Chamfered dodecahedron

Chamfered dodecahedron
File:Truncated rhombic triacontahedron.png
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
File: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. 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.

File:Chamfered dodecahedron.stl

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 rhombic triacontahedron
  G(2,0)
• Truncated icosahedron G(1,1)
  Truncated icosahedron
  G(1,1)
• cell-centered orthogonal projection of the 120-cell
  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
File:Conway polyhedron cD.png cD = G(2,0) cD	File:Conway polyhedron acD.png acD acD	File:Conway polyhedron tcD.png tcD tcD	File:Conway polyhedron zcD.png zcD = G(2,2) zcD	File:Conway polyhedron ecD.png ecD ecD	File:Conway polyhedron bcD.png bcD bcD	File:Conway polyhedron scD.png scD scD	File:Conway polyhedron dk6k5at5daD.png ccD = G(4,0) ccD	File:Goldberg polyhedron 4 2.png wcD = G(4,2) wcD
dual	join	needle	kis	ortho	medial	gyro	dual chamfer	dual whirl
File:Conway polyhedron dcD.png dcD dcD	File:Conway polyhedron jcD.png jcD jcD	File:Conway polyhedron ncD.png ncD ncD	File:Conway polyhedron kcD.png kcD kcD	File:Conway polyhedron ocD.png ocD ocD	File:Conway polyhedron mcD.png mcD mcD	File:Conway polyhedron gcD.png gcD gcD	File:Conway polyhedron k6k5at5daD.png dccD dccD	File:Geodesic polyhedron 4 2.png dwcD dwcD

Chamfered truncated icosahedron

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. File:Conway polyhedron kcD.png
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)