Icosahedral Group

Author

Eric W. Weisstein
September 18, 2019

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GroupTheory/IcosahedralGroup.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/IcosahedralGroup.html.

©2019 Wolfram Research, Inc. except for portions noted otherwise

I (pure rotation group)

$Version

bug(365947)

FiniteGroupData[{"PointGroup","I"},"Order"]

FiniteGroupData[60]

FiniteGroupData[{"AlternatingGroup",5},"IsomorphicGroups"]

FiniteGroupData["I"]

As designed since I is not a point group.

FiniteGroupData[{"CrystallographicPointGroup","I"}]

Notation

FiniteGroupData[{"PointGroup","I"},"AlternateStandardNames"]

bug(379614)

FiniteGroupData["I"]

FiniteGroupData[{"PointGroup","I"},"Notation"]

FiniteGroupData[{"PointGroup","I"},"ElementNames"]

Classes

FiniteGroupData[{"PointGroup","I"},"Classes"]

FiniteGroupData[{"PointGroup","I"},"ConjugacyClasses"]

FiniteGroupData[{"PointGroup","I"},"ConjugacyClassSizes"]

Cayley graph

FiniteGroupData[{"PointGroup","I"},"PermutationGroupRepresentation"]

CayleyGraph[System`AlternatingGroup[5]]

UndirectedGraph[%]

ToEntity[%]

Multipilication table

mt=FiniteGroupData[{"PointGroup","I"},"MultiplicationTable"]

ArrayPlot[FiniteGroupData[{"PointGroup","I"},"MultiplicationTable"],ColorFunction->#]&/@{Hue,GrayLevel}

spacerep=FiniteGroupData[{"PointGroup","I"},"SpaceRepresentation"]

matrixrep=FiniteGroupData[{"PointGroup","I"},"MatrixRepresentation"];

matrixrep==spacerep

permutationrep=FiniteGroupData[{"PointGroup","I"},"PermutationRepresentation"]

ArrayPlot[With[{m=spacerep},Simplify@Outer[Dot,m,m,1]/.Thread[m->Range[Length[m]]]]]

ArrayPlot[With[{m=permutationrep},Outer[Part,m,m,1]/.Thread[m->Range[Length[m]]]]]

I (pre-FiniteGroupData)

arules=Thread[AlternatingGroup[5]->Range[60]];

Group

GroupQ[AlternatingGroup[5]]//Timing

GroupQ[PointGroupMatrices["I"],Simplification->Expand]//Timing

Abelian

AbelianGroupQ[AlternatingGroup[5]]//Timing

Conjugacy Classes

ConjugacyClasses[AlternatingGroup[5]]/.rules

Length/@%

Subgroups

(subg=Subgroups[AlternatingGroup[5]]/.arules)//Timing

Length/@subg

Multiplication Table

i=Expand[PointGroup["I"]];

Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[i,Thread[i->Range[60]],Simplification->Expand][[1]],
DisplayFunction->Identity,
PlotLabel->TraditionalForm["\!\(\*
StyleBox[\"I\",\nFontSlant->\"Italic\"]\)"],ColorFunction->#,Frame->False,Mesh->True]&/@{Hue,GrayLevel}]]//Timing

ArrayPlot[Reverse@GroupMultiplicationTable[i,Thread[i->Range[60]],Simplification->Expand][[1]],
PlotLabel->Style["I",Italic],ColorFunction->Hue,ImageSize->200]

Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[AlternatingGroup[5],arules][[1]],
DisplayFunction->Identity,
PlotLabel->TraditionalForm[(Subscript[A, 5])],ColorFunction->#,Frame->False,Mesh->True]&/@{Hue,GrayLevel}]]

ArrayPlot[Reverse@GroupMultiplicationTable[AlternatingGroup[5],arules][[1]],
PlotLabel->Subscript[Style["A",Italic],5],ColorFunction->Hue,ImageSize->200]

Cycle Graph

Timing[m=MaximalGraphCycles[PointGroup["I"]]]

