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
Pentakis snub dodecahedron Pentakis snub dodecahedron The pentakis snub dodecahedron is a convex polyhedron with 140 triangular faces, 210 edges, and 72 vertices. It has chiral icosahedral symmetry. Its name comes from a topological construction from the snub dodecahedron with the kis operator applied to the pentagonal faces. In this construction, all the vertices are computed to be the same distance from the center. The 80 of the triangles are equilateral, and 60 triangles from the pentagons are isosceles. It is a (2,1) geodesic polyhedron, made of all triangles. The path between the valence-5 vertices is two edges in a row, and then