whereAistheequilibriummatrixtisavectorofinternalbarforcestensionsandliesinavectorspaceofdimensionbfisanassignmentofexternallyappliedforcesonetoeachjointandasthereare3jpossibleforcecomponents ID: 365641
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1IntroductionThispaperdealswithisostaticframeworks,i.e.,pin-jointedbarassemblies,commonlyreferredtoinengineeringliteratureastrussstructures,thatarebothkinematicallyandstaticallydeterminate.Suchsystemsareminimallyinnitesimallyrigidandmaximallystressfree:theycanbetermed`justrigid'.Ourultimategoalistoanswerthequestionposedinthetitle:whenaresymmetricpin-jointedframeworksisostatic?Asarststep,thepresentpaperprovidesaseriesofnecessaryconditionsobeyedbyisostaticframeworksthatpossesssymmetry,andalsosummarizesconjecturesandinitialresultsonsucientconditions.Frameworksprovideamodelthatisusefulinapplicationsrangingfromcivilengineering(Graver,2001)andthestudyofgranularmaterials(Donevetal.,2004)tobiochemistry(Whiteley2005).Manyofthesemodelframeworkshavesymmetry.Inapplications,bothpracticalandtheoreticaladvantagesaccruewhentheframeworkisisostatic.Inanumberofapplications,pointsymmetryoftheframeworkappearsnaturally,anditisthereforeofinteresttounderstandtheimpactofsymmetryontherigidityoftheframework.Maxwell(1864)formulatedanecessaryconditionforinnitesimalrigidity,acountingrulefor3Dpin-jointedstructures,withanobviouscounterpartin2D;thesewerelaterrenedbyCalladine(1978).Laman(1970)providedsucientcriteriaforinnitesimalrigidityin2D,buttherearewellknownproblemsinextendingthisto3D(Graveretal.,1993).TheMaxwellcountingrule,anditsextensions,canbere-casttotakeaccountofsymmetry(FowlerandGuest,2000)usingthelanguageofpoint-grouprepresentations(see,e.g.,Bishop,1973).Thesymmetry-extendedMaxwellrulegivesadditionalinformationfromwhichithasoftenbeenpos-sibletodetectandexplain`hidden'mechanismsandstatesofself-stressincaseswherethestandardcountingrulesgiveinsucientinformation(FowlerandGuest,2002;2005,Schulze2008a).Similarsymmetryextensionshavebeenderivedforotherclassicalcountingrules(CeulemansandFowler,1991;GuestandFowler,2005).Inthepresentpaper,wewillshowthatthesymmetry-extendedMaxwellrulecanbeusedtoprovidenecessaryconditionsforaniteframeworkpos-sessingsymmetrytobestress-freeandinnitesimallyrigid,i.e.,isostatic.Itturnsoutthatsymmetricisostaticframeworksmustobeysomesimplystatedrestrictionsonthecountsofstructuralcomponentsthatarexedbyvarioussymmetries.For2Dsystems,theserestrictionsimplythatisostaticstruc-turesmusthavesymmetriesbelongingtooneofonlysixpointgroups.For3Dsystems,allpointgroupsarepossible,asconvextriangulatedpolyhedra(isostaticbythetheoremsofCauchyandDehn(Cauchy1813,Dehn1916))2 