Disdyakis dodecahedron

In geometry, a disdyakis dodecahedron, (also hexoctahedron,[1] hexakis octahedron, octakis cube, octakis hexahedron, kisrhombic dodecahedron[2]), is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It resembles an augmented rhombic dodecahedron. Replacing each face of the rhombic dodecahedron with a flat pyramid creates a polyhedron that looks almost like the disdyakis dodecahedron, and is topologically equivalent to it. More formally, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron. The net of the rhombic dodecahedral pyramid also shares the same topology.

Disdyakis dodecahedron

(rotating and 3D model)
TypeCatalan solid
Conway notationmC
Coxeter diagram
Face polygon
scalene triangle
Faces48
Edges72
Vertices26 = 6 + 8 + 12
Face configurationV4.6.8
Symmetry groupOh, B3, [4,3], *432
Dihedral angle155° 4' 56"
Dual polyhedron
truncated cuboctahedron
Propertiesconvex, face-transitive

net

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.


Disdyakis
dodecahedron
Deltoidal
icositetrahedron
Rhombic
dodecahedron
Hexahedron
Octahedron

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.

Dimensions

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

The faces are scalene triangles. Their angles are , and .

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.

Projective
symmetry		 [4] [3] [2] [2] [2] [2] [2]+
Image
Dual
image
Polyhedra similar to the disdyakis dodecahedron are duals to the Bowtie octahedron and cube, containing extra pairs triangular faces .[3]

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.

See also

References
  1. https://etc.usf.edu/clipart/keyword/forms
  2. Conway, Symmetries of things, p.284
  3. Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons Craig S. Kaplan
  • 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 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, kisRhombic dodecahedron)
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