This demo shows how the cube 5 compound (c5c) envelopes the rhombic triacontahedron (RT). The c5c is one of the many stellations of RT.
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Manipulate[fi=GoldenRatio;alf=ArcTan[fi];bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];rt=PolyhedronData["RhombicTriacontahedron","Faces"];rtn=Rotate[rt,Pi,{0,0,1}];rtl=GraphicsComplex[PolyhedronData["RhombicTriacontahedron","VertexCoordinates"],Tube[PolyhedronData["RhombicTriacontahedron","FaceIndices"],0.03]];c5c=PolyhedronData["Cube5Compound","Faces"];c5clip=Graphics3D[{ClipPlanes{0,0,-1,shell},c5c}][[1]];Graphics3D[{Yellow,Scale[rtl,0.36],FaceForm[LightRed,Cyan],c5clip},BoxedFalse,SphericalRegionTrue,ViewPoint{100,73,61},ImageSize300],{{shell,-0.66,"shell"},-0.7,1},TrackedSymbols{shell},SaveDefinitions->True]
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The edges of c5c for pentagrams.
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fi=GoldenRatio;alf=ArcTan[fi];bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];c5ct=GraphicsComplex[PolyhedronData["CubeFiveCompound","VertexCoordinates"],Tube[PolyhedronData["CubeFiveCompound","FaceIndices"],0.016]];Graphics3D[{Cyan,c5c,Yellow,c5ct},BoxedFalse,SphericalRegionTrue,ViewPoint{0,0,100},ImageSize300]
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The pentagrams are on the faces of the enveloping dodecahedron.
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dod=PolyhedronData["Dodecahedron","Faces"];dodn=Rotate[dod,Pi,{0,0,1}];dodnn=Rotate[Scale[dod,1/fi{1,1,1},{0,0,0}],Pi,{0,0,1}];Graphics3D[{{Opacity[0.3],dodn},Cyan,c5c,Yellow,c5ct},BoxedFalse,SphericalRegionTrue,ViewPoint{0,0,100},ImageSize300]
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Graphics3D[{{Opacity[0.8],dodnn},Cyan,c5c,Yellow,c5ct},BoxedFalse,SphericalRegionTrue,ViewPoint{0,0,100},ImageSize300]
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Constructing c5c from five cubes.
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(*Tiltthecubebyanangle"bet"*)cub=PolyhedronData["Cube","Faces"];cubn=Rotate[cub,bet,{1,0,0}];Graphics3D[{cubn},BoxedFalse,SphericalRegionTrue,ViewPoint{100,10,20},ImageSize100]
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(*Rotatethetiltedcubefivetimes*)cubn5=Table[Rotate[cubn,i2Pi/5,{0,0,1}],{i,5}];Graphics3D[{cubn5},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint->{0,0,100}]
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Constructing c5c from 30 squares placed of the faces of RT. The size of squares are increased in the planes of the faces of RT by fi^3.
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ticot=TruncatedPolyhedron[Icosahedron[],1/fi^2];ticotn=Rotate[Scale[ticot,0.996fi^2{1,1,1},{0,0,0}],Pi,{0,0,1}];tv=ticot[[1]];sp=Sphere[{0,0,0},0.1];sp60=Rotate[Map[Translate[sp,#]&,fi^2tv],Pi,{0,0,1}];y=0.5+0.5fi;x=y/fi;rte=Sqrt[x^2+y^2];el=yfi;Manipulate[fi=GoldenRatio;alf=ArcTan[fi];bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];v=expand0.426;vA={v,-v,0};vB={v,v,0};vC={-v,v,0};vD={-v,-v,0};sq={Polygon[{vA,vB,vC,vD}]};sq1=Translate[{sq},{0,0,0.5+el-fi/2}];sq2=Rotate[sq1,Pi,{1,0,0}];sq3=Rotate[Rotate[{sq1,sq2},Pi/2,{1,0,0}],Pi/2,{0,0,1}];sq4=Rotate[Rotate[{sq1,sq2},Pi/2,{0,1,0}],Pi/2,{0,0,1}];sq30=Table[Rotate[Rotate[{sq1,sq2,sq3,sq4},bet,{1,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}];rt=PolyhedronData["RhombicTriacontahedron","Faces"];rt10={Green,Scale[rt,1.314]};Graphics3D[{sp60,rt10,LightRed,sq30},Boxed->False,SphericalRegion->True,PlotRange->3,ViewPoint->{0,0,100},ViewAngle->0.012],{{expand,5/fi^2,"expand"},1,fi^3},TrackedSymbols:>expand,SaveDefinitions->True]
When the increase is 5 /fi^2, then the vertices meet, and are located at the vertices of golden truncated icosahedron (ticot) fi^2 larger.
Spheres are placed at the vertices of “ticot” (tv).
The 30 squares coincide with the faces of SmallRhombicosidodecahedron when elevated.
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Inside the cube 5 compound
by Sandor Kabai
Wolfram Community, STAFF PICKS, October 4, 2023
https://community.wolfram.com/groups/-/m/t/3027031
by Sandor Kabai
Wolfram Community, STAFF PICKS, October 4, 2023
https://community.wolfram.com/groups/-/m/t/3027031