whereAistheequilibriummatrix;tisavectorofinternalbarforces(ten-sions),andliesinavectorspaceofdimensionb;fisanassignmentofex-ternallyappliedforces,onetoeachjoint,and,asthereare3jpossibleforcecomponents,fliesinavectorspaceofdimension3j(thisvectorspaceisthetensorproductofaj-dimensionalvectorspaceresultingfromassigningascalartoeachjoint,anda3-dimensionalvectorspaceinwhicha3Dforcevectorcanbedened).HenceAisa3jbmatrix.Astateofself-stressisasolutiontoAt=0,i.e.,avectorinthenullspaceofA;ifAhasrankr,thedimensionofthisnullspaceiss=b�r:(2)Further,thecompatibilityrelationshipcanbewrittenasCd=ewhereCisthecompatibilitymatrix;eisavectorofinnitesimalbarexten-sions,andliesinavectorspaceofdimensionb;disavectorofinnitesimalnodaldisplacements,thereare3jpossiblenodaldisplacementsandsodliesinavectorspaceofdimension3j.HenceCisab3jmatrix.Infact,itisstraightforwardtoshow(seee.g.,PellegrinoandCalladine,1986)thatCisidenticaltoAT.ThematrixCiscloselyrelatedtotherigiditymatrixcommonlyusedinthemathematicalliterature:therigiditymatrixisformedbymultiplyingeachrowofCbythelengthofthecorrespondingbar.OfparticularrelevancehereisthatfactthattherigiditymatrixandChaveanidenticalnullspace.AmechanismisasolutiontoATd=0,i.e.,avectorintheleft-nullspaceofA,andthedimensionofthisspaceis3j�r.However,thisspacehasabasiscomprisedofminternalmechanismsand6rigid-bodymechanisms,andhencem+6=3j�r:(3)Eliminatingrfrom(2)and(3)recoversMaxwell'sequation(1).Theabovederivationassumesthatthesystemis3-dimensional,butitcanbeappliedto2-dimensionalframeworks,simplyreplacing3j�6by2j�3:m�s=2j�b�3:(4)2.2Symmetry-extendedcountingruleThescalarformula(1)hasbeenshown(FowlerandGuest,2000)tobepartofamoregeneralsymmetryversionofMaxwell'srule.ForaframeworkwithpointgroupsymmetryG,4 andjointsfreely(sothatnobarorjointismappedontoitselfbyanysymme-tryoperation).BothbandjmustthenbemultiplesofjGj,theorderofthegroup:b=bjGj,j=jjGj.Cansuchaframeworkbeisostatic?AnyisostaticframeworkobeysthescalarMaxwellrulewithm�s=0asanecessarycondition.Inthreedimensions,wehaveb=3j�6,andhence:3D:b=3j�6 jGj:Intwodimensions,wehaveb=2j�3,andhence:2D:b=2j�3 jGj:Asbandjareintegers,jGjisrestrictedtovalues1,2,3and6in3D,and1and3in2D.Immediatelywehavethatifthepointgrouporderisnotoneofthesespecialvalues,itisimpossibletoconstructanisostaticframeworkwithallstructuralcomponentsplacedfreely:anyisostaticframeworkwithjGj6=1;3(2D)orjGj6=1;2;3;6(3D)musthavesomecomponentsinspecialpositions(componentsthatareunshiftedbysomesymmetryoperation).IntheSchoen iesnotation(Bishop,1973),thepointgroupsoforders1,2,3and6arejGj=1:C1jGj=2:C2,Cs,CijGj=3:C3jGj=6:C3h,C3v,D3,S6Furtherrestrictionsfollowfromthesymmetry-adaptedMaxwellrules(5),(6).Inahypotheticalframeworkwhereallbarsandjointsareplacedfreely,thebarandjointrepresentationsare�(b)=b�reg;�(j)=j�regwhere�registheregularrepresentationofGwithtracejGjundertheidentityoperation,and0underallotheroperations.Therepresentations�xyzand�xyhavetrace3and2,respectively,undertheidentityoperation,andhenceequations(5)and(6)become3D:�(m)��(s)=3j�reg�b�reg��xyz��RxRyRz;2D:�(m)��(s)=2j�reg�b�reg��xy��Rz;whichcanbewrittenas,6 Setting�(m)��(s)tozeroin(5)and(6),classbyclass,willgiveuptokindependentnecessaryconditionsfortheframeworktobeisostatic.Wewillcarryoutthisprocedureonceandforallforallpointgroups,asthereisaverylimitedsetofpossibleoperationstoconsider.Thetwo-dimensionalandthree-dimensionalcaseswillbeconsideredseparately.3.1Two-dimensionalisostaticframeworksInthissectionwetreatthetwo-dimensionalcase:bars,joints,andtheiras-sociateddisplacementsareallconnedtotheplane.