Disdyakis dodecahedron

Disdyakis dodecahedron
Click on picture for large version Spinning version
Type	Catalan
Face polygon	scalene triangle
Faces	48
Edges	72
Vertices	26 = 6 + 8 + 12
Face configuration	V4.6.8
Symmetry group	Oh, [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 $\tfrac{1}{7}\scriptstyle{\sqrt{3(2194+1513\sqrt{2})}}$.

See also

• Disdyakis triacontahedron
• Bisected hexagonal tiling
• Great rhombihexacron—A uniform dual polyhedron with the same surface topology

References

• 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-Strass, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, kisRhombic dodecahedron)

External links

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