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Chamfer (geometry)

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In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion: it moves the faces apart (outward), and adds a new face between each two adjacent faces; but contrary to expansion, it maintains the original vertices. (Equivalently: it separates the faces by reducing them, and adds a new face between each two adjacent faces; but it only moves the vertices inward.) For a polyhedron, this operation adds a new hexagonal face in place of each original edge.

Unchamfered, slightly chamfered, and chamfered cube
Historical crystal models of slightly chamfered Platonic solids
For the concept in machining and architecture, see chamfer.

In Conway polyhedron notation, chamfering 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 Platonic solids

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In the chapters below, the chamfers of the five Platonic solids are described in detail. Each is shown in an equilateral version where all edges have the same length, and in a canonical version where all edges touch the same midsphere. (They look noticeably different only for solids containing triangles.) The shown dual polyhedra are dual to the canonical versions.

Seed
Platonic
solid 		   
{3,3} 		 
{4,3} 		 
{3,4} 		 
{5,3} 		 
{3,5}
Chamfered
Platonic
solid
(equilateral
form) 		           

Chamfered tetrahedron

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Chamfered tetrahedron
 
(equilateral form)
Conway notation cT
Goldberg polyhedron GPIII(2,0) = {3+,3}2,0
Faces 4 congruent equilateral triangles
6 congruent equilateral* hexagons
Edges 24 (2 types:
triangle-hexagon,
hexagon-hexagon)
Vertices 16 (2 types)
Vertex configuration (12) 3.6.6
(4) 6.6.6
Symmetry group Tetrahedral (Td)
Dual polyhedron Alternate-triakis tetratetrahedron
Properties convex, equilateral*
 
Net
*for a certain chamfering/truncating depth

The chamfered tetrahedron or alternate truncated cube is a convex polyhedron constructed:

For a certain depth of chamfering/truncation, all (final) edges of the cT have the same length; then, the hexagons are equilateral, but not regular.

The dual of the chamfered tetrahedron is the alternate-triakis tetratetrahedron.

The cT is the Goldberg polyhedron GPIII(2,0) or {3+,3}2,0, containing triangular and hexagonal faces.

 
The truncated tetrahedron looks similar; but its hexagons correspond to the 4 faces, not to the 6 edges, of the yellow tetrahedron, i.e. to the 4 vertices, not to the 6 edges, of the red tetrahedron.
 
Historical drawings of truncated tetrahedron and slightly chamfered tetrahedron.[1]
Tetrahedral chamfers and their duals
 
chamfered tetrahedron
(canonical form) 		 
dual of the tetratetrahedron 		 
chamfered tetrahedron
(canonical form)
 
alternate-triakis tetratetrahedron 		 
tetratetrahedron 		 
alternate-triakis tetratetrahedron

Chamfered cube

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Chamfered cube
 
(equilateral form)
Conway notation cC = t4daC
Goldberg polyhedron GPIV(2,0) = {4+,3}2,0
Faces 6 congruent squares
12 congruent equilateral* hexagons
Edges 48 (2 types:
square-hexagon,
hexagon-hexagon)
Vertices 32 (2 types)
Vertex configuration (24) 4.6.6
(8) 6.6.6
Symmetry Oh, [4,3], (*432)
Th, [4,3+], (3*2)
Dual polyhedron Tetrakis cuboctahedron
Properties convex, equilateral*
 
Net (3 zones are shown by 3 colors for their hexagons — each square is in 2 zones —.)
*for a certain chamfering depth

The chamfered cube is constructed as a chamfer of a cube: the squares are reduced in size and new faces, hexagons, are added in place of all the original edges. The cC is a convex polyhedron with 32 vertices, 48 edges, and 18 faces: 6 congruent (and regular) squares, and 12 congruent flattened hexagons.
For a certain depth of chamfering, all (final) edges of the chamfered cube have the same length; then, the hexagons are equilateral, but not regular. They are congruent alternately truncated rhombi, have 2 internal angles of   and 4 internal angles of   while a regular hexagon would have all   internal angles.

The cC is also inaccurately called a truncated rhombic dodecahedron, although that name rather suggests a rhombicuboctahedron. The cC can more accurately be called a tetratruncated rhombic dodecahedron, because only the (6) order-4 vertices of the rhombic dodecahedron are truncated.

The dual of the chamfered cube is the tetrakis cuboctahedron.

Because all the faces of the cC have an even number of sides and are centrally symmetric, it is a zonohedron:

 
Chamfered cube (3 zones are shown by 3 colors for their hexagons — each square is in 2 zones —.)

The chamfered cube is also the Goldberg polyhedron GPIV(2,0) or {4+,3}2,0, containing square and hexagonal faces.

The cC is the Minkowski sum of a rhombic dodecahedron and a cube of edge length 1 when the eight order-3 vertices of the rhombic dodecahedron are at   and its six order-4 vertices are at the permutations of  

A topological equivalent to the chamfered cube, but with pyritohedral symmetry and rectangular faces, can be constructed by chamfering the axial edges of a pyritohedron. This occurs in pyrite crystals.

