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

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.

Contents

Chamfered Platonic solidsEdit

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 tetrahedronEdit

Chamfered tetrahedron
 
(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
 
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
 
chamfered tetrahedron (canonical)
 
dual of the tetratetrahedron
 
chamfered tetrahedron (canonical)
 
alternate-triakis tetratetrahedron
 
tetratetrahedron
 
alternate-triakis tetratetrahedron

Chamfered cubeEdit

Chamfered cube
 
(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
 
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  .

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 cuboctahedon

Chamfered octahedronEdit

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

Chamfered dodecahedronEdit

Chamfered dodecahedron
 
(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
 
chamfered dodecahedron (canonical)
 
rhombic triacontahedron
 
chamfered icosahedron (canonical)
 
pentakis icosidodecahedron
 
icosidodecahedron
 
triakis icosidodecahedron

Chamfered icosahedronEdit

Chamfered icosahedron
 
(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 tilingsEdit

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 cH cdaH

Relation to Goldberg polyhedraEdit

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 honeycombsEdit

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 alsoEdit

ReferencesEdit

  • 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 (ed.). 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 linksEdit