Uniform star polyhedron

See Prismatic uniform polyhedron. == Tetrahedral symmetry ==

There is one nonconvex form, the tetrahemihexahedron which has tetrahedral symmetry (with fundamental domain Möbius triangle (3 3 2)). There are two Schwarz triangles that generate unique nonconvex uniform polyhedra: one right triangle ( 3 2), and one general triangle ( 3 3). The general triangle ( 3 3) generates the octahemioctahedron which is given further on with its full octahedral symmetry. == Octahedral symmetry ==

There are 8 convex forms, and 10 nonconvex forms with octahedral symmetry (with fundamental domain Möbius triangle (4 3 2)). There are four Schwarz triangles that generate nonconvex forms, two right triangles ( 4 2), and ( 3 2), and two general triangles: ( 4 3), ( 4 4). == Icosahedral symmetry ==

There are 8 convex forms and 46 nonconvex forms with icosahedral symmetry (with fundamental domain Möbius triangle (5 3 2)). (or 47 nonconvex forms if Skilling's figure is included). Some of the nonconvex snub forms have reflective vertex symmetry. == Degenerate cases ==

Coxeter identified a number of degenerate star polyhedra by the Wythoff construction method, which contain overlapping edges or vertices. These degenerate forms include: • Small complex icosidodecahedron • Great complex icosidodecahedron • Small complex rhombicosidodecahedron • Great complex rhombicosidodecahedron • Complex rhombidodecadodecahedron Skilling's figure One further nonconvex degenerate polyhedron is the great disnub dirhombidodecahedron, also known as ''Skilling's figure'', which is vertex-uniform, but has pairs of edges which coincide in space such that four faces meet at some edges. It is counted as a degenerate uniform polyhedron rather than a uniform polyhedron because of its double edges. It has Ih symmetry. == See also ==