Chamfer (geometry)

A chamfered cube can be constructed by reducing the square faces, and connecting new edges to vertices at the original positions. Topologically: (v,e,f) --> (v+2e,4e,f+e)

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)...
GIV	C	cC	ccC	cccC
GV	D	cD	ccD	cccD	ccccD
GVI	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)...
GIV	tO	ctO	cctO
GV	tI	ctI	cctI
GVI	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)...
GIV	tkC	ctkC	cctkC
GV	tkD	ctkD	cctkD
GVI	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	GIII(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 (Td)
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	GIV(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	Oh, [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.

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.

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	Oh, [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	GV(2,0)
Fullerene	C80
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

A net for chamfered dodecahedron

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	Ih, [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.

See also

• Conway polyhedron notation

References

• Joseph D. Clinton, Clinton’s Equal Central Angle Conjecture [1]
• Antoine Deza, Michel Deza, Viatcheslav Grishukhin, Fullerenes and coordination polyhedra versus half-cube embeddings, 1998 PDF [2] (p. 72 Fig. 26. Chamfered tetrahedron)
• .

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