Trigyrate Rhombicosidodecahedron

Author

Eric W. Weisstein
September 8, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polyhedra/TrigyrateRhombicosidodecahedron.nb.

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Graphic

PolyhedronData

PolyhedronData["TrigyrateRhombicosidodecahedron","Orientations"]

PolyhedronData["TrigyrateRhombicosidodecahedron"]

PolyhedronData["TrigyrateRhombicosidodecahedron","VertexCoordinates"]

PolyhedronData (V12.1)

PolyhedronData["TrigyrateRhombicosidodecahedron"]

PolyhedronData["TrigyrateRhombicosidodecahedron","VertexCoordinates"]

Netlib

PolyhedronName[JohnsonSolid[75]]

Show[Graphics3D[Polyhedron[JohnsonSolid[75]]],Boxed->False]

Net

PolyhedronData

PolyhedronData["TrigyrateRhombicosidodecahedron", "Net"]

PolyhedronData["TrigyrateRhombicosidodecahedron", "Net","Colored"]

Netlib

Show[Graphics[Net[JohnsonSolid[75]]],AspectRatio->Automatic]

Properties

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname="TrigyrateRhombicosidodecahedron","Polyhedron"]

vol=PolyhedronData[pname,"Volume"]

AdjacentFaceIndices

AdjacentFaceIndices[p]

Centroid

Graphics3D[{AbsolutePointSize[10],
Point[{0,0,0}],{Blue,Point[PolyhedronData[pname,"Centroid"]]},Opacity[.2],PolyhedronData[pname,"GraphicsComplex"]}]

PolyhedronData[pname,"Centroid"]

(c=RegionCentroid[p])//Timing

N[c,10^4]//Quiet

MapIndexed[RootApproximant[N[#,1000],8]&,c,{2}]//Quiet

PolyhedronCentroid[p]

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]//FullSimplify

(sphere=MyCircumsphere[p])//Timing

Graphics3D[{{Opacity[.2],Yellow,sphere},p},Boxed->False]

Convex

ConvexPolyhedronQ[p]

DehnInvariant

PolyhedronData[pname,"DehnInvariant"]

(dehn=DehnInvariant[p,Debug->True]//FullSimplify)//Timing

ToConwayBasis[dehn]

Complement[Select[PolyhedronData[All],PolyhedronData[#,"DehnInvariant"]==dehn&&PolyhedronData[#,"Volume"]==vol&],{pname}]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

DihedralAngles[p]//FullSimplify//Tally

ByteCount[dih=SimplifyNumericallyDistinct[DihedralAngles[p],Debug->True]]//Timing

Sort[N[dih,20]]-SortBy[dih0,N]//Chop//Union

dih

EdgeLengths

PolyhedronData[pname,"EdgeLengths"]

PolyhedronData[pname,"Lines"]/.Line[l_]:>EuclideanDistance@@l//FullSimplify//Union//Quiet

GeneralizedDiameter

PolyhedronData["TrigyrateRhombicosidodecahedron","GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify//Quiet

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

(i=((MomentOfInertia[p,{0,0,0}])/(Volume[p])))//Timing

r=Map[RootApproximant[N[#,1000],8]&,i,{2}]//FullSimplify//Quiet

Equal@@Tr[r,List]

(3 (3885+1601 Sqrt[5]))/12100 IdentityMatrix[3]==r

N[i-r,10^4]//Quiet

PolyhedronInertiaTensor[p]//Timing

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

sphere=MyInsphere[p]

MeanCylindricalRadius

((NIntegrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//Timing

(i=((Integrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"])))//Timing

N[i]

MeanSquareCylindricalRadius

((NIntegrate[(x^2)+(y^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//Timing

(i=((Integrate[(x^2)+(y^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//FullSimplify)//Timing

N[i]

MeanSphericalRadius

((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//Timing

(i=((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"])))//Timing

N[i]

MeanSquareSphericalRadius

((NIntegrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//Timing

(i=((Integrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//FullSimplify)//Timing

N[i]

