Disdyakis dodecahedron

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

Cartesian coordinates

Let ~ a = \frac{1}{1 + 2 \sqrt{2}} ~ {\color{Gray} \approx 0.261}, ~~ b = \frac{1}{2 + 3 \sqrt{2}} ~ {\color{Gray} \approx 0.160}, ~~ c = \frac{1}{3 + 3 \sqrt{2}} ~ {\color{Gray} \approx 0.138}.

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)

Dimensions

If its smallest edges have length a, its surface area and volume are :\begin{align} A &= \tfrac67\sqrt{783+436\sqrt 2},a^2 \ V &= \tfrac17\sqrt{3\left(2194+1513\sqrt 2\right)}a^3\end{align}

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

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 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.

Notes

References

• (Section 3-9)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, kisRhombic dodecahedron)

References

1. "Keyword: "forms" {{!".
2. Conway, Symmetries of things, p.284
3. [http://www.cgl.uwaterloo.ca/csk/papers/bridges2001.html Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons] Craig S. Kaplan