Disdyakis dodecahedron

Disdyakis dodecahedron
Disdyakis dodecahedron (rotating and 3D model)
Type	Catalan solid
Conway notation	mC
Coxeter diagram	Template:CDD
Face polygon	File:DU11 facets.png 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" $\arccos (-\frac{71+12\sqrt{2}}{97})$
Dual polyhedron	File:Polyhedron great rhombi 6-8 max.png truncated cuboctahedron
Properties	convex, face-transitive
Disdyakis dodecahedron net

In geometry, a disdyakis dodecahedron, (also hexoctahedron, hexakis octahedron, octakis cube, octakis hexahedron, kisrhombic dodecahedron) or d48, 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 results in the Kleetope of the rhombic dodecahedron, which looks almost like the disdyakis dodecahedron, and is topologically equivalent to it.Template:Efn The net of the rhombic dodecahedral pyramid also shares the same topology.

Symmetry

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 mid-edge triangulation of the regular cube and octahedron, and rhombic dodecahedron.

File:Disdyakis 12.png Disdyakis dodecahedron

File:Disdyakis 12 in deltoidal 24.png Deltoidal icositetrahedron

File:Disdyakis 12 in rhombic 12.png Rhombic dodecahedron

File:Disdyakis 12 in Platonic 6.png Hexahedron

File:Disdyakis 12 in Platonic 8.png Octahedron

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. A spherical disdyakis dodecahedron can be thought of as the barycentric subdivision of the spherical cube or of the spherical octahedron.

Cartesian coordinates

Let $a=\frac{1}{1+2\sqrt{2}}\approx 0.261,b=\frac{1}{2+3\sqrt{2}}\approx 0.160,c=\frac{1}{3+3\sqrt{2}}\approx 0.138$.
Then the Cartesian coordinates for the vertices of a disdyakis dodecahedron centered at the origin are:

Template:Ifsubst style="color:#eb2424">●   permutations of (±a, 0, 0)   (vertices of an octahedron)
Template:Ifsubst style="color:#3061d6">●   permutations of (±b, ±b, 0)   (vertices of a cuboctahedron)
Template:Ifsubst style="color:#f9b900">●   (±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 scaled by $1/a$ result in Cartesian coordinates of unit circumradius, which are visualized in the figure below:
Combining an octahedron, cube, and cuboctahedron to form the disdyakis dodecahedron

Dimensions

If its smallest edges have length a, its surface area and volume are

$\begin{array}{rl}A& =\frac{6}{7}\sqrt{783+436\sqrt{2}}{a}^2\\ V& =\frac{1}{7}\sqrt{3(2194+1513\sqrt{2})}{a}^3\end{array}$

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

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.

Projective symmetry	[4]	[3]	[2]	[2]	[2]	[2]	[2]+
Image	File:Dual cube t012 B2.png	File:Dual cube t012.png	File:Dual cube t012 f4.png	File:Dual cube t012 e46.png	File:Dual cube t012 e48.png	File:Dual cube t012 e68.png	File:Dual cube t012 v.png
Dual image	File:3-cube t012 B2.svg	File:3-cube t012.svg	File:Cube t012 f4.png	File:Cube t012 e46.png	File:Cube t012 e48.png	File:Cube t012 e68.png	File:Cube t012 v.png

Related polyhedra and tilings

File:Conway polyhedron m3O.png	File:Conway polyhedron m3C.png
Polyhedra similar to the disdyakis dodecahedron are duals to the Bowtie octahedron and cube, containing extra pairs triangular faces .

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 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. Template:Omnitruncated table

See also

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

References

• Template:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, Template:ISBN [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, kisRhombic dodecahedron)

External links

• Template:Mathworld2
• Disdyakis Dodecahedron (Hexakis Octahedron) Interactive Polyhedron Model