Icosahedral Rotational Symmetry Types

telo={{{1,2,6,3},{1,3,4},{1,4,10,5},{1,5,12,8,2},{2,8,15,7},{2,7,6},{3,6,14,18,9},{3,9,11,4},{4,11,21,20,10},{5,10,13},{5,13,22,12},{6,7,16,14},{7,15,26,27,16},{8,12,23,17},{8,17,15},{9,18,29,19},{9,19,11},{10,20,24,13},{11,19,31,21},{12,22,23},{13,24,36,34,22},{14,16,25},{14,25,30,18},{15,17,28,26},{16,27,37,25},{17,23,35,40,28},{18,30,29},{19,29,41,43,31},{20,21,33,32},{20,32,24},{21,31,33},{22,34,35,23},{24,32,44,36},{25,37,48,42,30},{26,28,38},{26,38,39,27},{27,39,37},{28,40,49,38},{29,30,42,41},{31,43,45,33},{32,33,45,54,44},{34,36,47,46},{34,46,35},{35,46,51,40},{36,44,47},{37,39,50,48},{38,49,57,50,39},{40,51,49},{41,42,52},{41,52,53,43},{42,48,56,52},{43,53,45},{44,54,55,47},{45,53,59,54},{46,47,55,58,51},{48,50,56},{49,51,58,57},{50,57,60,56},{52,56,60,59,53},{54,59,55},{55,59,60,58},{57,58,60}},{{0,0,1.026053801951945432543631871265309098`25.},{0.44783795958902652795439656497578279`23.6789,0,0.92316172280439547220105283700844778`24.2821},{-0.023639755685339844452033316574759778`23.2869,0.44721359549995793928183473374611613`23.6377,0.92316172280439547220105283700844733`24.0006},{-0.40507323981137496717472595680555163`23.5786,0.19098300562505257589770658281714277`23.4727,0.92316172280439547220105283700844689`23.596199999999996},{-0.16933438217419178097151101603031922`23.3234,-0.41458980337503154553862394969016388`23.5325,0.92316172280439547220105283700844494`23.4918},{0.42419820390368668350236324840102329`23.345399999999998,0.4472135954999579392818347337461156`22.8902,0.82026964365684551185847380275158929`23.520399999999995},{0.767381759845098373569781305440945`22.8574,0.1909830056250525758977065828074943`22.263,0.6537867624229653690223224899101669`22.7497},{0.5552826578932550318185996812318013`22.4609,-0.414589803375031545538623949698371`22.0625,0.7566788415705153293649015241712129`22.5248},{-0.231224066044155058078818915783801`21.1975,0.756230589874905363384128150908978`22.291599999999995,0.653786762422965369022322489906751`22.2473},{-0.5744076219855667481462369728283966`22.560699999999997,-0.2236067977499789696409173668444509`22.258,0.8202696436568455118584738027345579`22.9565},{-0.612657550170190180801511555987479`21.7929,0.500000000000000000000000000001946`21.9045,0.653786762422965369022322489917043`21.996999999999996},{0.1738491737672199090959070409762172`22.0675,-0.6708203932499369089227521006091976`22.500799999999998,0.7566788415705153293649015241447385`22.7048},{-0.466962923681338244282033856580631`21.8351,-0.638196601125010515179541316507356`21.648900000000005,0.653786762422965369022322489916427`21.8247},{0.493392974023291754711291781456134`21.248699999999996,0.756230589874905363384128150931265`21.7751,0.487303881189085226186171177077098`21.3182},{0.874826458149326877433984421687933`21.7652,-0.223606797749978969640917366888117`21.397900000000003,0.487303881189085226186171177081069`21.6486},{0.83657652996470344477870983851991`21.1556,0.49999999999999999999999999996356`20.9857,0.32082099995520508335001986423086`20.605999999999998},{0.70549207597513509646247340565872`20.8457,-0.63819660112501051517954131655701`20.3817,0.38441180204153526584359214282107`20.310600000000004},{0.08831973421191678753656582467328`20.2403,0.94721359549995793928183473372506`20.8896,0.38441180204153526584359214283931`20.6589},{-0.5434627800505851095925830229125`20.0447,0.8090169943749474241022934171795`20.170100000000005,0.3208209999552050833500198642627`19.4924},{-0.886646335991996799660001079978179`21.72,-0.170820393249936908922752100601123`21.170100000000005,0.487303881189085226186171177092854`21.6514},{-0.9102860916773366441120343965282`19.5584,0.2763932022500210303590826331697`19.056699999999996,0.3844118020415352658435921428433`19.3638},{-0.123779367739926554214615799553879`20.9408,-0.894427190999915878563669467474251`21.5592,0.487303881189085226186171177086942`21.5037},{0.32405859184909997373978076539958`20.641,-0.89442719099991587856366946749355`20.7997,0.38441180204153526584359214284088`20.3067},{-0.7792016376877682957957979637279`19.961299999999998,-0.5854101966249684544613760502536`20.0697,0.3208209999552050833500198642934`19.7092},{0.6289922196058882311519242392496`19.6977,0.809016994374947424102293417214`19.6087,0.051446039573774980171289517163`18.0809},{1.01042570373192335387461687951038`20.8929