Chamfered dodecahedron

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Chamfered dodecahedron
Truncated rhombic triacontahedron.png
Type Goldberg polyhedron (GV(2,0) = {5+,3}2,0)
Fullerene (C80) [1]
Near-miss Johnson solid
Faces 12 pentagons
30 irregular hexagons
Edges 120 (2 types)
Vertices 80 (2 types)
Vertex configuration 60 (5.6.6)
20 (6.6.6)
Conway notation cD = t5daD = dk5aD
Symmetry group Icosahedral (Ih)
Dual polyhedron Pentakis icosidodecahedron
Properties convex, equilateral-faced
Net
Truncated rhombic triacontahedron net.png

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.

Contents

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 D2 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 GV(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.

Chemistry

The chamfered dodecahedron is the shape of the fullerene C80. Occasionally, this shape is denoted C80(Ih), describing its icosahedral symmetry and distinguishing it from other less-symmetric 80-vertex fullerenes. [2] 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. [3]

chamfered dodecahedron Chamfered dodecahedron.stl
chamfered dodecahedron

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

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.

References

  1. "C80 Isomers". Archived from the original on 2014-08-12. Retrieved 2014-08-05.
  2. Gelisgen, Ozcan; Yavuz, Serhat (2019). "A Note About Isometry Groups of Chamfered Dodecahedron and Chamfered Icosahedron Spaces" (PDF). International Journal of Geometry. 8 (2): 33–45.
  3. Deza, Antoine; Deza, Michel; Grishukhin, Viatcheslav (1998). "Fullerenes and coordination polyhedra versus half-cube embeddings". Discrete Mathematics. 192 (1–3): 41–80. doi:10.1016/S0012-365X(98)00065-X.

Bibliography