Chamfer (geometry)

Geometric operation which truncates the edges of polyhedra From Wikipedia, the free encyclopedia

Chamfer (geometry)

In geometry, a chamfer or edge-truncation is a topological operator that modifies one polyhedron into another. It separates the faces by reducing them, and adds a new face between each two adjacent faces (moving the vertices inward). Oppositely, 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.

Unchamfered, slightly chamfered, and chamfered cube
Historical crystal models of slightly chamfered Platonic solids

For a polyhedron, this operation adds a new hexagonal face in place of each original edge.

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

Summarize
Perspective

Chamfers of five Platonic solids are described in detail below. 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. The shown dual polyhedra are dual to the canonical versions.

More information SeedPlatonic solid, ChamferedPlatonicsolid(equilateral form) ...
Seed
Platonic
solid

{3,3}

{4,3}

{3,4}

{5,3}

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

Chamfered tetrahedron

More information Chamfered tetrahedron ...
Chamfered tetrahedron

(equilateral form)
Conway notationcT
Goldberg polyhedronGPIII(2,0) = {3+,3}2,0
Faces4 congruent equilateral triangles
6 congruent equilateral* hexagons
Edges24 (2 types:
triangle-hexagon,
hexagon-hexagon)
Vertices16 (2 types)
Vertex configuration(12) 3.6.6
(4) 6.6.6
Symmetry groupTetrahedral (Td)
Dual polyhedronAlternate-triakis tetratetrahedron
Propertiesconvex, equilateral*

Net
*for a certain chamfering/truncating depth
Close

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

More information Chamfered cube ...
Chamfered cube

(equilateral form)
Conway notationcC = t4daC
Goldberg polyhedronGPIV(2,0) = {4+,3}2,0
Faces6 congruent squares
12 congruent equilateral* hexagons
Edges48 (2 types:
square-hexagon,
hexagon-hexagon)
Vertices32 (2 types)
Vertex configuration(24) 4.6.6
(8) 6.6.6
SymmetryOh, [4,3], (*432)
Th, [4,3+], (3*2)
Dual polyhedronTetrakis cuboctahedron
Propertiesconvex, equilateral*

Net (3 zones are shown by 3 colors for their hexagons — each square is in 2 zones —.)
*for a certain chamfering depth
Close

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.

Uses

The DaYan Gem 7 is a twisty puzzle in the shape of a chamfered cube. [2]

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

More information Chamfered octahedron ...
Chamfered octahedron

(equilateral form)
Conway notationcO = t3daO
Faces8 congruent equilateral triangles
12 congruent equilateral* hexagons
Edges48 (2 types:
triangle-hexagon,
hexagon-hexagon)
Vertices30 (2 types)
Vertex configuration(24) 3.6.6
(6) 6.6.6.6
SymmetryOh, [4,3], (*432)
Dual polyhedronTriakis cuboctahedron
Propertiesconvex, equilateral*
*for a certain truncating depth
Close

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

More information Chamfered dodecahedron ...
Chamfered dodecahedron

(equilateral form)
Conway notationcD = t5daD = dk5aD
Goldberg polyhedronGPV(2,0) = {5+,3}2,0
FullereneC80[3]
Faces12 congruent regular pentagons
30 congruent equilateral* hexagons
Edges120 (2 types:
pentagon-hexagon,
hexagon-hexagon)
Vertices80 (2 types)
Vertex configuration(60) 5.6.6
(20) 6.6.6
Symmetry groupIcosahedral (Ih)
Dual polyhedronPentakis icosidodecahedron
Propertiesconvex, equilateral*
*for a certain chamfering depth
Close

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

More information Chamfered icosahedron ...
Chamfered icosahedron

(equilateral form)
Conway notationcI = t3daI
Faces20 congruent equilateral triangles
30 congruent equilateral* hexagons
Edges120 (2 types:
triangle-hexagon,
hexagon-hexagon)
Vertices72 (2 types)
Vertex configuration(24) 3.6.6
(12) 6.6.6.6.6
SymmetryIh, [5,3], (*532)
Dual polyhedronTriakis icosidodecahedron
Propertiesconvex, equilateral*
*for a certain truncating depth
Close

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

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

Relation to Goldberg polyhedra

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)...

More information GP(1,0), GP(2,0) ...
GP(1,0) GP(2,0) GP(4,0) GP(8,0) GP(16,0) ...
GPIV
{4+,3}
Thumb
C
Thumb
cC
Thumb
ccC
Thumb
cccC

ccccC
...
GPV
{5+,3}
Thumb
D
Thumb
cD
Thumb
ccD
Thumb
cccD
Thumb
ccccD
...
GPVI
{6+,3}
Thumb
H
Thumb
cH
Thumb
ccH

cccH

ccccH
...
Close

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)...

More information GP(1,1), GP(2,2) ...
GP(1,1) GP(2,2) GP(4,4) ...
GPIV
{4+,3}
Thumb
tO
Thumb
ctO
Thumb
cctO
...
GPV
{5+,3}
Thumb
tI
Thumb
ctI
Thumb
cctI
...
GPVI
{6+,3}
Thumb
Thumb
ctΔ

cctΔ
...
Close

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

More information GP(3,0), GP(6,0) ...
GP(3,0) GP(6,0) GP(12,0) ...
GPIV
{4+,3}
Thumb
tkC
Thumb
ctkC

cctkC
...
GPV
{5+,3}
Thumb
tkD
Thumb
ctkD

cctkD
...
GPVI
{6+,3}
Thumb
tkH
Thumb
ctkH

cctkH
...
Close

Chamfered polytopes and honeycombs

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

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

Thumb
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

References

Sources

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