For the concept in machining and architecture, see chamfer.
In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion: it moves the faces apart (outward), and adds a new face between each two adjacent faces; but contrary to expansion, it maintains the original vertices. (Equivalently: it separates the faces by reducing them, and adds a new face between each two adjacent faces; but it only moves the vertices inward.) For a polyhedron, this operation adds a new hexagonal face in place of each original edge.
In Conway polyhedron notation, chamfering 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 an equilateral version where all edges have the same length, and in a canonical version where all edges touch the same midsphere. (They look noticeably different only for solids containing triangles.) The shown dual polyhedra are dual to the canonical versions.
or by alternately truncating a (regular) cube: replacing 4 of its 8 vertices with congruent equilateral-triangle faces.
For a certain depth of chamfering/truncation, all (final) edges of the cT have the same length; then, the hexagons are equilateral, but not regular.
The dual of the chamfered tetrahedron is the alternate-triakis tetratetrahedron.
The cT is the Goldberg polyhedron GPIII(2,0) or {3+,3}2,0, containing triangular and hexagonal faces.
The truncated tetrahedron looks similar; but its hexagons correspond to the 4 faces, not to the 6 edges, of the yellow tetrahedron, i.e. to the 4 vertices, not to the 6 edges, of the red tetrahedron.Historical drawings of truncated tetrahedron and slightly chamfered tetrahedron.[1]
Net (3 zones are shown by 3 colors for their hexagons — each square is in 2 zones —.)
*for a certain chamfering depth
The chamfered cube is constructed as a chamfer of a cube: the squares are reduced in size and new faces, hexagons, are added in place of all the original edges. The cC is a convex polyhedron with 32 vertices, 48 edges, and 18 faces: 6 congruent (and regular) squares, and 12 congruent flattened hexagons. For a certain depth of chamfering, all (final) edges of the chamfered cube have the same length; then, the hexagons are equilateral, but not regular. They are congruent alternately truncatedrhombi, have 2 internal angles of and 4 internal angles of while a regular hexagon would have all internal angles.
The cC is also inaccurately called a truncated rhombic dodecahedron, although that name rather suggests a rhombicuboctahedron. The cC can more accurately be called a tetratruncated rhombic dodecahedron, because only the (6) order-4 vertices of the rhombic dodecahedron are truncated.
Chamfered cube (3 zones are shown by 3 colors for their hexagons — each square is in 2 zones —.)
The chamfered cube is also the Goldberg polyhedron GPIV(2,0) or {4+,3}2,0, containing square and hexagonal faces.
The cC is the Minkowski sum of a rhombic dodecahedron and a cube of edge length 1 when the eight order-3 vertices of the rhombic dodecahedron are at and its six order-4 vertices are at the permutations of
A topological equivalent to the chamfered cube, but 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 of axis shallower and deeper truncations of pyritohedron
The truncated octahedron looks similar; but its hexagons correspond to the 8 faces, not to the 12 edges, of the octahedron, i.e. to the 8 vertices, not to the 12 edges, of the cube.
In geometry, the chamfered octahedron is a convex polyhedron constructed by truncating the 8 order-3 vertices of the rhombic dodecahedron. These truncated vertices become congruent equilateral triangles, and the original 12 rhombic faces become congruent flattened hexagons. For a certain depth of truncation, all (final) edges of the cO have the same length; then, the hexagons are equilateral, but not regular.
The chamfered octahedron can also be called a tritruncated rhombic dodecahedron.
The dual of the cO is the triakis cuboctahedron.
Historical drawings of rhombic dodecahedron and slightly chamfered octahedron
Historical models of triakis cuboctahedron and slightly chamfered octahedron
The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 12 congruent regular pentagons and 30 congruent flattened hexagons. It is constructed as a chamfer of a regular dodecahedron. The pentagons are reduced in size and new faces, flattened hexagons, are added in place of all the original edges. For a certain depth of chamfering, all (final) edges of the cD have the same length; then, the hexagons are equilateral, but not regular.
The cD is also inaccurately called a truncated rhombic triacontahedron, although that name rather suggests a rhombicosidodecahedron. The cD can more accurately be called a pentatruncated rhombic triacontahedron, because only the (12) order-5 vertices of the rhombic triacontahedron are truncated.
The cD is the Goldberg polyhedron GPV(2,0) or {5+,3}2,0, containing pentagonal and hexagonal faces.
The truncated icosahedron looks similar, but its hexagons correspond to the 20 faces, not to the 30 edges, of the icosahedron, i.e. to the 20 vertices, not to the 30 edges, of the dodecahedron.
The chamfer operation applied in series creates progressively larger polyhedra with new faces, hexagonal, replacing the edges of the current one. The chamfer operator transforms GP(m,n) to GP(2m,2n).
A regular polyhedron, GP(1,0), creates a Goldberg polyhedra sequence: GP(1,0), GP(2,0), GP(4,0), GP(8,0), GP(16,0)...
Like the expansion operation, chamfer can be applied to any dimension.
For polygons, it triples the number of vertices. Example:
A chamfered square (See also the previous version of this figure.)
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.[something may be wrong in this passage]
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