(Notethatframeworksthatareisostaticintheplanemayhaveout-of-planemechanismswhencon-sideredin3-space.)Therelevantsymmetryoperationsare:theidentity(E),rotationby2=naboutapoint(Cn),andre ectioninaline().Thepos-siblegroupsarethegroupsCnandCnvforallnaturalnumbersn.CnisthecyclicgroupgeneratedbyCn,andCnvisgeneratedbyafCn;gpair.ThegroupC1visusuallycalledCs.Alltwo-dimensionalcasescanbetreatedinasinglecalculation,asshowninTable1.Eachentryinthetableisthetrace(character)oftheappropriaterepresentation(indicatedintheleftcolumn)ofthesymmetry(indicatedinthetopline).Charactersarecalculatedforfouroperations:wedistinguishC2fromtheCnoperationwithn�2.Eachlineinthetablerepresentsastageintheevaluationof(6).SimilartabularcalculationsarefoundinFowlerandGuest(2000)andsubsequentpapers.Totreatalltwo-dimensionalcasesinasinglecalculation,weneedanota-tionthatkeepstrackofthefateofstructuralcomponentsunderthevariousoperations,whichinturndependsonhowthejointsandbarsareplacedwithrespecttothesymmetryelements.ThenotationusedinTable1isasfollows.jisthetotalnumberofjoints;jcisthenumberofjointslyingonthepointofrotation(C�n2orC2)(notethat,asweareconsideringonlycaseswherealljointsaredistinct,jc=0or1);jisthenumberofjointslyingonagivenmirrorline;bisthetotalnumberofbars;b2isthenumberofbarsleftunshiftedbyaC2operation(seeFigure1(a)andnotethatCnwithn�2shiftsallbars);bisthenumberofbarsunshiftedbyagivenmirroroperation(seeFig-ure1(b):theunshiftedbarmayliein,orperpendicularto,themirrorline).8 (i)Trivially,all2Dframeworkshavetheidentityelementand(7)simplyrestatesthescalarMaxwellrule(4)withm�s=0.(ii)PresenceofaC2elementimposeslimitationsontheplacementofbarsandjoints.Asbothjcandb2mustbenon-negativeintegers,(8)hastheuniquesolutionb2=1,jc=0.Inotherwords,anisostatic2DframeworkwithaC2elementofsymmetryhasnojointonthepointofrotation,butexactlyonebarcentredatthatpoint.(iii)Similarly,presenceofamirrorlineimplies,by(9),thatb=1forthatline,butplacesnorestrictiononthenumberofjointsinthesameline,andhenceallowsthisbartolieeitherin,orperpendicularto,themirror.(iv)DeductionoftheconditionimposedbyarotationofhigherorderC�n2proceedsasfollows.Equation(10)with =2=nimplies(jc�1)cos2 n=1 