In geometry, a disdyakis dodecahedron, (also hexoctahedron, hexakis octahedron, octakis cube, octakis hexahedron, kisrhombic dodecahedron), is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It resembles an augmented rhombic dodecahedron. Replacing each face of the rhombic dodecahedron with a flat pyramid creates a polyhedron that looks almost like the disdyakis dodecahedron, and is topologically equivalent to it. More formally, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron, and the barycentric subdivision of the cube or of the regular octahedron. The net of the rhombic dodecahedral pyramid also shares the same topology. Symmetry It has Oh octahedral symmetry. Its collective edges represent the reflection planes of the symmetry. It can also be seen in the corner and mid-edge triangulation of the regular cube and octahedron, and rhombic dodecahedron. The edges of a spherical disdyakis dodecahedron belong to 9 great circles. Three of them form a spherical octahedron (gray in the images below). The remaining six form three square hosohedra (red, green and blue in the images below). They all correspond to mirror planes - the former in dihedral [2,2], and the latter in tetrahedral [3,3] symmetry. Cartesian coordinates Let a = 1/(1 + 2), b = 1/(2 + 3) and c = 1/. Then the Cartesian coordinates for the vertices of a disdyakis dodecahedron centered at the origin are all permutations of (±a, 0, 0) all permutations of (±b, ±b, 0) (±c, ±c, ±c). These numbers are based on the dual truncated cuboctahedron with an edge length of 2. The convex hulls for these vertices scaled by result in octahedron Cartesian coordinates of unit circumradius, which are visualized in the figure below: Dimensions If its smallest edges have length a, its surface area and volume are The faces are scalene triangles. Their angles are , and . Orthogonal projections The truncated cuboctahedron and its dual, the disdyakis dodecahedron can be drawn in a number of symmetric orthogonal projective orientations. Between a polyhedron and its dual, vertices and faces are swapped in positions, and edges are perpendicular. Related polyhedra and tilings The disdyakis dodecahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron. It is a polyhedra in a sequence defined by the face configuration V4.6.2n. This group is special for having all even number