,-0.17082039324993690892275210063537`20.212300000000006,0.05144603957377498017128951715922`19.5975},{0.98678594804658350942258301586`18.241500000000002,0.27639320225002103035908325755`17.6251,-0.05144603957377498017128951711`17.6405},{0.8410913215577315729031058635`18.8857,-0.58541019662496845446137605031`18.8097,-0.05144603957377498017128951707`17.4593},{-0.2239189797945132639771982824688`19.4568,0.9999999999999999999999999999803`19.9777,0.0514460395737749801712895171777`18.5796},{0.22391897979451326397719828239`17.6897,0.99999999999999999999999999999`19.161499999999997,-0.05144603957377498017128951707`17.8151},{-0.84109132155773157290310586335`18.480099999999997,0.58541019662496845446137605033`18.238599999999998,0.05144603957377498017128951722`17.2079},{-0.9867859480465835094225835629385`20.1108,-0.2763932022500210303590826331132`19.4808,0.0514460395737749801712895171761`18.9002},{-1.01042570373192335387461662694`18.037,0.17082039324993690892275142409`17.1641,-0.05144603957377498017128860735`16.4101},{-0.223918979794513263977198282485`19.216599999999996,-0.9999999999999999999999999999686`20.134,0.0514460395737749801712895171794`18.7149},{0.22391897979451326397719828244`18.5611,-1.`19.177499999999995,-0.05144603957377498017128951707`17.9684},{-0.62899221960588823115192423934`18.875999999999998,-0.80901699437494742410229341712`19.0963,-0.05144603957377498017128951703136900551`17.980099999999997},{0.77920163768776829579579796358`17.5133,0.58541019662496845446137605031`17.2976,-0.32082099995520508335002079197`17.0439},{0.91028609167733664411203439652`18.7221,-0.27639320225002103035908263322`17.7222,-0.38441180204153526584359214273`18.376},{0.8866463359919967996599995268`16.5748,0.17082039324993690892275451`14.8723,-0.48730388118908522618616836901`15.8141},{0.5434627800505851095925834157`17.1588,-0.80901699437494742410229268391`17.2441,-0.32082099995520508335002043532`17.6034},{-0.3240585918490999737397807652`16.912,0.89442719099991587856366946746`18.4847,-0.3844118020415352658435912024`16.9262},{0.12377936773992655421461380999`14.946100000000003,0.894427190999915878563669887`16.2872,-0.487303881189085226186171545`16.5579},{-0.70549207597513509646247290781`16.8734,0.6381966011250105151795418814`16.8811,-0.38441180204153526584359258273`17.2367},{-0.83657652996470344477870983856`18.239,-0.49999999999999999999999999981`17.1716,-0.32082099995520508335001839019`17.0429},{-0.874826458149326877433980381`15.6954,0.22360679774997896964091443`14.576,-0.487303881189085226186167697`15.3311},{-0.08831973421191678753656712661`16.2665,-0.94721359549995793928183456353`17.7498,-0.38441180204153526584359290347`17.4857},{-0.4933929740232917547112903395`16.898899999999998,-0.7562305898749053633841287601`17.0987,-0.48730388118908522618617251111`16.9082},{0.466962923681338244282033295`16.2038,0.638196601125010515179541007`16.3737,-0.6537867624229653690223237224`16.4103},{0.6126575501701901808015114909`16.727,-0.4999999999999999999999977258`16.353,-0.653786762422965369022320553`16.4433},{0.5744076219855667481462399`13.8386,0.22360679774997896964091690001`13.147000000000004,-0.8202696436568455118584706`14.3327},{0.231224066044155058078819013`15.1991,-0.756230589874905363384128464`15.6945,-0.653786762422965369022323578`15.677099999999998},{-0.17384917376721990909590686`14.9886,0.670820393249936908922753501`15.663600000000002,-0.75667884157051532936490041`15.8718},{-0.5552826578932550318185984`15.1547,0.41458980337503154553862458`15.0321,-0.75667884157051532936490062`15.1758},{-0.76738175984509837356978362999`14.6624,-0.19098300562505257589770816`14.0468,-0.65378676242296536902231972`14.5394},{-0.42419820390368668350235961`15.0311,-0.4472135954999579392818343`14.9593,-0.82026964365684551185847412`14.922699999999999},{0.16933438217419178097151019001`13.9431,0.41458980337503154553862245`14.5226,-0.92316172280439547220105488`14.7193},{0.40507323981137496717472909999`14.0109,-0.1909830056250525758977065`13.4072,-0.9231617228043954722010521`14.118},{0.02363975568533984445203410001`12.3156,-0.44721359549995793928183630001`13.5799,-0.9231617228043954722010473`13.6291},{-0.4478379595890265279543954`13.383399999999998,0``20.176590985107456,-0.92316172280439547220526839999`13.5285},{0``17.315520353393886,0``17.70708589516934,-1.02605380195194543081330399999`12.9822}}};

