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Concept
Triacontaèdre rhombique tronqué
Résumé
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 (edge-truncation) 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.
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 D_2 symmetry. The angles at the two vertices with vertex configuration 6.6.6 are and at the remaining four vertices with 5.6.6, they are 121.717° each.
It is the Goldberg polyhedron G_V(2,0), 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).
This is the shape of the fullerene C_80; sometimes this shape is denoted C_80(I_h) to describe its icosahedral symmetry and distinguish 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_1 space.
This polyhedron looks very similar to the uniform truncated icosahedron which has 12 pentagons, but only 20 hexagons.
Image:Truncated rhombic triacontahedron.png|'''Truncated rhombic triacontahedron'''G(2,0)
Image:Truncated icosahedron.png|[[Truncated icosahedron]]G(1,1)
File:Ortho solid 120-cell.png|cell-centered [[orthogonal projection]] of the [[120-cell]]
The chamfered dodecahedron creates more polyhedra by basic Conway polyhedron notation. The zip chamfered dodecahedron makes a chamfered truncated icosahedron, and Goldberg (2,2).
Source officielle
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