The icosahedron – the icosidodecahedron

If you then draw a regular triangle inside of each hexagon, so that the tips of the triangles touch the middle of three of the edges of the hexagon, the outline that the edges of those triangles inscribe on the surface of the polyhedron is an icosidodecahedron. (You can also arrive at the icosidodecahedron by inscribing each face of a dodecahedron with a pentagon that has been rotated 72 degrees, and then slicing off the resulting triangles from each vertex.)

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