In geometry, an omnitruncated polyhedron is a truncated quasiregular polyhedron. When they are alternated, they produce the snub polyhedra.
All omnitruncated polyhedra are considered as zonohedra. They have Wythoff symbol p q r | and vertex figures as 2p.2q.2r.
More generally, an omnitruncated polyhedron is a bevel operator in Conway polyhedron notation.
List of convex omnitruncated polyhedra
There are three convex forms. These forms can be seen as red faces of one regular polyhedron, yellow or green faces of the dual polyhedron, and blue faces at the truncated vertices of the quasiregular polyhedron.
Wythoffsymbol p q r Omnitruncated polyhedron Regular/quasiregular polyhedra 3 3 2 Truncated octahedron Tetrahedron/Octahedron 4 3 2 Truncated cuboctahedron Cube/Cuboctahedron/Octahedron 5 3 2 Truncated icosidodecahedron Dodecahedron/Icosidodecahedron/Icosahedron
List of nonconvex omnitruncated polyhedra
There are 5 nonconvex uniform omnitruncated polyhedra.
Wythoffsymbol p q r Omnitruncated star polyhedronWythoffsymbol p q r Omnitruncated star polyhedronRight triangle domains (r=2)General triangle domains 3 4/3 2 Great truncated cuboctahedron 4 4/3 3 Cubitruncated cuboctahedron 3 5/3 2 Great truncated icosidodecahedron 5 5/3 3 Icositruncated dodecadodecahedron 5 5/3 2 Truncated dodecadodecahedron
Other even-sided nonconvex polyhedra
There are 8 nonconvex forms with mixed Wythoff symbols p q (r s) |, and bow-tie shaped vertex figures, 2p.2q.-2q.-2p. They are not true omnitruncated polyhedra. Instead, the true omnitruncates p q r | or p q s | have coinciding 2r-gonal or 2s-gonal faces that must be removed respectively to form a proper polyhedron. All these polyhedra are one-sided, i.e. non-orientable. The p q r | degenerate Wythoff symbols are listed first, followed by the actual mixed Wythoff symbols.
Omnitruncated polyhedron Image Wythoff symbol Cubohemioctahedron 3/2 2 3 2 3 (3/2 3/2) Small rhombihexahedron 3/2 2 4 2 4 (3/2 4/2) Great rhombihexahedron 4/3 3/2 2 2 4/3 (3/2 4/2) Small rhombidodecahedron 2 5/2 5 2 5 (3/2 5/2) Small dodecicosahedron 3/2 3 5 3 5 (3/2 5/4) Rhombicosahedron 2 5/2 3 2 3 (5/4 5/2) Great dodecicosahedron 5/2 5/3 3 3 5/3 (3/2 5/2) Great rhombidodecahedron 3/2 5/3 2 2 5/3 (3/2 5/4)
General omnitruncations (bevel) Omnitruncations are also called cantitruncations or truncated rectifications (tr), and Conway's bevel (b) operator. When applied to nonregular polyhedra, new polyhedra can be generated, for example these 2-uniform polyhedra: CoxetertrrCtrrDtrtTtrtCtrtOtrtIConwaybaObaDbtTbtCbtObtIImage
See also Uniform polyhedron
References
Har'El, Z. Uniform Solution for Uniform Polyhedra., Geometriae Dedicata 47, 57-110, 1993. Zvi Har'El, Kaleido software, Images, dual images Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993.