Dodecahedron regular dodecahedron but less symmetrical: the pyritohedron with pyritohedral symmetry, and the tetartoid with tetrahedral symmetry: A pyritohedron is a dodecahedron with pyritohedral (T) symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not constrained to be regular, and the underlying atomic arrangement has no true fivefold symmetry axes. Its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of rotational symmetry are three mutually perpendicular twofold axes and four threefold axes. Although regular
Dodecahedron of the wedge-shaped "roof" above the faces of the cube. When "h" = 1, the six cross-edges degenerate to points and a rhombic dodecahedron is formed. When "h" = 0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When "h" = , the multiplicative inverse of the golden ratio, the result is a regular dodecahedron. When "h" = , the conjugate of this value, the result is a regular great stellated dodecahedron. A reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give