Split[Sort[Flatten[Rest/@m]]]

ShowLabeledGraph[RadialEmbedding[CycleGraph[PointGroup["I"]]]]

Polyhedra using I

Polyhedra

Vector[Icosidodecahedron]={0,0,1};
Vector[Dodecahedron]={1,GoldenRatio+1,0};
Vector[Icosahedron]={1,0,GoldenRatio};
Vector[TruncatedDodecahedron]={GoldenRatio,2,GoldenRatio+1};
Vector[TruncatedIcosahedron]={GoldenRatio,2 GoldenRatio+1,2};
Vector[SmallRhombicosidodecahedron]={GoldenRatio,-1,2};
Vector[SnubDodecahedron]={1,Root[-1-7 #1-14 ((#1)^2)-9 ((#1)^3)-7 ((#1)^4)+6 ((#1)^5)+((#1)^6)&,4],Root[-1+#1+2 ((#1)^2)+((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4]};

{Root[-1-7 #1-14 ((#1)^2)-9 ((#1)^3)-7 ((#1)^4)+6 ((#1)^5)+((#1)^6)&,4]//N,NRoots[-1-7 #1-14 ((#1)^2)-9 ((#1)^3)-7 ((#1)^4)+6 ((#1)^5)+((#1)^6)&[x]==0,x]}

{Root[-1+#1+2 ((#1)^2)+((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4]//N,NRoots[-1+#1+2 ((#1)^2)+((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&[x]==0,x]}

polys=#[[1,1,1]]&/@DownValues[Vector]

<<ConvexHull`

Show[GraphicsArray[Partition[
Graphics3D[#,Boxed->False]&/@Function[v,ConvexHull3D[Union[Chop/@Simplify[#.v&/@PointGroupMatrices["I"]]]]]/@(N/@Vector/@polys),
4,4,{1,1},{}]
]]//Timing

(Subscript[I, h])

FiniteGroupData["Ih"]

FiniteGroupData["Icosahedral","IsomorphicGroups"]

As designed since (Subscript[I, h]) is not a point group.

FiniteGroupData[{"CrystallographicPointGroup","Ih"}]

FiniteGroupData[120]

FiniteGroupData[{"CrystallographicPointGroup",32}]

FiniteGroupData[{"CrystallographicPointGroup",32},"IsomorphicGroups"]

Notation

FiniteGroupData["Icosahedral","AlternateStandardNames"]

FiniteGroupData["Icosahedral","Notation"]

FiniteGroupData["Icosahedral","ElementNames"]

Classes

FiniteGroupData["Icosahedral","Classes"]

FiniteGroupData["Icosahedral","ConjugacyClasses"]

FiniteGroupData["Icosahedral","ConjugacyClassSizes"]

Cayley graph

FiniteGroupData["Icosahedral","CayleyGraph"]

FiniteGroupData["Icosahedral","PermutationGroupRepresentation"]

(changed between V12.0 and V12.1)

CayleyGraph[FiniteGroupData["Icosahedral","PermutationGroupRepresentation"]]

UndirectedGraph[%]

ToEntity[%]

Multipilication table

ArrayPlot[FiniteGroupData["Icosahedral","MultiplicationTable"],ColorFunction->#]&/@{Hue,GrayLevel}

spacerep=FiniteGroupData["Icosahedral","SpaceRepresentation"]

matrixrep=FiniteGroupData["Icosahedral","MatrixRepresentation"];

matrixrep==spacerep

permutationrep=FiniteGroupData["Icosahedral","PermutationRepresentation"]

mt=FiniteGroupData["Icosahedral","MultiplicationTable"]

ArrayPlot[With[{m=spacerep},Simplify@Outer[Dot,m,m,1]/.Thread[m->Range[Length[m]]]]]

ArrayPlot[With[{m=permutationrep},Outer[Part,m,m,1]/.Thread[m->Range[Length[m]]]]]

