Disdyakis dodecahedron


Disdyakis dodecahedron
Disdyakis dodecahedron
Disdyakis dodecahedron
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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
Disdyakis dodecahedron
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.

Contents

Symmetry

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

Octahedral reflection domains.png Disdyakis dodecahedron.png

Dimensions

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

See also

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