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.
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. The shown duals are dual to the canonical versions.| Seed | tetrahedron| | cube| | octahedron| | dodecahedron| | icosahedron| |
| Chamfered |
Chamfered tetrahedron
The chamfered tetrahedron 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, containing triangular and hexagonal faces.
looks similar, but its hexagons correspond to 4 of the tetrahedrons vertices, rather than to its 6 edges.
chamfered tetrahedron | dual of the tetratetrahedron | chamfered tetrahedron |
alternate-triakis tetratetrahedron | tetratetrahedron | alternate-triakis tetratetrahedron |
Chamfered cube
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 or 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.
looks similar, but its hexagons correspond to the 8 vertices of the cube, rather than to its 12 edges.
chamfered cube | rhombic dodecahedron | chamfered octahedron |
tetrakis cuboctahedron | cuboctahedron | triakis cuboctahedron |
Chamfered octahedron
In geometry, the chamfered octahedron is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 8 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.
Chamfered dodecahedron
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.
looks similar, but its hexagons correspond to the 20 vertices of the dodecahedron, rather than to its 30 edges.
chamfered dodecahedron | rhombic triacontahedron | chamfered icosahedron |
pentakis icosidodecahedron | icosidodecahedron | triakis icosidodecahedron |
Chamfered icosahedron
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
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 to GP.A regular polyhedron, GP, create a Goldberg polyhedra sequence: GP, GP, GP, GP, GP...
| GP | GP | GP | GP | GP... | |
| GPIV | C | cC | ccC | cccC | |
| GPV | D | cD | ccD | cccD | ccccD |
| GPVI | H | cH | ccH | cccH | ccccH |
The truncated octahedron or truncated icosahedron, GP creates a Goldberg sequence: GP, GP, GP, GP....
| GP | GP | GP... | |
| GPIV | tO | ctO | cctO |
| GPV | tI | ctI | cctI |
| GPVI | tH | ctH | cctH |
A truncated tetrakis hexahedron or pentakis dodecahedron, GP, creates a Goldberg sequence: GP, GP, GP...
| GP | GP | GP... | |
| GPIV | tkC | ctkC | cctkC |
| GPV | tkD | ctkD | cctkD |
| GPVI | tkH | ctkH | cctkH |