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 (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.

Structure

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 symmetry. The angles at the two vertices with vertex configuration are $\arccos\left(\frac\right) = 116.565^$ 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

This is the shape of the fullerene ; sometimes this shape is denoted 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 space.

Related polyhedra

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).

Chamfered truncated icosahedron

In geometry, the chamfered truncated icosahedron is a convex polyhedron with 240 vertices, 360 edges, and 122 faces, 110 hexagons and 12 pentagons.

It is constructed by a chamfer operation to the truncated icosahedron, adding new hexagons in place of original edges. It can also be constructed as a zip (= dk = dual of kis of) operation from the ''chamfered dodecahedron''. In other words, raising pentagonal and hexagonal pyramids on a chamfered dodecahedron (kis operation) will yield the (2,2) geodesic polyhedron. Taking the dual of that yields the (2,2) Goldberg polyhedron, which is the chamfered truncated icosahedron, and is also Fullerene C₂₄₀.

Dual

Its dual, the hexapentakis chamfered dodecahedron has 240 triangle faces (grouped as 60 (blue), 60 (red) around 12 5-fold symmetry vertices and 120 around 20 6-fold symmetry vertices), 360 edges, and 122 vertices.

Hexapentakis chamfered dodecahedron

References

Bibliography

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*{{cite web , title=Mathematical Impressions: Goldberg Polyhedra , first=George , last=Hart , authorlink = George W. Hart , date=June 18, 2013 , url=https://www.simonsfoundation.org/multimedia/mathematical-impressions-goldberg-polyhedra/ , publisher= Simons Science News

External links

Vertex- and edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra
Livio Zefiro


( Conway polyhedron notation)

Goldberg polyhedra
Polyhedra
Mathematical notation