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p,"Simplification"->FullSimplify,Debug->True])//Timing

Graphics3D[{{Opacity[.5],p},{Red,sphere}},Boxed->False]

RegionFunction

reg=PolyhedronData["TrigyrateRhombicosidodecahedron","RegionFunction"]

bounds=RegionBounds[ImplicitRegion[Evaluate[N[reg[x,y,z]]],{x,y,z}]]

ranges=Flatten/@Transpose[{{x,y,z},.1{-1,1}+#&/@bounds}]

RegionPlot3D[Evaluate[N[reg[x,y,z]]],Evaluate[Sequence@@ranges],BoxRatios->Automatic,PlotPoints->40]

ByteCount[ineq=PolyhedronInequalities[p,{#1,#2,#3}]]

ByteCount[ineq2=FullSimplify[ineq]]//Timing

ineq2

SimplePolyhedron

SimplePolyhedronQ[p]

SkeletonCoordinates

vc=PolyhedronData["TrigyrateRhombicosidodecahedron","V"]

g=Graph[Range[vc],UndirectedEdge@@@PolyhedronData["TrigyrateRhombicosidodecahedron","EdgeIndices"]]

r=RecognizeGraph[g]

GraphData[r,"Graph","All"]

ToCommonEdges[GraphData[r],g]

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

RootReduce[%]

SurfaceArea[p]//RootReduce//Timing

Total[Area/@PolyhedronData[pname,"Polygons"]]//RootReduce//Timing

VertexSubsetHulls

PolyhedronData[pname,"VertexSubsetHulls"]

Platonic

(h=VertexSubsetHulls[p,"Platonic","Skip"->{}])//Timing

Volume

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify//Timing

PolyhedronVolume[p,Method->"GaussLaw"]//FullSimplify//Timing

Construction

$Version

Construct from {"SmallRhombicosidodecahedron", "C3"} by rotating 3 cupolas

PolyhedronData/@{{"SmallRhombicosidodecahedron","C3"},"TrigyrateRhombicosidodecahedron"}

f=PolyhedronData[{"SmallRhombicosidodecahedron","C3"},"FaceIndices"]

SmallRhombicosidodecahedronFaceIndices={{36,23,27},{37,28,24},{40,8,7},{35,5,6},{38,29,25},{39,26,30},{10,14,2},{9,1,13},{12,4,16},{11,15,3},{54,45,41},{55,42,46},{58,19,20},{53,18,17},{56,43,47},{57,48,44},{34,32,22},{33,21,31},{59,51,49},{60,50,52},{27,31,21,36},{23,36,1,9},{10,2,37,24},{37,22,32,28},{8,40,30,26},{25,29,40,7},{35,27,23,5},{6,24,28,35},{3,38,25,11},{21,33,29,38},{39,30,34,22},{12,26,39,4},{55,14,10,42},{41,9,13,54},{57,44,12,16},{15,11,43,56},{45,54,59,49},{50,60,55,46},{48,58,20,44},{43,19,58,47},{53,17,41,45},{46,42,18,53},{59,56,47,51},{52,48,57,60},{31,32,34,33},{17,18,6,5},{1,3,15,13},{14,16,4,2},{7,8,20,19},{51,52,50,49},{3,1,36,21,38},{22,37,2,4,39},{29,33,34,30,40},{27,35,28,32,31},{42,10,24,6,18},{41,17,5,23,9},{20,8,26,12,44},{11,25,7,19,43},{56,59,54,13,15},{57,16,14,55,60},{58,48,52,51,47},{49,50,46,53,45}};

