Disdyakis dodecahedron(rotating and 3D model)TypeCatalan solidConway notationmCCoxeter diagramFace polygonscalene triangleFaces48Edges72Vertices26 = 6 + 8 + 12Face configurationV4.6.8Symmetry groupOh, B3, [4,3], *432Dihedral angle155° 4' 56"Dual polyhedron truncated cuboctahedronPropertiesconvex, face-transitivenet In geometry, a disdyakis dodecahedron, (also hexoctahedron, hexakis octahedron, octakis cube, octakis hexahedron, kisrhombic dodecahedronConway, Symmetries of things, p.284) or d48, 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 results in the Kleetope of the rhombic dodecahedron, which looks almost like the disdyakis dodecahedron, and is topologically equivalent to it. The net of the rhombic dodecahedral pyramid also shares the same topology.
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.
Disdyakisdodecahedron Deltoidalicositetrahedron Rhombicdodecahedron Hexahedron Octahedron
Spherical polyhedron (see rotating model) Orthographic projections from 2-, 3- and 4-fold axes
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. A spherical disdyakis dodecahedron can be thought of as the barycentric subdivision of the spherical cube or of the spherical octahedron.
Stereographic projections 2-fold 3-fold 4-fold
Cartesian coordinates Let . Then the Cartesian coordinates for the vertices of a disdyakis dodecahedron centered at the origin are:
permutations of (±a, 0, 0) (vertices of an octahedron) permutations of (±b, ±b, 0) (vertices of a cuboctahedron) (±c, ±c, ±c) (vertices of a cube)
Convex hullsCombining an octahedron, cube, and cuboctahedron to form the disdyakis dodecahedron. The convex hulls for these vertices scaled by result in Cartesian coordinates of unit circumradius, which are visualized in the figure below:
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. Projectivesymmetry+ImageDualimage
Related polyhedra and tilings Polyhedra similar to the disdyakis dodecahedron are duals to the Bowtie octahedron and cube, containing extra pairs of triangular faces.Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons Craig S. Kaplan
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 First stellation of rhombic dodecahedron Disdyakis triacontahedron Kisrhombille tiling Great rhombihexacron—A uniform dual polyhedron with the same surface topology
Notes
References
(Section 3-9) The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, kisRhombic dodecahedron)
External links
Disdyakis Dodecahedron (Hexakis Octahedron) Interactive Polyhedron Model