2(11)andasjciseither0or1,thisimpliesthatjc=0andn=3.Thus,a2DisostaticframeworkcannothaveaCnrotationalelementwithn�3,andwheneitheraC2orC3rotationalelementispresent,nojointmaylieatthecentreofrotation.Insummary,a2Disostaticframeworkmayhaveonlysymmetryopera-tionsdrawnfromthelistfE;C2;C3;g,andhencethepossiblesymmetrygroupsGare6innumber:C1,C2,C3,Cs,C2v,C3v.Groupbygroup,theconditionsnecessaryfora2Dframeworktobeisostaticarethenasfollows.C1:b=2j�3.C2:b=2j�3withb2=1andjc=0,andasallotherbarsandjointsoccurinpairs,jisevenandbisodd.C3:b=2j�3withjc=0,andhencealljointsandbarsoccurinsetsof3.Cs:b=2j�3withb=1andallotherbarsoccurringinpairs.Symmetrydoesnotrestrictj.C2v:b=2j�3withjc=0andb2=b=1.Acentralbarliesinoneofthetwomirrorlines,andperpendiculartotheother.Anyadditionalbarsmustlieinthegeneralposition,andhenceoccurinsetsof4,withjointsinsetsof2and4.Hencebisoddandjiseven.10 Figure2:Examples,foreachofthepossiblegroups,ofsmall2Disostaticframeworks,withbarswhichareequivalentundersymmetrymarkedwiththesamesymbol:(a)C1;(b)C2;(c)C3;(d)CsC1v;(e)C2v;(f)C3v.Mirrorlinesareshowndashed,androtationaxesareindicatedbyacirculararrow.ForeachofCsandC3v,twoexamplesaregiven:(i)whereeachmirrorhasabarcenteredat,andperpendicularto,themirrorline;(ii)whereabarliesineachmirrorline.ForC2v,thebarlyingatthecentremustlieinonemirrorline,andperpendiculartotheother.12 Figure3:Possibleplacementofabarunshiftedbyaproperrotationaboutanaxis:(a)foranyCn2;(b)forC2alone.b2isthenumberofbarsunshiftedbytheC2rotation:suchbarsmustlieeitheralong,orperpendiculartoandcenteredon,theaxis(seeFig-ure3(a)and(b));bisthenumberofbarsunshiftedbyagivenmirroroperation(seeFig-ure5(a)and(b)).Again,eachofthecountsreferstoaparticularsymmetryelement,andso,forinstancethejointcountedinjcalsocontributestoj,jnandj.FromTable2,thesymmetrytreatmentofthe3DMaxwellequationre-ducestoscalarequationsofsixtypes.If�(m)��(s)=0,thenE:3j�b=6(12):b=j(13)i:3jc+bc=0(14)C2:j2+b2=2(15)C�n2:(jn�2)(2cos +1)=bn(16)S�n2:jc(2cos �1)=bnc(17)whereagivenequationapplieswhenthecorrespondingsymmetryoperationispresentinG.13 ECn6=2( )C2iSn6=2( ) �(j) jjnj2jjcjc�xyz 32cos +1�11�32cos �1 =�(j)�xyz 3j(2cos +1)jn�j2j�3jc(2cos �1)jc��(b) �b�bn�b2�b�bc�bnc�(�xyz+�RxRyRz) �6�4cos �22000 =�(m)��(s) 3j�b�6(2cos +1)(jn�2)�bn�j2�b2+2j�b�3jc�bc(2cos �1)jc�bncTable2:Calculationsofcharactersforrepresentationsinthe3Dsymmetry-extendedMaxwellequation(5). 14 Someobservationson3Disostaticframeworks,arisingfromtheabove,are:(i)From(12),theframeworkmustsatisfythescalarMaxwellrule(1)withm�s=0.(ii)From(13),eachmirrorthatispresentcontainsthesamenumberofjointsasbarsthatareunshiftedunderre ectioninthatmirror.(iii)From(14),acentro-symmetricframeworkhasneitherajointnorabarcenteredattheinversioncentre.(iv)ForaC2axis,(15)hassolutions(j2;b2)=(2;0);(1;1);(0;2):Thecountb2referstobothbarsthatliealong,andthosethatlieperpendicularto,theaxis.However,ifabarweretoliealongtheC2axis,itwouldcontribute1tob2and2toj2thusgeneratingacontradictionof(15),sothatinfactallbarsincludedinb2mustlieperpendiculartotheaxis.