Length[f]

v=PolyhedronData[{"SmallRhombicosidodecahedron","C3"},"VertexCoordinates"]

v={{-(1/2),(1/(2 (Sqrt[3]))),-((3+2 (Sqrt[5]))/(2 (Sqrt[3])))},{-(1/2),-((2+(Sqrt[5]))/(2 (Sqrt[3]))),(Sqrt[(7/4)+((2 (Sqrt[5]))/3)])},{-(1/2),(Sqrt[(3/4)+((Sqrt[5])/3)]),-((4+(Sqrt[5]))/(2 (Sqrt[3])))},{-(1/2),-(1/(2 (Sqrt[3]))),(1/2) (Sqrt[(29/3)+4 (Sqrt[5])])},{-(1/2),-((4+(Sqrt[5]))/(2 (Sqrt[3]))),-((2+(Sqrt[5]))/(2 (Sqrt[3])))},{-(1/2),-((3+2 (Sqrt[5]))/(2 (Sqrt[3]))),-(1/(2 (Sqrt[3])))},{-(1/2),(Sqrt[(5/3)])+((Sqrt[3])/2),(1/(2 (Sqrt[3])))},{-(1/2),(Sqrt[(7/4)+((2 (Sqrt[5]))/3)]),(Sqrt[(3/4)+((Sqrt[5])/3)])},{0,-(1/(Sqrt[3])),-(1/2) (Sqrt[(29/3)+4 (Sqrt[5])])},{0,-(Sqrt[(2/3) (3+(Sqrt[5]))]),(Sqrt[(3/4)+((Sqrt[5])/3)])},{0,(Sqrt[(2/3) (3+(Sqrt[5]))]),-((2+(Sqrt[5]))/(2 (Sqrt[3])))},{0,(1/(Sqrt[3])),(Sqrt[(5/3)])+((Sqrt[3])/2)},{(1/2),(1/(2 (Sqrt[3]))),-((3+2 (Sqrt[5]))/(2 (Sqrt[3])))},{(1/2),-((2+(Sqrt[5]))/(2 (Sqrt[3]))),(Sqrt[(7/4)+((2 (Sqrt[5]))/3)])},{(1/2),(Sqrt[(3/4)+((Sqrt[5])/3)]),-((4+(Sqrt[5]))/(2 (Sqrt[3])))},{(1/2),-(1/(2 (Sqrt[3]))),(1/2) (Sqrt[(29/3)+4 (Sqrt[5])])},{(1/2),-((4+(Sqrt[5]))/(2 (Sqrt[3]))),-((2+(Sqrt[5]))/(2 (Sqrt[3])))},{(1/2),-((3+2 (Sqrt[5]))/(2 (Sqrt[3]))),-(1/(2 (Sqrt[3])))},{(1/2),(Sqrt[(5/3)])+((Sqrt[3])/2),(1/(2 (Sqrt[3])))},{(1/2),(Sqrt[(7/4)+((2 (Sqrt[5]))/3)]),(Sqrt[(3/4)+((Sqrt[5])/3)])},{(1/4) (-5-(Sqrt[5])),(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))]),-(Sqrt[(3/4)+((Sqrt[5])/3)])},{(1/4) (-5-(Sqrt[5])),-(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))]),(Sqrt[(3/4)+((Sqrt[5])/3)])},{(1/4) (-1-(Sqrt[5])),-(1/2) (Sqrt[(5/6) (3+(Sqrt[5]))]),-(Sqrt[(7/4)+((2 (Sqrt[5]))/3)])},{(1/4) (-1-(Sqrt[5])),-((7+3 (Sqrt[5]))/(4 (Sqrt[3]))),((Sqrt[(5/3)])/2)},{(1/4) (-1-(Sqrt[5])),(Sqrt[(47/24)+((7 (Sqrt[5]))/8)]),-((Sqrt[(5/3)])/2)},{(1/4) (-1-(Sqrt[5])),(1/2) (Sqrt[(5/6) (3+(Sqrt[5]))]),(Sqrt[(7/4)+((2 (Sqrt[5]))/3)])},{(1/2) (-1-(Sqrt[5])),-(Sqrt[(1/6) (3+(Sqrt[5]))]),-(Sqrt[(3/4)+((Sqrt[5])/3)])},{(1/2) (-1-(Sqrt[5])),(2/((Sqrt[3]) (-3+(Sqrt[5])))),(1/(2 (Sqrt[3])))},{(1/2) (-1-(Sqrt[5])),(Sqrt[(1/6) (7+3 (Sqrt[5]))]),-(1/(2 (Sqrt[3])))},{(1/2) (-1-(Sqrt[5])),(Sqrt[(1/6) (3+(Sqrt[5]))]),(Sqrt[(3/4)+((Sqrt[5])/3)])},{-1-((Sqrt[5])/2),-(1/(2 (Sqrt[3]))),-((Sqrt[(5/3)])/2)},{-1-((Sqrt[5])/2),-((Sqrt[(5/3)])/2),(1/(2 (Sqrt[3])))},{-1-((Sqrt[5])/2),((Sqrt[(5/3)])/2),-(1/(2 (Sqrt[3])))},{-1-((Sqrt[5])/2),(1/(2 (Sqrt[3]))),((Sqrt[(5/3)])/2)},{(1/(-3+(Sqrt[5]))),-(1/2) (Sqrt[(5/6) (7+3 (Sqrt[5]))]),-((Sqrt[(5/3)])/2)},{(1/(-3+(Sqrt[5]))),(NumericalApproximation[Root, (RowDefault["-0.178", FEExpressionsNumericalApproximationElider]), Root[1-36 ((#1)^2)+144 ((#1)^4)&,2], ]),-((4+(Sqrt[5]))/(2 (Sqrt[3])))},{(1/(-3+(Sqrt[5]))),-((7+(Sqrt[5]))/(4 (Sqrt[3]))),(Sqrt[(3/4)+((Sqrt[5])/3)])},{(1/(-3+(Sqrt[5]))),((7+(Sqrt[5]))/(4 (Sqrt[3]))),-((2+(Sqrt[5]))/(2 (Sqrt[3])))},{(1/(-3+(Sqrt[5]))),(NumericalApproximation[Root, (RowDefault["0.178", FEExpressionsNumericalApproximationElider]), Root[1-36 ((#1)^2)+144 ((#1)^4)&,3], ]),(Sqrt[(7/4)+((2 (Sqrt[5]))/3)])},{(1/(-3+(Sqrt[5]))),(Sqrt[(35/24)+((5 (Sqrt[5]))/8)]),((Sqrt[(5/3)])/2)},{(1/4) (1+(Sqrt[5])),-(1/2) (Sqrt[(5/6) (3+(Sqrt[5]))]),-(Sqrt[(7/4)+((2 (Sqrt[5]))/3)])},{(1/4) (1+(Sqrt[5])),-((7+3 (Sqrt[5]))/(4 (Sqrt[3]))),((Sqrt[(5/3)])/2)},{(1/4) (1+(Sqrt[5])),(Sqrt[(47/24)+((7 (Sqrt[5]))/8)]),-((Sqrt[(5/3)])/2)},{(1/4) (1+(Sqrt[5])),(1/2) (Sqrt[(5/6) (3+(Sqrt[5]))]),(Sqrt[(7/4)+((2 (Sqrt[5]))/3)])},{(1/2) (1+(Sqrt[5])),-(Sqrt[(1/6) (3+(Sqrt[5]))]),-(Sqrt[(3/4)+((Sqrt[5])/3)])},{(1/2) (1+(Sqrt[5])),(2/((Sqrt[3]) (-3+(Sqrt[5])))),(1/(2 (Sqrt[3])))},{(1/2) (1+(Sqrt[5])),(Sqrt[(1/6) (7+3 (Sqrt[5]))]),-(1/(2 (Sqrt[3])))},{(1/2) (1+(Sqrt[5])),(Sqrt[(1/6) (3+(Sqrt[5]))]),(Sqrt[(3/4)+((Sqrt[5])/3)])},{(1/2) (2+(Sqrt[5])),-(1/(2 (Sqrt[3]))),-((Sqrt[(5/3)])/2)},{(1/2) (2+(Sqrt[5])),-((Sqrt[(5/3)])/2),(1/(2 (Sqrt[3])))},{(1/2) (2+(Sqrt[5])),((Sqrt[(5/3)])/2),-(1/(2 (Sqrt[3])))},{(1/2) (2+(Sqrt[5])),(1/(2 (Sqrt[3]))),((Sqrt[(5/3)])/2)},{(1/4) (3+(Sqrt[5])),-(1/2) (Sqrt[(5/6) (7+3 (Sqrt[5]))]),-((Sqrt[(5/3)])/2)},{(1/4) (3+(Sqrt[5])),(NumericalApproximation[Root, (RowDefault["-0.178", FEExpressionsNumericalApproximationElider]), Root[1-36 ((#1)^2)+144 ((#1)^4)&,2], ]),-((4+(Sqrt[5]))/(2 (Sqrt[3])))},{(1/4) (3+(Sqrt[5])),-((7+(Sqrt[5]))/(4 (Sqrt[3]))),(Sqrt[(3/4)+((Sqrt[5])/3)])},{(1/4) (3+(Sqrt[5])),((7+(Sqrt[5]))/(4 (Sqrt[3]))),-((2+(Sqrt[5]))/(2 (Sqrt[3])))},{(1/4) (3+(Sqrt[5])),(NumericalApproximation[Root, (RowDefault["0.178", FEExpressionsNumericalApproximationElider]), Root[1-36 ((#1)^2)+144 ((#1)^4)&,3], ]),(Sqrt[(7/4)+((2 (Sqrt[5]))/3)])},{(1/4) (3+(Sqrt[5])),(Sqrt[(35/24)+((5 (Sqrt[5]))/8)]),((Sqrt[(5/3)])/2)},{(1/4) (5+(Sqrt[5])),(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))]),-(Sqrt[(3/4)+((Sqrt[5])/3)])},{(1/4) (5+(Sqrt[5])),-(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))]),(Sqrt[(3/4)+((Sqrt[5])/3)])}};

