|Type||Conway polyhedron cD=t5daD|
|Edges||120 (2 types)|
|Vertices||80 (2 types)|
|Vertex configuration||(60) 5.6.6
|Symmetry group||Icosahedral (Ih)|
|Dual polyhedron||Pentakis icosidodecahedron|
The chamfered dodecahedron (also called truncated rhombic triacontahedron) is a convex polyhedron constructed as a truncation of the rhombic triacontahedron. It can more accurately be called a pentatruncated rhombic triacontahedron because only the order-5 vertices are truncated.
These 12 order-5 vertices can be truncated such that all edges are 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 D2 symmetry. The angles at the two vertices with vertex configuration 6.6.6 are arccos(-1/sqrt(5)) = 116.565 degrees, and at the remaining four vertices with 5.6.6, they are 121.717 degrees each.
It is the Goldberg polyhedron GV(2,0), containing pentagonal and hexagonal faces.
Related polyhedra and polytopes
This polyhedron looks very similar to the uniform truncated icosahedron which has 12 pentagons, but only 20 hexagons.
This polyhedron is also a part of a sequence of truncated rhombic polyhedra and tilings with [n,3] Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares. The truncated forms have regular n-gons at the truncated vertices, and nonregular hexagonal faces. The sequence has two vertex figures (n.6.6) and (6,6,6).
|Polyhedra||Euclidean tiling||Hyperbolic tiling|
Alternate truncated cube
Truncated rhombic dodecahedron
Truncated rhombic triacontahedron
If all 32 vertices of a rhombic triacontahedron are truncated, the resulting solid has 12 regular pentagons, 20 equilateral triangles and 30 irregular octagons, with 180 edges in all, and 120 vertices. Its dual is a triangular hecatonicosahedron known as the tripentakis icosidodecahedron, a solid formed by adding a low pyramid to each face of a uniform icosidodecahedron. This figure is to the dodecahedron as the truncated cube is to the tetrahedron.
This is the shape of the fullerene C80; sometimes this shape is denoted C80(Ih) to describe its icosahedral symmetry and distinguish it from other less-symmetric 80-vertex fullerenes. It is one of only four fullerenes found by Deza, Deza & Grishukhin (1998) to have a skeleton that can be isometrically embeddable into an L1 space.
- Deza, A.; Deza, M.; Grishukhin, V. (1998), "Fullerenes and coordination polyhedra versus half-cube embeddings", Discrete Mathematics 192 (1): 41–80, doi:10.1016/S0012-365X(98)00065-X.
- Antoine Deza, Michel Deza, Viatcheslav Grishukhin, Fullerenes and coordination polyhedra versus half-cube embeddings, 1998 PDF 
- VTML polyhedral generator Try "t5daD" (Conway polyhedron notation)
- Zometool model
- Fullerene C80
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