Disdyakis dodecahedron
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Type	Catalan Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....
Face polygon	scalene triangle
Faces	48
Edges	72
Vertices	26 = 6 + 8 + 12
Face configuration Face configuration In geometry, a face configuration is notational description of a face-transitive polyhedron. It represents a sequential count of the number of faces that exist at each vertex around a face....	V4.6.8
Symmetry group	Oh, [4,3], *432
Dihedral angle Dihedral angle In geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...	155° 4' 56"
Dual polyhedron Dual polyhedron In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...	truncated cuboctahedron Truncated cuboctahedron In geometry, the truncated cuboctahedron is an Archimedean solid. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges...
Properties	convex, face-transitive
Net Net (polyhedron) In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron...

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a disdyakis dodecahedron, or hexakis octahedron, is a Catalan solid

Catalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....

 and the dual to the Archimedean

Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...

 truncated cuboctahedron

Truncated cuboctahedron

In geometry, the truncated cuboctahedron is an Archimedean solid. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges...

. As such it is face-transitive but with irregular face polygons. It looks a bit like an inflated rhombic dodecahedron

Rhombic dodecahedron

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...

—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

Kleetope

In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope is another polyhedron or polytope formed by replacing each facet of with a shallow pyramid...

 of the rhombic dodecahedron.

Symmetry

It has O_h octahedral symmetry

Octahedral symmetry

150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...

. Its collective edges represent the reflection planes of the symmetry.

Dimensions

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

See also

• Disdyakis triacontahedron
  Disdyakis triacontahedron
  In geometry, a disdyakis triacontahedron, or hexakis icosahedron is a Catalan solid and the dual to the Archimedean truncated icosidodecahedron. As such it is face uniform but with irregular face polygons...
  
  
• Bisected hexagonal tiling
• Great rhombihexacron
  Great rhombihexacron
  In geometry, the great rhombihexacron is a nonconvex isohedral polyhedron. It is the dual of the uniform great rhombihexahedron . It has 24 identical bow-tie-shaped faces, 18 vertices, and 48 edges....
  
  —A uniform dual polyhedron with the same surface topology

External links

• Disdyakis Dodecahedron (Hexakis Octahedron) Interactive Polyhedron Model