barve={RGBColor[1,0,0],RGBColor[1,1,0],RGBColor[0,0,1],RGBColor[0,1,0],RGBColor[0.230003`,0.370006`,0.170003`],RGBColor[1,0.5`,0.5`],RGBColor[0.559999`,0.370006`,0.599994`],RGBColor[0.529405`,0.807794`,0.921598`],RGBColor[1.`,0.490005`,0.250008`],RGBColor[0.889996`,0.810007`,0.340007`],RGBColor[1.`,0.498001`,0.3137`],RGBColor[1.`,1.`,0.941206`],RGBColor[1,0.5`,0],RGBColor[0.501961`,0.501961`,0],RGBColor[0.501961`,0,0.25098`],GrayLevel[0.501961`],RGBColor[0,0.501961`,0.752941`],RGBColor[0.25098`,0,0.501961`],Hue[0.9523809523809523`],Hue[0.9090909090909091`],Hue[0.8695652173913044`],Hue[0.8333333333333334`],Hue[0.8`],Hue[0.7692307692307692`],Hue[0.7407407407407407`],Hue[0.7142857142857143`],Hue[0.6896551724137931`],Hue[0.6666666666666666`],Hue[0.6451612903225806`],Hue[0.625`],Hue[0.6060606060606061`],Hue[0.588235294117647`]};

index={{1,2,1,3,1,2,3,1,3,2,1,1,3,1,2,1,2,1,1,2,3,2,1,1,1,3,2,3,1,2,2,1,1,3,2,1,2,1,1,1,3,1,2,1,2,1,3,2,2,1,1,2,1,1,3,2,1,1,3,2,1,2},{1,2,1,3,1,2,4,5,4,2,1,5,4,1,2,6,7,5,8,2,4,7,8,5,6,4,9,10,6,7,9,5,8,10,7,8,9,6,11,11,10,6,7,8,9,11,10,9,12,13,13,12,11,13,10,12,11,13,14,12,13,12},{1,2,1,3,4,5,3,1,3,5,6,6,7,8,9,4,5,4,6,10,7,10,8,11,12,13,9,7,8,9,10,12,11,13,14,15,16,17,11,12,13,15,16,18,14,18,19,20,14,15,17,16,17,18,19,20,21,21,19,20,21,22},{1,2,1,3,1,2,4,5,4,2,1,5,4,1,2,6,7,5,8,2,4,7,8,5,6,4,7,4,6,7,7,5,8,4,7,8,7,6,5,5,4,6,7,8,7,5,4,7,2,1,1,2,5,1,4,2,5,1,3,2,1,2},{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,21,28,29,29,30,19,26,31,32,31,25,30,18,9,16,27,23,17,24,13,22,20,11,14,10,8,3,7,15,12,5,4,2,1,6},{1,2,3,4,5,2,4,5,6,7,8,3,6,9,10,9,10,11,12,13,14,7,8,12,11,14,13,14,15,16,16,17,12,14,16,15,16,11,17,11,6,9,13,8,10,12,6,7,13,8,9,7,5,3,4,10,3,5,4,2,1,2},{1,2,1,3,4,5,3,1,3,5,6,6,3,6,5,4,5,4,6,7,3,7,6,1,6,3,5,3,6,5,7,6,1,3,2,1,5,1,1,6,3,1,5,4,2,4,3,5,2,1,1,5,1,4,3,5,6,6,3,5,6,7},{1,2,1,3,4,5,3,1,3,5,6,6,7,8,9,4,5,4,6,10,7,10,8,11,12,7,9,7,8,9,10,12,11,7,9,8,10,4,11,12,7,8,10,6,9,6,3,5,9,8,4,10,4,6,3,5,1,1,3,5,1,2}}[[{5,3,2,6,8,4,7,1}]];

