Rhombic hexecontahedron

In geometry, a rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry. It was discovered in 1940 by Helmut Unkelbach.

The rhombic hexecontahedron can be dissected into 20 acute golden rhombohedra meeting at a central point. This gives the volume of a hexecontahedron of side length a to be ${\displaystyle V=(10+2{\sqrt {5}})a^{3}}$ and the area to be ${\displaystyle A=(24{\sqrt {5}})a^{2}}$.

A rhombic hexecontahedron can be constructed from a regular dodecahedron, by taking its vertices, its face centers and its edge centers and scaling them in or out from the body center to different extents. Thus, if the 20 vertices of a dodecahedron are pulled out to increase the circumradius by a factor of (ϕ+1)/2 ≈ 1.309, the 12 face centers are pushed in to decrease the inradius to (3-ϕ)/2 ≈ 0.691 of its original value, and the 30 edge centers are left unchanged, then a rhombic hexecontahedron is formed. (The circumradius is increased by 30.9% and the inradius is decreased by the same 30.9%.) Scaling the points by different amounts results in hexecontahedra with kite-shaped faces or other polyhedra.

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