Chamfered dodecahedron


In geometry, the chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the pentakis icosidodecahedron.
It is also called a truncated rhombic triacontahedron, constructed as a truncation of the rhombic triacontahedron. It can more accurately be called an order-5 truncated rhombic triacontahedron because only the order-5 vertices are truncated.

Structure

These 12 order-5 vertices can be truncated such that all edges are of equal length. The original 30 rhombic faces become non-regular hexagons, and the truncated vertices become regular pentagons.
The hexagon faces can be equilateral but not regular with D symmetry. The angles at the two vertices with vertex configuration are and at the remaining four vertices with, they are each.
It is the Goldberg polyhedron, containing pentagonal and hexagonal faces.
It also represents the exterior envelope of a cell-centered orthogonal projection of the 120-cell, one of six convex regular 4-polytopes.

Chemistry

The chamfered dodecahedron is the shape of the fullerene. Occasionally, this shape is denoted, describing its icosahedral symmetry and distinguishing it from other less-symmetric 80-vertex fullerenes. It is one of only four fullerenes found by to have a skeleton that can be isometrically embeddable into an L space.

Related polyhedra

This polyhedron looks very similar to the uniform truncated icosahedron, which has 12 pentagons, but only 20 hexagons.

Related polytopes

It represents the exterior envelope of a cell-centered orthogonal projection of the 120-cell, one of six convex regular 4-polytopes, into 3 dimensions.