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Disdyakis dodecahedron
Click on picture for large version Spinning version
Type	Catalan
Coxeter diagram	Template:CDD
Face polygon	scalene triangle
Faces	48
Edges	72
Vertices	26 = 6 + 8 + 12
Face configuration	V4.6.8
Symmetry group	Oh, BC3, [4,3], *432
Dihedral angle	155° 4' 56" $\arccos(-{\frac {71+12{\sqrt {2}}}{97}})$
Dual polyhedron	truncated cuboctahedron
Properties	convex, face-transitive
Net

In geometry, a disdyakis dodecahedron, or hexakis octahedron, is a Catalan solid and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It looks a bit like an inflated rhombic dodecahedron—if one replaces each face of the rhombic dodecahedron with a single vertex and four triangles in a regular fashion one ends up with a disdyakis dodecahedron. More formally, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron.

Symmetry

It has O_h octahedral symmetry. Its collective edges represent the reflection planes of the symmetry.

Dimensions

If its smallest edges have length 1, its surface area is ${\tfrac {6}{7}}\scriptstyle {\sqrt {783+436{\sqrt {2}}}}$ and its volume is ${\tfrac {1}{7}}\scriptstyle {\sqrt {3(2194+1513{\sqrt {2}})}}$.

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 $n\geq 7.$

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

• Disdyakis triacontahedron
• Bisected hexagonal 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-Strass, ISBN 978-1-56881-220-5 [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