Disdyakis dodecahedron
(rotating and3D 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	Oh, B3, [4,3], *432
Dihedral angle	155° 4' 56" ${\displaystyle \arccos (-\frac{71+12\sqrt{2}}{97})}$
Dual polyhedron	truncated cuboctahedron
Properties	convex,face-transitive
net

Ingeometry, adisdyakis dodecahedron, (alsohexoctahedron,^[1]hexakis octahedron,octakis cube,octakis hexahedron,kisrhombic dodecahedron^[2]) ord48, is aCatalan solid with 48 faces and the dual to theArchimedeantruncated cuboctahedron. As such it isface-transitive but with irregular face polygons. It resembles an augmentedrhombic dodecahedron. Replacing each face of the rhombic dodecahedron with a flat pyramid results in theKleetope of the rhombic dodecahedron, which looks almost like the disdyakis dodecahedron, and istopologically equivalent to it.^[a] The net of therhombic dodecahedral pyramid also shares the same topology.

It has O_hoctahedral 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.

Disdyakis dodecahedron

Deltoidal icositetrahedron

Rhombic dodecahedron

Hexahedron

Octahedron

Spherical polyhedron
(seerotating model)	Orthographic projections from 2-, 3- and 4-fold axes

The edges of a spherical disdyakis dodecahedron belong to 9great circles. Three of them form a spherical octahedron (gray in the images below). The remaining six form three squarehosohedra (red, green and blue in the images below). They all correspond tomirror planes - the former indihedral [2,2], and the latter intetrahedral [3,3] symmetry. A spherical disdyakis dodecahedron can be thought of as thebarycentric subdivision of thespherical cube or of thespherical octahedron.^[3]

Stereographic projections
	2-fold	3-fold	4-fold

Let${\displaystyle a=\frac{1}{1+2\sqrt{2}}\approx 0.261,b=\frac{1}{2+3\sqrt{2}}\approx 0.160,c=\frac{1}{3+3\sqrt{2}}\approx 0.138}$.
Then theCartesian 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 acuboctahedron)
●   (±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${\displaystyle 1/a}$ result in Cartesian coordinates of unitcircumradius, which are visualized in the figure below:

If its smallest edges have lengtha, its surface area and volume are

The faces are scalene triangles. Their angles are${\displaystyle \arccos (\frac{1}{6}-\frac{1}{12}\sqrt{2})\approx {87.201}^{\circ }}$,${\displaystyle \arccos (\frac{3}{4}-\frac{1}{8}\sqrt{2})\approx {55.024}^{\circ }}$ and${\displaystyle \arccos (\frac{1}{12}+\frac{1}{2}\sqrt{2})\approx {37.773}^{\circ }}$.

The truncated cuboctahedron and its dual, thedisdyakis 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 theBowtie 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{31,1}	t{3,4} t{31,1}	{3,4} {31,1}	rr{4,3} s2{3,4}	tr{4,3}	sr{4,3}	h{4,3} {3,3}	h2{4,3} t{3,3}	s{3,4} s{31,1}
		=	=	=				= or	= or	=
Duals to uniform polyhedra
V43	V3.82	V(3.4)2	V4.62	V34	V3.43	V4.6.8	V34.4	V33	V3.62	V35

It is a polyhedra in a sequence defined by theface 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 anyn ≥ 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 asymmetry group with order 2,3,n mirrors at each triangle face vertex.

*n32 symmetry mutation of omnitruncated tilings: 4.6.2n v t e
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 v t e
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.∞

• First stellation of rhombic dodecahedron
• Disdyakis triacontahedron
• Kisrhombille tiling
• Great rhombihexacron—A uniform dual polyhedron with the same surface topology

• Williams, Robert (1979).The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc.ISBN 0-486-23729-X. (Section 3-9)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss,ISBN 978-1-56881-220-5[1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, kisRhombic dodecahedron)

• Weisstein, Eric W., "Disdyakis dodecahedron" ("Catalan solid") atMathWorld.
• Disdyakis Dodecahedron (Hexakis Octahedron) Interactive Polyhedron Model

• Catalan solids

• Articles with short description
• Short description is different from Wikidata