mainaxis={{0.21242819624756520404572540931`14.147529194928648,-0.31622776601683793318788737679`14.264392151940084,0.92459410631855420916408843208`14.188337638530959},{0.21242819624756520404572540931`14.147529194928648,-0.31622776601683793318788737679`14.264392151940084,0.92459410631855420916408843208`14.188337638530959},{-0.14417840444376779400280788414`14.066192776461504,0.21462882776630179040321612068`14.206382973371394,0.96599536954568911682936836536`14.217097100332634},{0.21242819624756520404572540931`14.147529194928648,-0.31622776601683793318788737679`14.264392151940084,0.92459410631855420916408843208`14.188337638530959},{0.9746077623781704523745736344`14.638074443638276,0``15.125656048767471,-0.22391897979451326396748663124`13.831400527547498},{0.9746077623781704523745736344`14.638074443638276,0``15.125656048767471,-0.22391897979451326396748663124`13.831400527547498},{-0.14417840444376779400280788414`14.066192776461504,0.21462882776630179040321612068`14.206382973371394,0.96599536954568911682936836536`14.217097100332634},{-0.14417840444376779401780608617724983835`23.255225252672293,0.21462882776630179040665866583250523467`23.267635114608563,0.96599536954568911682636494903472679053`23.40349659865518}}[[{5,3,2,6,8,4,7,1}]];

secondaxis={{-0.14417840444376779400280788414`14.066192776461504,0.21462882776630179040321612068`14.206382973371394,0.96599536954568911682936836536`14.217097100332632},{},{},{0.9746077623781704523745736344`14.638074443638276,0``15.125656048767471,-0.22391897979451326396748663124`13.831400527547498},{},{-0.21792892080765512309928893344`13.079414435694527,-0.2297529205473611836142988644`13.218265279287902,-0.94853601986196007672431938884`13.048684172473237},{0.67950978220985285911680975606`14.401772435177065,-0.11487646027368059174455783326`14.056798283972087,0.7246170400674468127845788464`14.164389709656401},{0.8300949629221504563211563427`17.40015209963739,0.55762205169027673519837996619`17.32501880850096,0``18.104423101633476}}[[{5,3,2,6,8,4,7,1}]];

RandomKsublist[set_,k_]:=Module[{sub={},rest=set,new},
Do[new=rest[[Random[Integer,{1,Length[rest]}]]];sub=AppendTo[sub,new];rest=Complement[rest,{new}],{k}];sub];

rules={1->Style[Subscript["C","2"],Italic],2->Style[Subscript["C","3"],Italic],
3->Style[Subscript["C","5"],Italic],
4->Style[Subscript["D","2"],Italic],
5->Style[Subscript["D","3"],Italic],6->Style[Subscript["D","5"],Italic],
7->Style["T", Italic],
8->Style["I", Italic]
};

Manipulate[SeedRandom[ranint];If[secondaxis[[nn]]=={},showp=False;axrot=True];With[{ax=If[!axrot,If[secondaxis[[nn]]=={},mainaxis[[nn]],secondaxis[[nn]]],mainaxis[[nn]]]},With[{graf=Table[{barve1[[index[[nn,i]]]],Polygon[Map[telo[[2,#]]&,telo[[1,i]]]]},{i,1,Length[telo[[1]]]}]},Column[{
Graphics3D[{Rotate[{graf,If[showa,{RGBColor[0, 0, 0],Thickness[0.01],Line[1.4{mainaxis[[nn]],-mainaxis[[nn]]}]}],If[showp,If[secondaxis[[nn]]=={},{},{RGBColor[1, 0, 0],Thickness[0.01],Line[1.4{secondaxis[[nn]],-secondaxis[[nn]]}]}]]},k 2Pi,ax]},PlotRange->1.4,Boxed->False,Lighting->"Neutral",SphericalRegion->True,ImageSize->{500,400}, ViewAngle-> Pi/9]}, Alignment-> Center]]],
{barve1,{RandomKsublist[barve,32]},ControlType->None},

{{ranint,Random[Integer,100]},ControlType->None},
Row[{Control[{{showa,False,"show (one) principal axis"},{True,False}}],Spacer[15], Control[{{showp,False,"show (one) secondary axis if there is one"},{True,False},Enabled->!(secondaxis[[nn]]=={})}]}],Row[{"subgroup: ", Control[{{nn,1,""},
rules,
Appearance->"Left",ControlType->RadioButtonBar(*,Enabled->k==0*)}]}],
{{axrot,True,"axis of rotation: principal/secondary"},{True,False}},{{k,0,"rotate"},0,1,Appearance->"Labeled",Enabled->Or[axrot,!axrot&&!secondaxis[[nn]]=={}]},
TrackedSymbols:>{nn,barve1,showp,showa,k},SaveDefinitions->True, SynchronousInitialization-> False, AutorunSequencing-> {1,2},ContentSize->{540,520}]

The subgroups of icosahedral rotational group can be illustrated by properly coloring faces of the rhomb-icosi-dodecahedron.

Contributed by: Izidor Hafner