In geometry, chamfering or edgetruncation 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 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 notation  cT 
Goldberg polyhedron  GP_{III}(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 (T_{d}) 
Dual polyhedron  Alternatetriakis tetratetrahedron 
Properties  convex, equilateralfaced 
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 G_{III}(2,0), containing triangular and hexagonal faces.
chamfered tetrahedron (canonical) 
dual of the tetratetrahedron 
chamfered tetrahedron (canonical) 
alternatetriakis tetratetrahedron 
tetratetrahedron 
alternatetriakis tetratetrahedron 
Chamfered cube
Chamfered cube  

(with equal edge length)  
Conway notation  cC = t4daC 
Goldberg polyhedron  GP_{IV}(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  O_{h}, [4,3], (*432) T_{h}, [4,3^{+}], (3*2) 
Dual polyhedron  Tetrakis cuboctahedron 
Properties  convex, zonohedron, equilateralfaced 
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 order4 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 GP_{IV}(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.


chamfered cube (canonical) 
rhombic dodecahedron 
chamfered octahedron (canonical) 
tetrakis cuboctahedron 
cuboctahedron 
triakis cuboctahedon 
Chamfered octahedron
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  O_{h}, [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 order3 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.
Chamfered dodecahedron
Chamfered dodecahedron  

(with equal edge length)  
Conway notation  cD] = t5daD = dk5aD 
Goldberg polyhedron  G_{V}(2,0) = {5+,3}_{2,0} 
Fullerene  C_{80}^{[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 (I_{h}) 
Dual polyhedron  Pentakis icosidodecahedron 
Properties  convex, equilateralfaced 
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 order5 vertices are truncated.
chamfered dodecahedron (canonical) 
rhombic triacontahedron 
chamfered icosahedron (canonical) 
pentakis icosidodecahedron 
icosidodecahedron 
triakis icosidodecahedron 
Chamfered icosahedron
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  I_{h}, [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 order3 vertices. The hexagonal faces can be made equilateral but not regular.
It can also be called a tritruncated rhombic triacontahedron, a truncation of the order3 vertices of the rhombic triacontahedron.
Chamfered regular 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)...  

GP_{IV} {4+,3} 
C 
cC 
ccC 
cccC 

GP_{V} {5+,3} 
D 
cD 
ccD 
cccD 
ccccD 
GP_{VI} {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)...  

GP_{IV} {4+,3} 
tO 
ctO 
cctO 
GP_{V} {5+,3} 
tI 
ctI 
cctI 
GP_{VI} {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)...  

GP_{IV} {4+,3} 
tkC 
ctkC 
cctkC 
GP_{V} {5+,3} 
tkD 
ctkD 
cctkD 
GP_{VI} {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
 Goldberg, Michael (1937). "A class of multisymmetric 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:10.1007/9780387927145_9.
 Hart, George (June 18, 2013). "Mathematical Impressions: Goldberg Polyhedra". Simons Science News.
 Antoine Deza, Michel Deza, Viatcheslav Grishukhin, Fullerenes and coordination polyhedra versus halfcube embeddings, 1998 PDF [2] (p. 72 Fig. 26. Chamfered tetrahedron)
 Deza, A.; Deza, M.; Grishukhin, V. (1998), "Fullerenes and coordination polyhedra versus halfcube embeddings", Discrete Mathematics, 192 (1): 41–80, doi:10.1016/S0012365X(98)00065X, archived from the original on 20070206.
External links
 Chamfered Tetrahedron
 Chamfered Solids
 Vertex and edgetruncation of the Platonic and Archimedean solids leading to vertextransitive polyhedra Livio Zefiro
 VRML polyhedral generator (Conway polyhedron notation)
 VRML model Chamfered cube
 3.2.7. Systematic numbering for (C80Ih) [5,6] fullerene
 Fullerene C80
 How to make a chamfered cube