Chamfer (geometry)

Platonic solids

• 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 GP_III or _2,0, containing triangular and hexagonal faces. Its dual is the alternate-triakis tetratetrahedron.
• chamfered cube: from a cube, the resulting polyhedron has twelve hexagonal and six square centrally symmetric faces, a zonohedron. This is also an example of the Goldberg polyhedron GP_IV or _2,0. Its dual is the tetrakis cuboctahedron. A twisty puzzle of the DaYan Gem 7 is the shape of a chamfered cube.
• chamfered octahedron or tritruncated rhombic dodecahedron: from a regular octahedron by chamfering, 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 GP_V or _2,0. Its dual is triakis cuboctahedron.
• 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. GP_V = _2,0. The structure resembles C₆₀ fullerene. Its dual is the pentakis icosidodecahedron.
• 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.

  Regular tilings

Relation to Goldberg polyhedra