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
Regular dodecahedron such a regular dodecahedron is 1 : , or : 1, or (5 + ) : 4. For example, an embedded cube with a volume of 64 (and edge length of 4), will nest within a regular dodecahedron of volume 64 + 32"ϕ" (and edge length of 4"ϕ" − 4). Thus, the difference in volume between the encompassing regular dodecahedron and the enclosed cube is always one half the volume of the cube times "ϕ". From these ratios are derived simple formulas for the volume of a regular dodecahedron with edge length "a" in terms of the golden mean: Golden