Disdyakis dodecahedron  

(rotating and 3D model)  
Type  Catalan solid 
Conway notation  mC 
Coxeter diagram  
Face polygon  scalene triangle 
Faces  48 
Edges  72 
Vertices  26 = 6 + 8 + 12 
Face configuration  V4.6.8 
Symmetry group  O_{h}, B_{3}, [4,3], *432 
Dihedral angle  155° 4' 56" 
Dual polyhedron  truncated cuboctahedron 
Properties  convex, facetransitive 
net 
In geometry, a disdyakis dodecahedron, (also hexoctahedron,^{[1]} hexakis octahedron, octakis cube, octakis hexahedron, kisrhombic dodecahedron^{[2]}), is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is facetransitive 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.^{[3]} The net of the rhombic dodecahedral pyramid also shares the same topology.
It has O_{h} octahedral symmetry. Its collective edges represent the reflection planes of the symmetry. It can also be seen in the corner and midedge triangulation of the regular cube and octahedron, and rhombic dodecahedron.
Disdyakis dodecahedron 
Deltoidal icositetrahedron 
Rhombic dodecahedron 
Hexahedron 
Octahedron 
Spherical polyhedron  

(see rotating model)  Orthographic projections from 2, 3 and 4fold axes 
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.
Stereographic projections  

2fold  3fold  4fold  
Let .
Then the Cartesian coordinates for the vertices of a disdyakis dodecahedron centered at the origin are:
● permutations of (±a, 0, 0) (vertices of an octahedron)
● permutations of (±b, ±b, 0) (vertices of a cuboctahedron)
● (±c, ±c, ±c) (vertices of a cube)
Convex hulls 

Combining an octahedron, cube, and cuboctahedron to form the disdyakis dodecahedron. The convex hulls for these vertices^{[4]} scaled by result in Cartesian coordinates of unit circumradius, which are visualized in the figure below: 
If its smallest edges have length a, its surface area and volume are
The faces are scalene triangles. Their angles are , and .
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.
Projective symmetry 
[4]  [3]  [2]  [2]  [2]  [2]  [2]^{+} 

Image  
Dual image 
Polyhedra similar to the disdyakis dodecahedron are duals to the Bowtie octahedron and cube, containing extra pairs triangular faces .^{[5]} 
The disdyakis dodecahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.
Uniform octahedral polyhedra  

Symmetry: [4,3], (*432)  [4,3]^{+} (432) 
[1^{+},4,3] = [3,3] (*332) 
[3^{+},4] (3*2)  
{4,3}  t{4,3}  r{4,3} r{3^{1,1}} 
t{3,4} t{3^{1,1}} 
{3,4} {3^{1,1}} 
rr{4,3} s_{2}{3,4} 
tr{4,3}  sr{4,3}  h{4,3} {3,3} 
h_{2}{4,3} t{3,3} 
s{3,4} s{3^{1,1}} 
= 
= 
= 
= or 
= or 
=  




 
Duals to uniform polyhedra  
V4^{3}  V3.8^{2}  V(3.4)^{2}  V4.6^{2}  V3^{4}  V3.4^{3}  V4.6.8  V3^{4}.4  V3^{3}  V3.6^{2}  V3^{5} 
It is a polyhedra in a sequence defined by the face configuration V4.6.2n. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any n ≥ 7.
With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors.
Each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3,n mirrors at each triangle face vertex.
*n32 symmetry mutation of omnitruncated tilings: 4.6.2n  

Sym. *n32 [n,3] 
Spherical  Euclid.  Compact hyperb.  Paraco.  Noncompact hyperbolic  
*232 [2,3] 
*332 [3,3] 
*432 [4,3] 
*532 [5,3] 
*632 [6,3] 
*732 [7,3] 
*832 [8,3] 
*∞32 [∞,3] 
[12i,3] 
[9i,3] 
[6i,3] 
[3i,3]  
Figures  
Config.  4.6.4  4.6.6  4.6.8  4.6.10  4.6.12  4.6.14  4.6.16  4.6.∞  4.6.24i  4.6.18i  4.6.12i  4.6.6i 
Duals  
Config.  V4.6.4  V4.6.6  V4.6.8  V4.6.10  V4.6.12  V4.6.14  V4.6.16  V4.6.∞  V4.6.24i  V4.6.18i  V4.6.12i  V4.6.6i 
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n  

Symmetry *n42 [n,4] 
Spherical  Euclidean  Compact hyperbolic  Paracomp.  
*242 [2,4] 
*342 [3,4] 
*442 [4,4] 
*542 [5,4] 
*642 [6,4] 
*742 [7,4] 
*842 [8,4]... 
*∞42 [∞,4]  
Omnitruncated figure 
4.8.4 
4.8.6 
4.8.8 
4.8.10 
4.8.12 
4.8.14 
4.8.16 
4.8.∞ 
Omnitruncated duals 
V4.8.4 
V4.8.6 V4.8.8 
V4.8.10 
V4.8.12 
V4.8.14 
V4.8.16 
V4.8.∞ 