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
First stellation of rhombic dodecahedron First stellation of rhombic dodecahedron In geometry, the first stellation of the rhombic dodecahedron is a stellation of the rhombic dodecahedron. This polyhedron is sometimes called "Escher's solid"; it appears in M. C. Escher's works "Waterfall" and in a study for "Stars" (although "Stars" itself features a different shape, the compound of three octahedra). It has 48 faces, all triangles. It has 26 vertices in total: 6 with degree 8, 8 with degree 6, and 12 with degree 4. It has 72 edges in total, giving an Euler characteristic of 26 + 48 − 72 = +2. Its vertices are