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In geometry, a chamfer or edge-truncation is a topological operator that modifies one polyhedron into another. It separates the faces by reducing them, and adds a new face between each two adjacent faces (moving the vertices inward). Oppositely, 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.
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.
Left to right: chamfered tetrahedron, cube, octahedron, dodecahedron, and icosahedron
Chamfers of five Platonic solids are described in detail below.
Historical drawings of truncated tetrahedron and slightly chamfered tetrahedron.[1]
chamfered tetrahedron or alternated truncated cube: from a regular tetrahedron, this replaces its six edges with congruent flattened hexagons; or alternately truncating a cube, replacing four of its eight vertices with congruent equilateral-triangle faces. This is an example of Goldberg polyhedron GPIII(2,0) or {3+,3}2,0, containing triangular and hexagonal faces. Its dual is the alternate-triakis tetratetrahedron.[2]
chamfered cube: from a cube, the resulting polyhedron has twelve hexagonal and six square centrally symmetric faces, a zonohedron.[3] This is also an example of the Goldberg polyhedron GPIV(2,0) or {4+,3}2,0. Its dual is the tetrakis cuboctahedron. A twisty puzzle of the DaYan Gem 7 is the shape of a chamfered cube.[4]
chamfered octahedron or tritruncated rhombic dodecahedron: from a regular octahedron by chamfering,[5] or by truncating the eight order-3 vertices of the rhombic dodecahedron, which become congruent equilateral triangles, and the original twelve rhombic faces become congruent flattened hexagons. It is a Goldberg polyhedron GPV(2,0) or {5+,3}2,0. Its dual is triakis cuboctahedron.[2]
chamfered dodecahedron: by chamfering a regular dodecahedron, the resulting polyhedron has 80 vertices, 120 edges, and 42 faces: 12 congruent regular pentagons and 30 congruent flattened hexagons. GPV(2,0) = {5+,3}2,0. The structure resembles C60fullerene.[6] Its dual is the pentakis icosidodecahedron.[2]
chamfered icosahedron or tritruncated rhombic triacontahedron: by chamfering a regular icosahedron, or truncating the twenty order-3 vertices of the rhombic triacontahedron. The hexagonal faces of the cI can be made equilateral, but not regular, with a certain depth of truncation. Its dual is the triakis icosidodecahedron.[2]
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)...
