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, the
great icosahedron, the
small snub icosicosidodecahedron, and the
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: • The
icosahedron,
snub cube and
snub dodecahedron are the only three
convex ones. They are obtained by snubification of the
truncated octahedron,
truncated cuboctahedron and the
truncated icosidodecahedron - the three convex truncated
quasiregular polyhedra. • The only snub polyhedron with the
chiral octahedral group of symmetries is the
snub cube. • Only the
icosahedron and the
great icosahedron are also
regular polyhedra. They are also
deltahedra. • Only the icosahedron, great icosahedron,
small snub icosicosidodecahedron,
small retrosnub icosicosidodecahedron,
great dirhombicosidodecahedron, and
great disnub dirhombidodecahedron also have reflective symmetries. 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. The exception is the tetrahedron, for which all the faces are derived as red snub triangles, as alternating the square bases of the cube results in degenerate
digons as faces.
Notes: • Two of these polyhedra may be constructed from the first two snub polyhedra in the list starting with the
icosahedron: the
pentagonal antiprism is a
parabidiminished icosahedron and a
pentagrammic crossed-antiprism is a parabidiminished great icosahedron, also known as a
parabireplenished great icosahedron.
Non-uniform Two
Johnson solids are snub polyhedra: the
snub disphenoid and the
snub square antiprism. Neither is chiral. == Bibliography ==