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 topological operator 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.

In Conway polyhedron notation it is represented by the letter c. 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 GP(m,n) to GP(2m,2n).

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

	GP(1,0)	GP(2,0)	GP(4,0)	GP(8,0)	GP(16,0)...
GP_IV {4+,3}	C	cC	ccC	cccC
GP_V {5+,3}	D	cD	ccD	cccD	ccccD
GP_VI {6+,3}	H	cH	ccH	cccH	ccccH

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

	GP(1,1)	GP(2,2)	GP(4,4)...
GP_IV {4+,3}	tO	ctO	cctO
GP_V {5+,3}	tI	ctI	cctI
GP_VI {6+,3}	tH	ctH	cctH

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

	GP(3,0)	GP(6,0)	GP(12,0)...
GP_IV {4+,3}	tkC	ctkC	cctkC
GP_V {5+,3}	tkD	ctkD	cctkD
GP_VI {6+,3}	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	GP_III(2,0) = {3+,3}_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	GP_IV(2,0) = {4+,3}_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) T_h, [4,3⁺], (32)
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 $(\pm 2,0,0)$ .

Variations

The chamfered cube can be constructed with pyritohedral symmetry and rectangular faces. This can be seen as a pyritohedron with 6 axial edges planned. This occurs in pyrite crystals.

Uses in tessellations

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

This polyhedron looks similar to the uniform truncated octahedron:

Chamfered cube	Truncated octahedron

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 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
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.
Antoine Deza, Michel Deza, Viatcheslav Grishukhin, Fullerenes and coordination polyhedra versus half-cube embeddings, 1998 PDF (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, archived from the original on 2007-02-06 .

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
Fullerene C80
1. (Number 7 -Ih)
How to make a chamfered cube

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