(Subscript[I, h]) construction

ConstructionoftheIand(Subscript[I, h]) group matrix representation and polyhedron generating vectors by
Ed Pegg, Jr.
February 21 and June 14, 2002

k = (1+(Sqrt[5]))/2;

verts = {{0,k,1},{0,k,-1},{0,-k,1},{0,-k,-1},{1,0,k},{-1,0,k},{1,0,-k},{-1,0,-k},{k,1,0},{k,-1,0},{-k,1,0},{-k,-1,0}};

tris = {{1,2,9},{1,2,11},{1,5,9},{1,5,6},{1,6,11},{2,7,9},{2,8,11},{5,9,10},{3,5,6},{6,11,12},{7,9,10},{2,7,8}, {8,11,12},{3,5,10}, {3,6,12}, {4,7,10},{4,7,8}, {4,8,12},{3,4,10},{3,4,12}};

face[a_List] := {verts[[a[[1]]]],verts[[a[[2]]]],verts[[a[[3]]]]}

face[tris[[1]]]

tri1 = Inverse[face[tris[[1]]]]//FullSimplify

triAll = Flatten[Map[Permutations,tris],1]

MatrixForm/@(icosab=Map[Last,Drop[Sort[Table[With[{temp=FullSimplify[tri1 . face[triAll[[qq]]]]},{Det[temp],temp}],{qq,1,120}]],60]]/.{(1/4) (1+(Sqrt[5]))-> α}/.{(1/4) (-1-(Sqrt[5]))-> -α}/.{(1/4) (-1+(Sqrt[5]))-> β}/.{(1/4) (1-(Sqrt[5]))-> -β})

icoshalf=icosab/.{α->(1/4) (1+(Sqrt[5])),β->(1/4) (-1+(Sqrt[5]))};

icos=Union[-icoshalf,icoshalf];

Union[FullSimplify[icos]]//Length

Length[icos]

(Subscript[I, h]) (pre-FiniteGroupData)

ih=Expand[PointGroup["Ih"]];

irules=Thread[ih->Range[120]];

Group

GroupQ[ih,Simplification->Expand]//Timing

Abelian

AbelianGroupQ[ih,Simplification->Expand]//Timing

Conjugacy Classes

(conj=ConjugacyClasses[ih,Simplification->Expand]/.rules)//Timing

Length/@conj

Subgroups

SubgroupQ[Sequence@@PointGroup/@{"I","Ih"},Simplification->Expand]//Timing

subg=Subgroups[ih,Simplification->Expand]/.rules

Multiplication Table

Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[ih,irules,Simplification->Expand][[1]],
PlotLabel->TraditionalForm["(Subscript[I, h])"],DisplayFunction->Identity,
ColorFunction->#,Frame->False,Mesh->False]&/@{Hue,GrayLevel}]]//Timing

ArrayPlot[Reverse@GroupMultiplicationTable[ih,irules,Simplification->Expand][[1]],
PlotLabel->Style[Subscript["I","h"],Italic],ColorFunction->Hue,Frame->False,Mesh->False,ImageSize->200]

Polyhedra using (Subscript[I, h])

Vector120[GreatRhombicosidodecahedron]={GoldenRatio,3,2GoldenRatio};

polys=#[[1,1,1]]&/@DownValues[Vector120]

<<ConvexHull`

Show[GraphicsArray[Partition[
Graphics3D[#,Boxed->False]&/@Function[v,ConvexHull3D[Union[Chop/@Simplify[#.v&/@PointGroupMatrices["Ih"]]]]]/@(N/@Vector120/@polys),
4,4,{1,1},{}]
]]//Timing

Triangle[HexakisIcosahedron]={{0,(Sqrt[(1/10) (5+(Sqrt[5]))]),(Sqrt[(1/10) (5-(Sqrt[5]))])},{0,1,0},{(Sqrt[(1/6) (3-(Sqrt[5]))]),(Sqrt[(1/6) (3+(Sqrt[5]))]),0}};

Show[Graphics3D[Polygon[Simplify[Triangle[HexakisIcosahedron].#]]&/@PointGroupMatrices["Ih"],Boxed-> False]]