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. Cartesian coordinates The 62 vertices of the disdyakis triacontahedron fall in three sets: Twelve vertices are of the form and its cyclic permutations, and they form a regular icosahedron among themselves. Twenty vertices are of the form or and its cyclic permutations, that together form a regular dodecahedron. When the above 32 vertices are taken together, they form the vertices of a rhombic triacontahedron, whose 30 face centers form an icosidodecahedron with coordinates and and their cyclic permutations. If those 30 face centers are scaled outwards from the origin by a factor, yielding and then they and their cyclic permutations form the last 30 vertices of the disdyakis triacontahedron. These hulls are visualized in the figure below scaled so the icosahedron has a unit circumradius: Faces The faces of a disdyakis triacontahedron are scalene triangles. If is the golden ratio then their angles are equal to , and . Symmetry The edges of the polyhedron projected onto a sphere form 15 great circles, and represent all 15 mirror planes of reflective Ih icosahedral symmetry. Combining pairs of light and dark triangles define the fundamental domains of the nonreflective (I) icosahedral symmetry. The edges of a compound of five octahedra also represent the 10 mirror planes of icosahedral symmetry. Orthogonal