Chamfer (geometry)

In geometry, chamfering or edge-truncation is a Conway polyhedron notation operation that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintain the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge.

A polyhedron with e edges will have a chamfered form containing 2e new vertices, 3e new edges, and e new hexagonal faces.

Chamfered regular and quasiregular polyhedra

Class	Regular					Quasiregular
Seed	{3,3}	{4,3}	{3,4}	{5,3}	{3,5}	aC	aD
Chamfered	cT	cC	cO	cD	cI	caC	caD

Relation to Goldberg polyhedra

The chamfer operation applied in series creates progressively larger polyhedra with new hexagonal faces replacing edges from the previous one. The chamfer operator transforms G(m,n) to G(2m,2n).

A regular polyhedron, G(1,0), create a Goldberg polyhedra sequence: G(1,0), G(2,0), G(4,0), G(8,0), G(16,0)...

	G(1,0)	G(2,0)	G(4,0)	G(8,0)	G(16,0)...
G_IV	C	cC	ccC	cccC
G_V	D	cD	ccD	cccD	ccccD
G_VI	H	cH	ccH	cccH	ccccH

The truncated octahedron or truncated icosahedron, G(1,1) creates a Goldberg sequence: G(1,1), G(2,2), G(4,4), G(8,8)....

	G(1,1)	G(2,2)	G(4,4)...
G_IV	tO	ctO	cctO
G_V	tI	ctI	cctI
G_VI	tH	ctH	cctH

A truncated tetrakis hexahedron or pentakis dodecahedron, G(3,0), creates a Goldberg sequence: G(3,0), G(6,0), G(12,0)...

	G(3,0)	G(6,0)	G(12,0)...
G_IV	tkC	ctkC	cctkC
G_V	tkD	ctkD	cctkD
G_VI	tkH	ctkH	cctkH

Chamfered regular tilings

Chamfered regular tilings
Square tiling, Q {4,4}	Triangular tiling, Δ {3,6}	Hexagonal tiling, H {6,3}

cQ	cΔ	cH

Chamfered polytopes and honeycombs

Like the expansion operation, chamfer can be applied to any dimension. For polygons, it triples the number of vertices. For polychora, new cells are created around the original edges, The cells are prisms, containing two copies of the original face, with pyramids augmented onto the prism sides.

Chamfered tetrahedron

Chamfered tetrahedron

Conway notation	cT
Goldberg polyhedron	G_III(2,0)
Faces	4 triangles 6 hexagons
Edges	24 (2 types)
Vertices	16 (2 types)
Vertex configuration	(12) 3.6.6 (4) 6.6.6
Symmetry group	Tetrahedral (T_d)
Dual polyhedron	Alternate-triakis octahedron
Properties	convex, equilateral-faced

The chamfered tetrahedron (or alternate truncated cube) is a convex polyhedron constructed as an alternately truncated cube or chamfer operation on a tetrahedron, replacing its 6 edges with hexagons.

It is the Goldberg polyhedron G_III(2,0), containing triangular and hexagonal faces.

It can look a little like a truncated tetrahedron, , which has 4 hexagonal and 4 triangular faces, which is the related Goldberg polyhedron: G_III(1,1).

Net

Chamfered cube

Chamfered cube

Conway notation	cC = t4daC
Goldberg polyhedron	G_IV(2,0)
Faces	6 squares 12 hexagons
Edges	48 (2 types)
Vertices	32 (2 types)
Vertex configuration	(24) 4.6.6 (8) 6.6.6
Symmetry	O_h, [4,3], (*432)
Dual polyhedron	Tetrakis cuboctahedron
Properties	convex, zonohedron, equilateral-faced
net

In geometry, the chamfered cube (also called truncated rhombic dodecahedron) is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 6 (order 4) vertices.

The 6 vertices are truncated such that all edges are equal length. The original 12 rhombic faces become flattened hexagons, and the truncated vertices become squares.

The hexagonal faces are equilateral but not regular. They are formed by a truncated rhombus, have 2 internal angles of about 109.47 degrees ( $\cos ^{-1}(-{\frac {1}{3}})$ ) and 4 internal angles of about 125.26 degrees, while a regular hexagon would have all 120 degree angles.