(v)Equation(16)canbewritten,with =2=n,as(jn�2)2cos2 n+1=bnwithn�2.Thenon-negativeintegersolutionjn=2,bn=0,ispossibleforalln.Forn�2thefactor(2cos(2=n)+1)isrationalatn=3;4;6,butgeneratesafurtherdistinctsolutiononlyforn=3:n=30(j3�2)=b3andsohereb3=0,butj3isunrestricted.n=4j4�2=b4C4impliesC24=C2aboutthesameaxis,andhenceb4=0,andj4=j2=2.n=62(j6�2)=b6C6impliesC36=C2andC26=C3aboutthesameaxis,andhenceb6=b3=0,andj6=j3=j2=2.16 (a)(b)Figure6:Aregularoctahedron(a),andaconvexpolyhedron(b)generatedbyaddingatwistedoctahedrontoeveryfaceoftheoriginaloctahedron.Thepolyhedronin(b)hastherotationbutnotthere ectionsymmetriesofthepolyhedronin(a).Ifaframeworkisconstructedfromeitherpolyhedronbyplacingbarsalongedges,andjointsatvertices,theframeworkwillbeisostatic.AnexampleofthecappingofaregularoctahedronisshowninFigure6.Similartechniquescanbeappliedtocreatepolyhedraforanyofthepointgroups.Oneinterestingpossibilityarisesfromconsiderationofgroupsthatcon-tainC3axes.Equation(16)allowsanunlimitednumberofjoints,thoughnotbars,alonga3-foldsymmetryaxis.Thus,startingwithanisostaticframework,jointsmaybeaddedsymmetricallyalongthe3-foldaxes.TopreservetheMaxwellcount,eachadditionaljointisaccompaniedby3newbars.Thus,forinstance,wecan`cap'everyfaceofanicosahedrontogivethecompoundicosahedron-plus-dodecahedron(thesecondstellationoftheicosahedron),asillustratedinFigure7,andthisprocesscanbecontinuedadinnitumaddingapileof`hats'consistingofanewjoint,linkedtoallthreejointsofanoriginalicosahedralface(Figure8).Similarconstructionsstartingfromcubicandtrigonallysymmetricisostaticframeworkscanbeenvisaged.Additionofasingle`hat'toatriangleofaframeworkisoneoftheHennenbergmoves(Tay&Whiteley1985):changesthatcanbemadetoanisostaticframeworkwithoutintroducingextramechanismsorstatesofselfstress.18 (a)(b)Figure7:Anicosahedron(a),andthesecondstellationoftheicosahedron(b).Ifaframeworkisconstructedfromeitherpolyhedronbyplacingbarsalongedges,andjointsatvertices,theframeworkwillbeisostatic.Theframework(b)couldbeconstructedfromtheframework(a)by`capping'eachfaceoftheoriginalicosahedronpreservingtheC3vsitesymmetry. Figure8:Aseriesof`hats'addedsymmetricallyalonga3-foldaxisofanisostaticframeworkleavesthestructureisostatic.19 4SucientConditionsforIsostaticRealisa-tions.4.1Conditionsfortwo-dimensionalisostaticframeworksForaframeworkwithpoint-groupsymmetryGtheprevioussectionhaspro-videdsomenecessaryconditionsfortherealizationtobeisostatic.Theseconditionsincludedsomeover-allcountsonbarsandjoints,alongwithsub-countsonspecialclassesofbarsandjoints(barsonmirrorsorperpendiculartomirrors,barscenteredontheaxisofrotation,jointsonthecentreofrota-tionetc.).Here,assumingthattheframeworkisrealizedwiththejointsinacongurationasgenericaspossible(subjecttothesymmetryconditions),weinvestigatewhethertheseconditionsaresucienttoguaranteethattheframeworkisisostatic.Thesimplestcaseistheidentitygroup(C1).Forthisbasicsituation,thekeyresultisLaman'sTheorem.Inthefollowing,wetakeG=fJ;Bgtodenetheconnectivityoftheframework,whereJisthesetofjjointsandBthesetofbbars,andwetakeptodenethepositionsofallofthejointsin2D.Theorem1(Laman,1970)Foragenericcongurationin2D,p,theframe-workG(p)isisostaticifandonlyifG=fJ;Bgsatisestheconditions:(i)b=2j�3;(ii)foranynon-emptysetofbarsB,whichcontactsjustthejointsinJ,withjBj=bandjJj=j,b2j�3.Ourgoalistoextendtheseresultstoothersymmetrygroups.Withtheappropriatedenitionof`generic'forsymmetrygroups(Schulze2008a),wecananticipatethatthenecessaryconditionsidentiedintheprevioussectionsforthecorrespondinggroupplustheLamanconditionidentiedinTheorem1,whichconsiderssubgraphsthatarenotnecessarilysymmetric,willbesuf-cient.Forthreeoftheplanesymmetrygroups,thishasbeenconrmed.Weusethepreviousnotationforthepointgroupsandtheidenticationofspecialbarsandjoints,anddescribeacongurationas`genericwithsym-metrygroupG'if,apartfromconditionsimposedbysymmetry,thepointsareinagenericposition(theconstraintsimposedbythelocalsitesymmetrymayremove0,1or2ofthetwobasicfreedomsofthepoint).Theorem2(Schulze2008b)Ifpisaplanecongurationgenericwithsym-metrygroupG,andG(p)isaframeworkrealizedwiththesesymmetries,thenthefollowingnecessaryconditionsarealsosucientforG(p)tobeisostatic:20 3j0�b0�6=0,weneedtoasserttheconditionscorrespondingtothesym-metryoperationsinG0aswell.Theseconditionsareclearlynecessary,andforallre ections,half-turns,and6-foldrotationsinG0,theydonotfollowfromtheglobalconditionsontheentiregraph(astheywouldintheplane).SeeSchulze(2008c)fordetails.Alloftheaboveconditionscombined,however,arestillnotsucientfora3-dimensionalframeworkG(p)whichisgenericwithpointgroupsymmetryGtobeisostatic,becauseevenifG(p)satisesalloftheseconditions,thesymmetryimposedbyGmayforcepartsofG(p)tobe` attened'sothataself-stressofG(p)iscreated.Formoredetailsonhow` atness'causedbysymmetrygivesrisetoadditionalnecessaryconditionsfor3-dimensionalframeworkstobeisostatic,wereferthereadertoSchulze,Watson,andWhiteley(2008).ReferencesAltmann,S.L.andHerzig,P.,1994.Point-GroupTheoryTables.ClarendonPress,Oxford.Atkins,P.W.,Child,M.S.andPhillips,C.S.G.,1970.TablesforGroupThe-ory.OxfordUniversityPress,Oxford.Bishop,D.M.,1973.GroupTheoryandChemistry.ClarendonPress,Ox-ford.Calladine,C.R.,1978.BuckminsterFuller's`Tensegrity'structuresandClerkMaxwell'srulesfortheconstructionofsti frames.InternationalJournalofSolidsandStructures14,161{172.Cauchy,A.L.,1813.Recherchesurlespolyedres|premiermemoire,Jour-naldel'EcolePolytechnique9,66{86.Ceulemans,A.andFowler,P.W.,1991.ExtensionofEuler'stheoremtothesymmetrypropertiesofpolyhedra.Nature353,52{54.Dehn,M.,1916.UberdieStarreitkonvexerPolyeder,MathematischeAn-nalen77,466{473.DonevA.,Torquato,S.,Stillinger,F.H.andConnelly,R.,2004.Jamminginhardsphereanddiskpackings.JournalofAppliedPhysics95,989{999.Fowler,P.W.andGuest,S.D.,2000.AsymmetryextensionofMaxwell'sruleforrigidityofframes.InternationalJournalofSolidsandStructures37,1793{1804.22 necessaryconditionsforindependence,inpreparation.Tay,T-S.andWhiteley,W.,1985.GeneratingIsostaticFrameworks.Struc-turalTopology11,20{69.Whiteley,W.,1991Vertexsplittinginisostaticframeworks,StructuralTopol-ogy16,23{30.Whiteley,W.,2005.Countingoutthe exibilityofproteins.PhysicalBiology2,116{126.24