Disdyakis dodecahedron

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.

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

| last1 = Langer | first1 = Joel C.

| last2 = Singer | first2 = David A.

| doi = 10.1007/s00032-010-0124-5

| issue = 2

| journal = Milan Journal of Mathematics

| mr = 2781856

| pages = 643–682

| title = Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem

| volume = 78

| year = 2010}}

==Cartesian coordinates==

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

Then the Cartesian coordinates for the vertices of a disdyakis dodecahedron centered at the origin are:

{{color|#eb2424|●}}   permutations of (±a, 0, 0)   (vertices of an octahedron)

{{color|#3061d6|●}}   permutations of (±b, ±b, 0)   (vertices of a cuboctahedron)

{{color|#f9b900|●}}   (±c, ±c, ±c)   (vertices of a cube)

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

:

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

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.

The disdyakis dodecahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.

{{Octahedral truncations}}

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.

{{Omnitruncated table}}

{{Omnitruncated4 table}}

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

{{notelist}}

{{reflist}}

• {{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, {{ISBN|978-1-56881-220-5}} [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)

• {{Mathworld2 |urlname=DisdyakisDodecahedron |title=Disdyakis dodecahedron |urlname2=CatalanSolid |title2=Catalan solid}}
• [https://web.archive.org/web/20070701185441/http://polyhedra.org/poly/show/37/hexakis_octahedron Disdyakis Dodecahedron (Hexakis Octahedron)] Interactive Polyhedron Model

{{Catalan solids}}

{{Polyhedron navigator}}

Category:Catalan solids