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
Type	Goldberg polyhedron (GV(2,0) = {5 ,3}2,0) Fullerene (C80) 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

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 $\arccos \left({\frac {-1}{\sqrt {5}}}\right)=116.565^{\circ }$ 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.

Chemistry

This is the shape of the fullerene C₈₀; sometimes this shape is denoted C₈₀(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 Deza, Deza & Grishukhin (1998) to have a skeleton that can be isometrically embeddable into an L₁ space.

chamfered dodecahedron

Related polyhedra

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

• 
  Truncated rhombic triacontahedron
  G(2,0)
• 
  Truncated icosahedron
  G(1,1)
• 
  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 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

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

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

• Deza, Antoine; Deza, Michel; Grishukhin, Viatcheslav (October 1998). "Fullerenes and coordination polyhedra versus half-cube embeddings". Discrete Mathematics. 192 (1–3): 41–80. doi:10.1016/S0012-365X(98)00065-X.
• Goldberg, Michael (1937). "A class of multi-symmetric polyhedra". Tohoku Mathematical Journal. 43: 104–108.
• Hart, George (2012). "Goldberg Polyhedra". In Senechal, Marjorie (ed.). Shaping Space (2nd ed.). Springer. pp. 125–138. doi:10.1007/978-0-387-92714-5_9. ISBN 978-0-387-92713-8.
• Hart, George (June 18, 2013). "Mathematical Impressions: Goldberg Polyhedra". Simons Science News.

External links

• Vertex- and edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra Livio Zefiro
• VRML polyhedral generator (Conway polyhedron notation)