Chamfered dodecahedron

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Chamfered dodecahedron
Chamfered dodecahedron
Conway notation cD = t5daD = dk5aD
Goldberg polyhedron GV(2,0) = {5+,3}2,0
Fullerene C80[1]
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
Truncated rhombic triacontahedron net.png
net


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 (geometry) (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[edit]

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°, 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[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.

chamfered dodecahedron

Related polyhedra[edit]

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
Conway polyhedron cD.png
cD = G(2,0)
cD
Conway polyhedron acD.png
acD
acD
Conway polyhedron tcD.png
tcD
tcD
Conway polyhedron zcD.png
zcD = G(2,2)
zcD
Conway polyhedron ecD.png
ecD
ecD
Conway polyhedron bcD.png
bcD
bcD
Conway polyhedron scD.png
scD
scD
Conway polyhedron dk6k5at5daD.png
ccD = G(4,0)
ccD
Goldberg polyhedron 4 2.png
wcD = G(4,2)
wcD
dual join needle kis ortho medial gyro dual chamfer dual whirl
Conway polyhedron dcD.png
dcD
dcD
Conway polyhedron jcD.png
jcD
jcD
Conway polyhedron ncD.png
ncD
ncD
Conway polyhedron kcD.png
kcD
kcD
Conway polyhedron ocD.png
ocD
ocD
Conway polyhedron mcD.png
mcD
mcD
Conway polyhedron gcD.png
gcD
gcD
Conway polyhedron k6k5at5daD.png
dccD
dccD
Geodesic polyhedron 4 2.png
dwcD
dwcD

Chamfered truncated icosahedron[edit]

Chamfered truncated icosahedron
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[edit]

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.

Conway polyhedron kcD.png
Hexapentakis chamfered dodecahedron

References[edit]

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

External links[edit]