title: Omnitruncated polyhedron
text: # Omnitruncated polyhedron
  
  In geometry, an omnitruncated polyhedron is a truncated quasiregular polyhedron. When they are alternated, they produce the snub polyhedra.
  
  All omnitruncated polyhedra are 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. They 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.
  
  | Wythoff symbol p q r | | Omnitruncated polyhedron | Regular/quasiregular polyhedra |
  | - | - | - | - |
  | 3 3 2 | | Truncated octahedron | Tetrahedron/Octahedron/Tetrahedron |
  | 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.
  
  | Wythoff symbol p q r | | Omnitruncated star polyhedron | Wythoff symbol p q r | | Omnitruncated star polyhedron |
  | - | - | - | - | - | - |
  | Right triangle domains (r=2) | Right triangle domains (r=2) | General triangle domains | 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: the true omnitruncates p q r | or p q s | have coinciding 2r-gonal or 2s-gonal faces respectively that must be removed 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 | 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:
  
  | Coxeter | trrC | trrD | trtT | trtC | trtO | trtI |
  | - | - | - | - | - | - | - |
  | Conway | baO | baD | btT | btC | btO | btI |
  | Image | | | | | | |
  