 
 
Pyritohedron and its axis truncation
 
 
Historical crystallographic models of axis shallower and deeper truncations of pyritohedron
 
The truncated octahedron looks similar; but its hexagons correspond to the 8 faces, not to the 12 edges, of the octahedron, i.e. to the 8 vertices, not to the 12 edges, of the cube.
Octahedral chamfers and their duals
 
chamfered cube
(canonical form) 		 
rhombic dodecahedron 		 
chamfered octahedron
(canonical form)
 
tetrakis cuboctahedron 		 
cuboctahedron 		 
triakis cuboctahedron

Chamfered octahedron

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Chamfered octahedron
 
(equilateral form)
Conway notation cO = t3daO
Faces 8 congruent equilateral triangles
12 congruent equilateral* hexagons
Edges 48 (2 types:
triangle-hexagon,
hexagon-hexagon)
Vertices 30 (2 types)
Vertex configuration (24) 3.6.6
(6) 6.6.6.6
Symmetry Oh, [4,3], (*432)
Dual polyhedron Triakis cuboctahedron
Properties convex, equilateral*
*for a certain truncating depth

In geometry, the chamfered octahedron is a convex polyhedron constructed by truncating the 8 order-3 vertices of the rhombic dodecahedron. These truncated vertices become congruent equilateral triangles, and the original 12 rhombic faces become congruent flattened hexagons.
For a certain depth of truncation, all (final) edges of the cO have the same length; then, the hexagons are equilateral, but not regular.

The chamfered octahedron can also be called a tritruncated rhombic dodecahedron.

The dual of the cO is the triakis cuboctahedron.

 
Historical drawings of rhombic dodecahedron and slightly chamfered octahedron
 
 
Historical models of triakis cuboctahedron and slightly chamfered octahedron

Chamfered dodecahedron

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Chamfered dodecahedron
 
(equilateral form)
Conway notation cD = t5daD = dk5aD
Goldberg polyhedron GPV(2,0) = {5+,3}2,0
Fullerene C80[2]
Faces 12 congruent regular pentagons
30 congruent equilateral* hexagons
Edges 120 (2 types:
pentagon-hexagon,
hexagon-hexagon)
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*
*for a certain chamfering depth
Main article: Chamfered dodecahedron

The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 12 congruent regular pentagons and 30 congruent flattened hexagons.
It is constructed as a chamfer of a regular dodecahedron. The pentagons are reduced in size and new faces, flattened hexagons, are added in place of all the original edges. For a certain depth of chamfering, all (final) edges of the cD have the same length; then, the hexagons are equilateral, but not regular.

The cD is also inaccurately called a truncated rhombic triacontahedron, although that name rather suggests a rhombicosidodecahedron. The cD can more accurately be called a pentatruncated rhombic triacontahedron, because only the (12) order-5 vertices of the rhombic triacontahedron are truncated.

The dual of the chamfered dodecahedron is the pentakis icosidodecahedron.

The cD is the Goldberg polyhedron GPV(2,0) or {5+,3}2,0, containing pentagonal and hexagonal faces.

 
The truncated icosahedron looks similar, but its hexagons correspond to the 20 faces, not to the 30 edges, of the icosahedron, i.e. to the 20 vertices, not to the 30 edges, of the dodecahedron.
Icosahedral chamfers and their duals
 
chamfered dodecahedron
(canonical form) 		 
rhombic triacontahedron 		 
chamfered icosahedron
(canonical form)
 
pentakis icosidodecahedron 		 
icosidodecahedron 		 
triakis icosidodecahedron

Chamfered icosahedron

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Chamfered icosahedron
 
(equilateral form)
Conway notation cI = t3daI
Faces 20 congruent equilateral triangles
30 congruent equilateral* hexagons
Edges 120 (2 types:
triangle-hexagon,
hexagon-hexagon)
Vertices 72 (2 types)
Vertex configuration (24) 3.6.6
(12) 6.6.6.6.6
Symmetry Ih, [5,3], (*532)
Dual polyhedron Triakis icosidodecahedron
Properties convex, equilateral*
*for a certain truncating depth

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

The chamfered icosahedron can also be called a tritruncated rhombic triacontahedron.

The dual of the cI is the triakis icosidodecahedron.

Chamfered regular tilings

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Chamfered regular and quasiregular tilings
 
Square tiling, Q
{4,4} 		 
Triangular tiling, Δ
{3,6} 		 
Hexagonal tiling, H
{6,3} 		 
Rhombille, daH
dr{6,3}
       
cQ cΔ cH cdaH

Relation to Goldberg polyhedra

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

A regular polyhedron, GP(1,0), creates 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) ...
GPIV
{4+,3} 		 
C 		 
cC 		 
ccC 		 
cccC
ccccC 		...
GPV
{5+,3} 		 
D 		 
cD 		 
ccD 		 
cccD 		 
ccccD 		...
GPVI
{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) ...
GPIV
{4+,3} 		 
tO 		 
ctO 		 
cctO 		...
GPV
{5+,3} 		 
tI 		 
ctI 		 
cctI 		...
GPVI
{6+,3} 		 
tΔ 		 
ctΔ
cctΔ 		...

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) ...
GPIV
{4+,3} 		 
tkC 		 
ctkC
cctkC 		...
GPV
{5+,3} 		 
tkD 		 
ctkD
cctkD 		...
GPVI
{6+,3} 		 
tkH 		 
ctkH
cctkH 		...

Chamfered polytopes and honeycombs

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Like the expansion operation, chamfer can be applied to any dimension.

For polygons, it triples the number of vertices. Example:

 
A chamfered square
(See also the previous version of this figure.)

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.[something may be wrong in this passage]

See also

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References

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  1. ^ Spencer 1911, p. 575, or p. 597 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, TETRAHEDRAL CLASS, FIGS. 30 & 31.
  2. ^ "C80 Isomers". Archived from the original on 2014-08-12. Retrieved 2014-08-09.

Sources

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