ClosedLine[l_]:={Red,AbsoluteThickness[10],Line[Append[l,l[[1]]]]}

pentagonindices={53,55,61};

Graphics3D[{
,
MapIndexed[Text[#2[[1]],#]&,Mean[v[[#]]]&/@SmallRhombicosidodecahedronFaceIndices],
ClosedLine[v[[f[[#]]]]]&/@pentagonindices
}]

pentagons=SmallRhombicosidodecahedronFaceIndices[[pentagonindices]]

fnew=Select[SmallRhombicosidodecahedronFaceIndices,MatchQ[Length[Intersection[Flatten[pentagons],#]],0|5]&]

rotaxes=Mean[v[[SmallRhombicosidodecahedronFaceIndices[[#]]]]]&/@pentagonindices//FullSimplify

Transpose[{rotaxes,pentagonindices}]

rotpents=(Function[vv,RotationMatrix[π/5,#].vv]/@v[[SmallRhombicosidodecahedronFaceIndices[[#2]]]])&@@@Transpose[{rotaxes,pentagonindices}]//FullSimplify

vnew=ReplacePart[v,Rule@@@Flatten[Thread/@Transpose[{pentagons,rotpents}],1]]

By hand

Length[vnew]

Length[fnew]

PolyhedronData["TrigyrateRhombicosidodecahedron",{"V","F"}]

SmallestRotation[l_]:=Sort[NestList[RotateRight,l,Length[l]-1]][[1]]

faddl={};

f2=Sort[SmallestRotation/@Join[fnew,faddl]]

Graphics3D[{p=Polyhedron[vnew,f2],MapIndexed[Text[#2[[1]],#,Background->White]&,vnew]}]

ConvexHull

Show[hull=ConvexHullRegion[vnew],Boxed->True]

SmallestRotation[l_]:=Sort[NestList[RotateRight,l,Length[l]-1]][[1]]

newf=Sort[SmallestRotation/@MeshCells[hull,2][[All,1]]]

p=Polyhedron[vnew,newf]

ConvexPolyhedronQ[p]

SimplePolyhedronQ[p]

Volume[p//N]

PolyhedronData["TrigyrateRhombicosidodecahedron","Volume"]