A snub polyhedron is a polyhedron obtained by alternating a corresponding omnitruncated or truncated polyhedron, depending on the definition. Some but not all authors include antiprisms as snub polyhedra, as they obtained by this construction from a degenerate "polyhedron" with only two faces.
Contents
Chiral snub polyhedra do not always have reflection symmetry and hence sometimes have two enantiomorphous forms which are reflections of each other. Their symmetry groups are all point groups.
For example, the snub cube:
Snub polyhedra have Wythoff symbol | p q r and by extension, vertex configuration 3.p.3.q.3.r. Retrosnub polyhedra (a subset of the snub polyhedron, containing the great icosahedron, small retrosnub icosicosidodecahedron, and great retrosnub icosidodecahedron) still have this form of Wythoff symbol, but their vertex configurations are instead (3.−p.3.−q.3.−r)/2.
Among the snub polyhedra that cannot be otherwise generated, only the pentagonal antiprism, pentagrammic antiprism, pentagrammic crossed-antiprism, small snub icosicosidodecahedron and small retrosnub icosicosidodecahedron are known to occur in any non-prismatic uniform 4-polytope. The tetrahedron, octahedron, icosahedron, and great icosahedron appear commonly in non-prismatic uniform 4-polytopes, but not in their snub constructions. Every snub polyhedron however can appear in the polyhedral prism based on them.
Uniform
There are 12 uniform snub polyhedra, not including the antiprisms, the icosahedron as a snub tetrahedron, the great icosahedron as a retrosnub tetrahedron and the great disnub dirhombidodecahedron, also known as Skilling's figure.
When the Schwarz triangle of the snub polyhedron is isosceles, the snub polyhedron is not chiral. This is the case for the antiprisms, the icosahedron, great icosahedron, small snub icosicosidodecahedron, and small retrosnub icosicosidodecahedron.
In the pictures of the snub derivation (showing a distorted snub polyhedron, topologically identical to the uniform version, arrived at from geometrically alternating the parent uniform omnitruncated polyhedron) where green is not present, the faces derived from alternation are coloured red and yellow, while the snub triangles are blue. Where green is present (only for the snub icosidodecadodecahedron and great snub dodecicosidodecahedron), the faces derived from alternation are red, yellow, and blue, while the snub triangles are green.
Notes:
There is also the infinite set of antiprisms. They are formed from prisms, which are truncated hosohedra, degenerate regular polyhedra. Those up to hexagonal are listed below. In the pictures showing the snub derivation, the faces derived from alternation (of the prism bases) are coloured red, and the snub triangles are coloured yellow.
Notes:
Non-uniform
Two Johnson solids are snub polyhedra: the snub disphenoid and the snub square antiprism. Neither is chiral.