Because all its faces have an even number of sides with 180 degree rotation symmetry, it is a zonohedron. It is also the Goldberg polyhedron G_IV(2,0), containing square and hexagonal faces.

Coordinates

The chamfered cube is the Minkowski sum of a rhombic dodecahedron and a cube of side length 1 when eight vertices of the rhombic dodecahedron are at $(\pm 1,\pm 1,\pm 1)$ and its six vertices are at the permutations of $(\pm 2,0,0)$ .

This polyhedron looks similar to the uniform truncated octahedron:

Chamfered cube	Truncated octahedron

We can construct a truncated octahedron model by twenty four chamfered cube blocks.^[1] ^[2]

Chamfered octahedron

Chamfered octahedron

Conway notation	cO = t3daO
Faces	8 triangles 12 hexagons
Edges	48 (2 types)
Vertices	30 (2 types)
Vertex configuration	(24) 3.6.6 (6) 6.6.6
Symmetry	O_h, [4,3], (*432)
Dual polyhedron	Triakis cuboctahedron
Properties	convex

In geometry, the chamfered octahedron is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 8 (order 3) vertices.

The 8 vertices are truncated such that all edges are equal length. The original 12 rhombic faces become flattened hexagons, and the truncated vertices become triangles.

The hexagonal faces are equilateral but not regular.

Chamfered dodecahedron

Chamfered dodecahedron

Conway notation	cD=t5daD
Goldberg polyhedron	G_V(2,0)
Fullerene	C₈₀^[3]
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 (I_h)
Dual polyhedron	Pentakis icosidodecahedron
Properties	convex, equilateral-faced

The chamfered dodecahedron (also called truncated rhombic triacontahedron) is a convex polyhedron 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.

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.

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

Truncated rhombic triacontahedron
Truncated icosahedron
cell-centered orthogonal projection of the 120-cell

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 C₆₀; sometimes this shape is denoted C₆₀(I_h) to describe its icosahedral symmetry and distinguish it from other less-symmetric 60-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 icosahedron

Chamfered icosahedron

Conway notation	cI = t3daI
Faces	20 triangles 30 hexagons
Edges	120 (2 types)
Vertices	72 (2 types)
Vertex configuration	(24) 3.6.6 (12) 6.6.6
Symmetry	I_h, [5,3], (*532)
Dual polyhedron	triakis icosidodecahedron
Properties	convex

In geometry, the chamfered icosahedron is a convex polyhedron constructed from the rhombic triacontahedron by truncating the 20 order-3 vertices. The hexagonal faces can be made equilateral but not regular.

References

Goldberg, Michael (1937). "A class of multi-symmetric polyhedra". Tohoku Mathematical Journal.
Joseph D. Clinton, Clinton’s Equal Central Angle Conjecture [1]
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.
Hart, George (June 18, 2013). "Mathematical Impressions: Goldberg Polyhedra". Simons Science News.
Antoine Deza, Michel Deza, Viatcheslav Grishukhin, Fullerenes and coordination polyhedra versus half-cube embeddings, 1998 PDF [2] (p. 72 Fig. 26. Chamfered tetrahedron)
Deza, A.; Deza, M.; Grishukhin, V. (1998), "Fullerenes and coordination polyhedra versus half-cube embeddings", Discrete Mathematics, 192 (1): 41–80, doi:10.1016/S0012-365X(98)00065-X.

External links

Chamfered Tetrahedron
Chamfered Solids
Vertex- and edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra Livio Zefiro
VRML polyhedral generator (Conway polyhedron notation)
- VRML model Chamfered cube
3.2.7. Systematic numbering for (C80-Ih) [5,6] fullerene
Zometool model
Fullerene C80
1. [3] (Number 7 -Ih)
2. [4]
How to make a chamfered cube

[1] Gallery of Wooden Polyhedra

[2] "Wooden polyhedra(English edition)"

[3] C80 Isomers

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[2]

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