Chamfer (geometry)

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 maintains the original vertices. For polyhedra, 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

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 Platonic solids

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

Seed
{3,3}

{4,3}

{3,4}

{5,3}

{3,5}
Chamfered

Chamfered tetrahedron

Chamfered tetrahedron

(with equal edge length)
Conway notationcT
Goldberg polyhedronGPIII(2,0) = {3+,3}2,0
Faces4 triangles
6 hexagons
Edges24 (2 types)
Vertices16 (2 types)
Vertex configuration(12) 3.6.6
(4) 6.6.6
Symmetry groupTetrahedral (Td)
Dual polyhedronAlternate-triakis tetratetrahedron
Propertiesconvex, equilateral-faced

net

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 GIII(2,0), containing triangular and hexagonal faces.

The truncated tetrahedron looks similar, but its hexagons correspond to the 4 faces of the original tetrahedron, rather than to its 6 edges.
Tetrahedral chamfers and related solids

chamfered tetrahedron (canonical)

dual of the tetratetrahedron

chamfered tetrahedron (canonical)

alternate-triakis tetratetrahedron

tetratetrahedron

alternate-triakis tetratetrahedron

Chamfered cube

Chamfered cube

(with equal edge length)
Conway notationcC = t4daC
Goldberg polyhedronGPIV(2,0) = {4+,3}2,0
Faces6 squares
12 hexagons
Edges48 (2 types)
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-faced

net

The chamfered cube is a convex polyhedron with 32 vertices, 48 edges, and 18 faces: 12 hexagons and 6 squares. It is constructed as a chamfer of a cube. The squares are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the tetrakis cuboctahedron.

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

The hexagonal faces are equilateral but not regular. They are formed by a truncated rhombus, have 2 internal angles of about 109.47°, or , and 4 internal angles of about 125.26°, while a regular hexagon would have all 120° angles.

Because all its faces have an even number of sides with 180° rotation symmetry, it is a zonohedron. It is also the Goldberg polyhedron GPIV(2,0) or {4+,3}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 and its six vertices are at the permutations of .

A topological equivalent 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
The truncated octahedron looks similar, but its hexagons correspond to the 8 vertices of the cube, rather than to its 12 edges.
Octahedral chamfers and related solids

chamfered cube (canonical)

rhombic dodecahedron

chamfered octahedron (canonical)

tetrakis cuboctahedron

cuboctahedron

triakis cuboctahedron

Chamfered octahedron

Chamfered octahedron

(with equal edge length)
Conway notationcO = t3daO
Faces8 triangles
12 hexagons
Edges48 (2 types)
Vertices30 (2 types)
Vertex configuration(24) 3.6.6
(6) 6.6.6
SymmetryOh, [4,3], (*432)
Dual polyhedronTriakis cuboctahedron
Propertiesconvex

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

It can also be called a tritruncated rhombic dodecahedron, a truncation of the order-3 vertices of the rhombic dodecahedron.

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.

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

Chamfered dodecahedron

Chamfered dodecahedron

(with equal edge length)
Conway notationcD] = t5daD = dk5aD
Goldberg polyhedronGV(2,0) = {5+,3}2,0
FullereneC80[1]
Faces12 pentagons
30 hexagons
Edges120 (2 types)
Vertices80 (2 types)
Vertex configuration(60) 5.6.6
(20) 6.6.6
Symmetry groupIcosahedral (Ih)
Dual polyhedronPentakis icosidodecahedron
Propertiesconvex, equilateral-faced

The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the pentakis icosidodecahedron.

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

The truncated icosahedron looks similar, but its hexagons correspond to the 20 vertices of the dodecahedron, rather than to its 30 edges.
Icosahedral chamfers and related solids

chamfered dodecahedron (canonical)

rhombic triacontahedron

chamfered icosahedron (canonical)

pentakis icosidodecahedron

icosidodecahedron

triakis icosidodecahedron

Chamfered icosahedron

Chamfered icosahedron

(with equal edge length)
Conway notationcI = t3daI
Faces20 triangles
30 hexagons
Edges120 (2 types)
Vertices72 (2 types)
Vertex configuration(24) 3.6.6
(12) 6.6.6
SymmetryIh, [5,3], (*532)
Dual polyhedrontriakis icosidodecahedron
Propertiesconvex

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.

It can also be called a tritruncated rhombic triacontahedron, a truncation of the order-3 vertices of the rhombic triacontahedron.


Chamfered regular tilings

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

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)...
GPIV
{4+,3}

C

cC

ccC

cccC
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}

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)...
GPIV
{4+,3}

tkC

ctkC
cctkC
GPV
{5+,3}

tkD

ctkD
cctkD
GPVI
{6+,3}

tkH

ctkH
cctkH

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.

See also

References

  1. "C80 Isomers". Archived from the original on 2014-08-12. Retrieved 2014-08-09.
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