Rhombic Hexecontahedron

The rhombic hexecontahedron is a 60-faced polyhedron that can be obtained by stellating the rhombic triacontahedron by placing a plane along each edge which is perpendicular to the plane of symmetry in which the edge lies, and taking the solid bounded by these planes gives a hexecontahedron (Steinhaus 1999). It is therefore a rhombic triacontahedron stellation. It appears to have been first noted and illustrated by Unkelbach (1940) as one of 20 finite equilateral polyhedra whose edges lie in planes of symmetry and whose faces are convex polygons which do not penetrate one another.

The 60 faces of the rhombic hexecontahedron are golden rhombi (Kabai 2002, p. 179).

Amazingly, the rhombic hexecontahedron is inferred to exist in nature as the central core of a quasicrystal aggregate of produced by slow solidification (Guyot 1987).

The rhombic hexecontahedron is implemented in the Wolfram Language as PolyhedronData["RhombicHexecontahedron"]. It is also the logo for the Wolfram|Alpha web site https://www.wolframalpha.com/ (Weisstein 2009).

The rhombic hexecontahedron can be constructed by extending the long edges of each rhombic face of the rhombic triacontahedron to obtain rhombi on either side of the original that are a factor of the golden ratio larger that the original central rhombus (Kabai 2002, p. 181).

A tetrahedron 10-compound, octahedron 5-compound, cube 5-compound, icosahedron, dodecahedron, and icosidodecahedron can be inscribed in the vertices of a rhombic hexecontahedron, illustrated above (E. Weisstein, Dec. 24-27, 2009).

20 golden rhombohedra can be combined to form a solid rhombic hexecontahedron. An addition 180 regular dodecahedra can be placed face-to-face to lie along the edges of a rhombic hexecontahedron (Kabai 2011, Fig. 40).

The skeleton of the rhombic hexecontahedron is the deltoidal hexecontahedral graph, illustrated above.

The rhombic hexecontahedron with edge length has surface area and volume given by

			(1)
			(2)

and inertia tensor

(3)