CITE THIS NOTEBOOK: Common faces of icosahedron and truncated cube by Sándor Kabai. Wolfram Community NOV 2 2022.
Truncated cube (and Archimedean solid) has four faces coinciding with those of icosahedron. This demonstration starts with ICO and TC embedded in a cube. The edges are 1 (ICO), 1.618 (cube), 0.6702 (TC). The size of ICO is increased by 1.4920 = 1/6702 to bring its faces to those of TC.
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fi=GoldenRatio;bet=ArcTan[1/fi];ico=GraphicsComplex[PolyhedronData["Icosahedron","VertexCoordinates"],Polygon[PolyhedronData["Icosahedron","FaceIndices"]]];cubt=GraphicsComplex[PolyhedronData["Cube","VertexCoordinates"],Tube[PolyhedronData["Cube","FaceIndices"]]];cubtn=Scale[cubt,fi];tc=GraphicsComplex[PolyhedronData["TruncatedCube","VertexCoordinates"],Polygon[PolyhedronData["TruncatedCube","FaceIndices"]]];aff=fi/(1+2/Sqrt[2]);tcn=Scale[tc,aff];Manipulate[icon=Scale[Rotate[ico,-bet,{0,1,0}],en/aff];Graphics3D[{Pink,cubtn,Yellow,icon,Cyan,tcn},BoxedFalse,SphericalRegionTrue,PlotRange1.3,ViewPoint{4,2,2},ViewAngle0.23],{{en,0.67,"enlarge icosahedron"},0.67,0.999},TrackedSymbols->en,SaveDefinitions->True]
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