Disdyakis dodecahedron

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Disdyakis dodecahedron
Disdyakis dodecahedron
Click on picture for large version

Spinning version

Type Catalan solid
Conway notation mC
Coxeter diagram CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png
Face polygon DU11 facets.png
scalene triangle
Faces 48
Edges 72
Vertices 26 = 6 + 8 + 12
Face configuration V4.6.8
Symmetry group Oh, B3, [4,3], *432
Dihedral angle 155° 4' 56"
Dual polyhedron truncated cuboctahedron
Properties convex, face-transitive
Disdyakis dodecahedron
Net

In geometry, a disdyakis dodecahedron, or hexakis octahedron or kisrhombic dodecahedron[1]), 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 superficially resembles 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[edit]

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.png
Disdyakis
dodecahedron
Spherical disdyakis dodecahedron2.png
Spherical
Disdyakis dodecahedron cubic.png
Cubic
Disdyakis dodecahedron octahedral.png
Octahedral
Rhombic dodeca.png
Rhombic
dodecahedral

Seen in stereographic projection the edges of the disdyakis dodecahedron form 9 circles (or centrally radial lines) in the plane. The 9 circles can be divided into two groups of 3 and 6 (drawn in purple and red), representing in two orthogonal subgroups: [2,2], and [3,3]:

Orthogonal Stereographic
Spherical disdyakis dodecahedron Disdyakis dodecahedron stereographic D4.png Disdyakis dodecahedron stereographic D3.png Disdyakis dodecahedron stereographic D2.png
[4] [3] [2]

Dimensions[edit]

If its smallest edges have length 1, its surface area is and its volume is .

Orthogonal projections[edit]

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 cube t012.png Dual cube t012 B2.png Dual cube t012 f4.png Dual cube t012 e46.png Dual cube t012 e48.png Dual cube t012 e68.png Dual cube t012 v.png
Dual
image
3-cube t012.svg 3-cube t012 B2.svg Cube t012 f4.png Cube t012 e46.png Cube t012 e48.png Cube t012 e68.png Cube t012 v.png

Related polyhedra and tilings[edit]

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

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[edit]

References[edit]

  1. ^ Conway, Symmetries of things, p.284
  • 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[edit]