Chamfer (geometry)

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Unchamfered, slightly chamfered and chamfered cube

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.

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[edit]

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 Polyhedron 4a.png Polyhedron 4b.png
{3,3}
Polyhedron 6.png
{4,3}
Polyhedron 8.png
{3,4}
Polyhedron 12.png
{5,3}
Polyhedron 20.png
{3,5}
Chamfered Polyhedron chamfered 4a edeq.png Polyhedron chamfered 4b edeq.png Polyhedron chamfered 6 edeq.png Polyhedron chamfered 8 edeq.png Polyhedron chamfered 12 edeq.png Polyhedron chamfered 20 edeq.png

Chamfered tetrahedron[edit]

Chamfered tetrahedron
Polyhedron chamfered 4a edeq max.png
(with equal edge length)
Conway notation cT
Goldberg polyhedron GPIII(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 (Td)
Dual polyhedron Alternate-triakis tetratetrahedron
Properties convex, equilateral-faced
Polyhedron chamfered 4a net.svg
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 4 of the tetrahedrons vertices, rather than to its 6 edges.
Tetrahedral chamfers and related solids
Polyhedron chamfered 4a.png
chamfered tetrahedron (canonical)
Polyhedron 4-4 dual.png
dual of the tetratetrahedron
Polyhedron chamfered 4b.png
chamfered tetrahedron (canonical)
Polyhedron chamfered 4a dual.png
alternate-triakis tetratetrahedron
Polyhedron 4-4.png
tetratetrahedron
Polyhedron chamfered 4b dual.png
alternate-triakis tetratetrahedron

Chamfered cube[edit]

Chamfered cube
Polyhedron chamfered 6 edeq max.png
(with equal edge length)
Conway notation cC = t4daC
Goldberg polyhedron GPIV(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 Oh, [4,3], (*432)
Th, [4,3+], (3*2)
Dual polyhedron Tetrakis cuboctahedron
Properties convex, zonohedron, equilateral-faced
Truncated rhombic dodecahedron net.png
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° 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 .

The chamfered cube can be constructed with pyritohedral symmetry and rectangular faces (right). This can be seen as a pyritohedron (left) with 6 axial edges planned. This occurs in pyrite crystals.
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
Polyhedron chamfered 6.png
chamfered cube (canonical)
Polyhedron 6-8 dual.png
rhombic dodecahedron
Polyhedron chamfered 8.png
chamfered octahedron (canonical)
Polyhedron chamfered 6 dual.png
tetrakis cuboctahedron
Polyhedron 6-8.png
cuboctahedron
Polyhedron chamfered 8 dual.png
triakis cuboctahedon

Chamfered octahedron[edit]

Chamfered octahedron
Polyhedron chamfered 8 edeq max.png
(with equal edge length)
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.

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 representation of a chamfered octahedron (right)

Chamfered dodecahedron[edit]

Chamfered dodecahedron
Polyhedron chamfered 12 edeq max.png
(with equal edge length)
Conway notation cD] = t5daD = dk5aD
Goldberg polyhedron GV(2,0) = {5+,3}2,0
Fullerene C80[1]
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

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
Polyhedron chamfered 12.png
chamfered dodecahedron (canonical)
Polyhedron 12-20 dual.png
rhombic triacontahedron
Polyhedron chamfered 20.png
chamfered icosahedron (canonical)
Polyhedron chamfered 12 dual.png
pentakis icosidodecahedron
Polyhedron 12-20.png
icosidodecahedron
Polyhedron chamfered 20 dual.png
triakis icosidodecahedron

Chamfered icosahedron[edit]

Chamfered icosahedron
Polyhedron chamfered 20 edeq max.png
(with equal edge length)
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.

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


Chamfered regular tilings[edit]

Chamfered regular tilings
Uniform tiling 44-t0.svg
Square tiling, Q
{4,4}
Uniform tiling 63-t2.svg
Triangular tiling, Δ
{3,6}
Uniform tiling 63-t0.svg
Hexagonal tiling, H
{6,3}
Chamfered square tiling.png Chamfered triangular tiling.png Chamfered hexagonal tiling.png Truncated rhombille tiling.png Chamfered hexagonal tiling2.png
cQ cH

Relation to Goldberg polyhedra[edit]

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}
Uniform polyhedron-43-t0.svg
C
Truncated rhombic dodecahedron2.png
cC
Octahedral goldberg polyhedron 04 00.svg
ccC
Octahedral goldberg polyhedron 08 00.svg
cccC
GPV
{5+,3}
Uniform polyhedron-53-t0.svg
D
Truncated rhombic triacontahedron.png
cD
Chamfered chamfered dodecahedron.png
ccD
Chamfered chamfered chamfered dodecahedron.png
cccD
Chamfered chamfered chamfered chamfered dodecahedron.png
ccccD
GPVI
{6+,3}
Uniform tiling 63-t0.svg
H
Truncated rhombille tiling.png
cH
Chamfered chamfered hexagonal tiling.png
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}
Uniform polyhedron-43-t12.svg
tO
Chamfered truncated octahedron.png
ctO
Chamfered chamfered truncated octahedron.png
cctO
GPV
{5+,3}
Uniform polyhedron-53-t12.svg
tI
Chamfered truncated icosahedron.png
ctI
Chamfered chamfered truncated icosahedron.png
cctI
GPVI
{6+,3}
Uniform tiling 63-t12.svg
tH
Chamfered truncated triangular tiling.png
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}
Octahedral goldberg polyhedron 03 00.svg
tkC
Octahedral goldberg polyhedron 06 00.svg
ctkC
cctkC
GPV
{5+,3}
Conway polyhedron Dk6k5tI.png
tkD
Chamfered truncated pentakis dodecahedron.png
ctkD
cctkD
GPVI
{6+,3}
Truncated hexakis hexagonal tiling.png
tkH
Chamfered truncated hexakis hexagonal tiling.png
ctkH
cctkH

Chamfered polytopes and honeycombs[edit]

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[edit]

References[edit]

  • Goldberg, Michael (1937). "A class of multi-symmetric polyhedra". Tohoku Mathematical Journal.
  • Joseph D. Clinton, Clinton’s Equal Central Angle Conjecture [1]
  • 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 [2] (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[edit]