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.

Chamfered dodecahedron
TypeGoldberg polyhedron (GV(2,0) = {5+,3}2,0)
Fullerene (C80)[1]
Near-miss Johnson solid
Faces12 pentagons
30 irregular hexagons
Edges120 (2 types)
Vertices80 (2 types)
Vertex configuration60 (5.6.6)
20 (6.6.6)
Conway notationcD = t5daD = dk5aD
Symmetry groupIcosahedral (Ih)
Dual polyhedronPentakis icosidodecahedron
Propertiesconvex, equilateral-faced
Net

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

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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

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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.

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

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 dodecahedron polyhedra
"seed" ambo truncate zip expand bevel snub chamfer whirl
 
cD = G(2,0)
cD
 
acD
acD
 
tcD
tcD
 
zcD = G(2,2)
zcD
 
ecD
ecD
 
bcD
bcD
 
scD
scD
 
ccD = G(4,0)
ccD
 
wcD = G(4,2)
wcD
dual join needle kis ortho medial gyro dual chamfer dual whirl
 
dcD
dcD
 
jcD
jcD
 
ncD
ncD
 
kcD
kcD
 
ocD
ocD
 
mcD
mcD
 
gcD
gcD
 
dccD
dccD
 
dwcD
dwcD

Chamfered truncated icosahedron

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Chamfered truncated icosahedron
 
Goldberg polyhedron GV(2,2) = {5+,3}2,2
Conway notation ctI
Fullerene C240
Faces 12 pentagons
110 hexagons (3 types)
Edges 360
Vertices 240
Symmetry Ih, [5,3], (*532)
Dual polyhedron Hexapentakis chamfered dodecahedron
Properties convex

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

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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

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  1. ^ "C80 Isomers". Archived from the original on 2014-08-12. Retrieved 2014-08-05.

Bibliography

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