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
Tetrahedrally diminished dodecahedron triangles are always equilateral, while the quadrilaterals are irregular, although the two adjacent edges that meet at the vertices of a tetrahedron are equal. As a self-dual hexadecahedron, it is one of 302404 forms, 1476 with at least order 2 symmetry, and the only one with tetrahedral symmetry. As a diminished regular dodecahedron, with 4 vertices removed, the quadrilaterals faces are trapezoids. As a stellation of the regular icosahedron it is one of 32 stellations defined with tetrahedral symmetry. It has kite faces. In Conway polyhedron notation, it can represented as pT, applying George W. Hart's propeller operator to a