Chamfered dodecahedron

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Chamfered dodecahedron
Chamfered dodecahedron
Type Conway polyhedron cD=t5daD
Goldberg polyhedron GV(2,0)
Faces 12 pentagons
30 hexagons
Edges 120 (2 types)
Vertices 80 (2 types)
Vertex configuration (60) 5.6.6
(20) 6.6.6
Symmetry group Icosahedral (Ih)
Dual polyhedron Pentakis icosidodecahedron
Properties convex, equilateral-faced

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[edit]

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

It also represents the exterior envelope of a cell-centered orthogonal projection of the 120-cell, one of six (convex regular 4-polytopes).

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
[3,3] [4,3] [5,3] [6,3] [7,3] [8,3]
Hexahedron.svg
Cube
Rhombicdodecahedron.jpg
Rhombic dodecahedron
Rhombictriacontahedron.jpg
Rhombic triacontahedron
Rhombic star tiling.png
Rhombille
Order73 qreg rhombic til.png Uniform dual tiling 433-t01-yellow.png
Alternate truncated cube.png
Chamfered tetrahedron
Truncated rhombic dodecahedron2.png
Chamfered cube
Truncated rhombic triacontahedron.png
Chamfered dodecahedron
Truncated rhombille tiling.png
Hexagonal tiling

Rhombic triacontahedron[edit]

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.

Chemistry[edit]

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.

See also[edit]

References[edit]

External links[edit]