Chamfered dodecahedron

Chamfered dodecahedron
Conway notation	cD] = t5daD
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
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 a pentatruncated 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(-1/sqrt(5)) = 116.565 degrees, and at the remaining four vertices with 5.6.6, they are 121.717 degrees 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.

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
Edges	360
Vertices	240
Symmetry	Ih, [5,3], (*532)
Dual polyhedron	Kised truncated icosahedron
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 operation from the chamfered dodecahedron.

It is Goldberg polyhedron G(2,2) and Fullerene C₂₄₀.

References

• Goldberg, Michael (1937). "A class of multi-symmetric polyhedra". Tohoku Mathematical Journal. 
• Hart, George (2012). "Goldberg Polyhedra". In Senechal, Marjorie. Shaping Space (2nd ed.). Springer. pp. 125–138. doi:10.1007/978-0-387-92714-5_9. 
• 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)