenotactuallyfaces,sincetheyhaveholesinthemiddleofthem.Allyouneedtodoisdividetheseregionsintobonadefaceswiththehelpofsomeextraedges.SinceEuler'sformuladoesn'tworkforthisshape,therearetwothingswecando—starttocryortrytomakeitwork.Mathematicianswouldgenerallypreferthelatterapproach.TomakeEuler'sformulaworkformoregeneralshapes,yousimplyneedtonotethatV�E+F=0inthisparticularcaseand,infact,V�E+F=0foranyshapewhichhasoneholethroughthemiddleofit.Infact,ifyoutrythesamethingforshapeswithgholes,you'lleventuallydiscoverthatV�E+F=2�2g.SoV�E+Fseemstochangewhenyoutalkaboutverydifferentgeometricobjects—forexample,objectswithdifferentnumbersofholes—butseemstobethesamewhenyoutalkaboutsimilargeometricobjects—forexample,objectswiththesamenumberofholes.Anotherobservationisthatifwetakethegeometricobjectpicturedaboveanddrawitsothatitlooksabitcurvier,thenthatdoesn'tchangethenumberofvertices,edgesandfaces,soV�E+Fdoesn'tchangeatall.Infact,wecanbend,stretch,warp,morphordeformitandthevalueofV�E+Fwouldn'tchange.TheEarthSupposethatyoulivedareallyreallylongtimeago.Thenyouwouldprobablybelieve,asdidmostpeo-ple,thatthesurfaceoftheEarthisabigatplane.Andwhatmakesyouthinkthat?Well,it'ssimplyduetothefactthateverywhereyoustand,youno-ticethatthere'saprettyatpieceofearthimmedi-atelysurroundingyourfeet.Andsinceabigatplaneseemstohavethissameproperty,you'vesim-plyjumpedtotheconclusionthattheEarthmustbebigatplane.But,asyouknow,thisisratherfool-ishthinking.Therearemanygeometricobjectsapartfromtheplanewhichhavethisproperty.Thesphereisjustonemoreexample,andwe'regoingtoexploreothershapesthatcanarise.Thisidealeadsustostudythingscalledsurfaceswhichwe'llsoontalkabout. TopologyThetwostoriesabove—oneaboutEuler'sformulaandoneabouttheEarth—motivateustoconsidertopology,whichveryroughlystudiesintrinsicfundamentalpropertiesofgeometricobjectsandwhichdoesn'tcareaboutlength,size,angle,andsoforth.Whenyoustudyobjectsinmathematics,youalwaysneedtohavesomenotionofwhentwoofthoseobjectsarethesame.Forexample,inEuclideangeometrywehavecongruence,ingrouptheorywehaveisomorphism,ingraphtheorywehaveisomorphism,andintopologywehavethenotionofhomeomorphism.Intuitivelyspeaking,twogeometricobjectsarehomeomorphicifit'spossibletobend,stretch,warp,morphordeformonesothatitbecomestheotherone.Notethatyouarenotallowedtocutandgluehere.5 3.POLYHEDRA,GRAPHSANDSURFACES3.2.PlatonicSolidsandBeyond Amoremathematicallyprecisedenitionofhomeomorphismisasfollows.Twogeometricobjectsarehomeomorphicifthereexistsabijection—inotherwords,aone-to-onecorrespondence—fromonetotheotherwhichiscontinuousandhasacontinuousinverse.IftwogeometricobjectsAandBarehomeomorphic,thenwewriteA=B,andwecallabijectionfromAtoBwhichiscontinuousandhasacontinuousinverseahomeomorphism.2Example.Thesphereandthecubearecertainlyhomeomorphictoeachother.Thisisbecauseyoucantakethesphereandsquashitintoaboxuntilitlookslikeacube.Or,ontheotherhand,youcouldtakeacubeandblowituplikeaballoonuntilitlookslikeasphere.Thesesortsofdeformationsarecertainlyallowedandshowthatthetwoshapesarehomeomorphic.Forthesamesortofreason,adiskandasquarearealsohomeomorphictoeachother.Youcanprobablyseewhytopologyissometimesinformallycalledrubbersheetgeometry.Anexplicithomeomorphisminthiscaseisn'ttoodifculttodescribeand,ifyouwerereallykeen,youcouldevenwritedownanequationforone.Theideaistostickthesphereinsideofthecubeandsupposethatthesphereisalightbulbwithasourceoflightatitscentre.Anypointonthespherenowcastsashadowonthecube,andthisgivesamapfromthespheretothecubewhichisacontinuousbijectionwithacontinuousinverse.Example.Consideranunknottedpieceofstringliketheoneshownbelowleftandaknottedpieceofastringliketheoneshownbelowright. Believeitornot,thesetwoarehomeomorphictoeachother.Sure,youneedtocuttheknotopenandgluetheendstogetheragaintomaketheunknottedloop—butthat'sonlyifyouhappentoliveinthreedimensions.Topology,unlikewemeremortals,doesn'tliveinanynumberofdimensions.Intuitively,youcandeformtheunknottedloopintoaknotifyoumakethosedeformationsinfour-dimensionalspace.Thereasonbeingthatlinescanjustmovepasteachotherinfour-dimensionalspacewithouthitting.It'sjustthehigherdimensionalanalogueofthefactthattwopointsonthelinecan'tmovepasteachotherwithouthittingyetintwodimensions,theycandosowithease.Ifthisdoesn'tconvinceyou,thenyoucanalwaysgobacktothemathematicaldenitionofhomeomorphism.Supposethatyouwrotethenumbersfrom1to100alongtheunknottedloop,inorder.Youcouldalsodothesamethingaroundtheknottedloopandconvinceyourselfthatthere'safunctionfromonetotheotherwhichmatchesupthenumbers.It'seasytoseethatsuchafunctioncanbemadetobeabijectionwhichiscontinuousandhasacontinuousinverse—hence,thetwoarehomeomorphic. 2ThewordhomeomorphismcomesfromtheancientGreekwordsmeaningsimilarshape.6 3.POLYHEDRA,GRAPHSANDSURFACES3.2.PlatonicSolidsandBeyond Problem.IfaplanargraphwithEedgesdividestheplaneintoFfaces,provethatF2E 3.Proof.Forthisproblem,weneedtoassume—asisusual—thatthegraphcontainsnoloopsormultipleedges.NotethatthedualgraphhasFverticesand,sinceeveryfaceoftheoriginalgraphhasatleastthreesides,everyvertexofthedualgraphhasdegreeatleastthree.Sothesumofthedegreesoftheverticesinthedualgraphisatleast3F.However,thenumberofedgesinthedualgraphisequaltoE.Sowecannowinvokethehandshakinglemmatodeducethat2E3F,whichrearrangestogivethedesiredresult. Problem.Considerapolyhedronallofwhosefacesaretrianglessuchthatfourfacesmeetateveryvertex.Determinethenumberofvertices,edgesandfacesofthepolyhedron.Proof.IfyouwanttoworkoutthethreeunknownquantitiesV,EandF,thenitmakessensetolookforthreerelationsthatthesenumbersobeys.Onerelationwillcomefromusingthehandshakinglemma,anotherwillcomefromusingthehandshakinglemmaonthedual,andanotherisgiventousbyEuler'sformula. Sincefourfacesmeetateveryvertex,weknowthateveryvertexinthepolyhedronmusthavedegreefour.Thehandshakinglemmaassertsthatthesumofthedegreesisequaltotwicethenumberofedges,sowehavetheequation4V=2Eorequivalently,V=1 2E.ThismeansthatifweknowthevalueofE,thenwealsoknowthevalueofV. Sinceeveryfaceisatriangle,weknowthateveryvertexinthedualmusthavedegreethree.Thehandshakinglemmaonthedualassertsthatthesumofthenumbersofedgesaroundeachfaceisequaltotwicethenumberofedges,sowehavetheequation3F=2Eorequivalently,F=2 3E.ThismeansthatifweknowthevalueofE,thenwealsoknowthevalueofF. NowwecanuseEuler'sformulaV�E+F=2andsubstituteforVandF.Ifyoudothisproperly,youshouldobtain1 2E�E+2 3E=2)E=12.Fromthis,wecaneasilydeducethatV=1 2E=6andF=2 3E=8.Infact,anexampleofsuchapolyhedronisgivenbythePlatonicsolidknownastheoctahedron. 8