Disdyakis triacontahedron

In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place. If the bipyramids, the gyroelongated bipyramids, and the trapezohedra are excluded, the disdyakis triacontahedron has the most faces of any other strictly convex polyhedron where every face of the polyhedron has the same shape. Projected into a sphere, the edges of a disdyakis triacontahedron define 15 great circles. Buckminster Fuller used these 15 great circles, along with 10 and 6 others in two other polyhedra to define his 31 great circles of the spherical icosahedron. Being a Catalan solid with triangular faces, the disdyakis triacontahedron's three face angles and common dihedral angle must obey the following constraints analogous to other Catalan solids: The above four equations are solved simultaneously to get the following face angles and dihedral angle: where is the golden ratio. As with all Catalan solids, the dihedral angles at all edges are the same, even though the edges may be of different lengths. The 62 vertices of a disdyakis triacontahedron are given by: Twelve vertices and their cyclic permutations, Eight vertices , Twelve vertices and their cyclic permutations, Six vertices and their cyclic permutations. Twenty-four vertices and their cyclic permutations, where and is the golden ratio.