page_title: Uniform star polyhedron
section_title: Octahedral symmetry
hierarchical_section_title: Uniform star polyhedron / Octahedral symmetry
caption_attribution_description: English: KaleidoTile, Topology and Geometry Software, Jeff Weeks Spherical domains for octahedral symmetry
context_page_description: In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures or both.
  The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler–Poinsot polyhedra, 5 quasiregular ones, and 48 semiregular ones.
  There are also two infinite sets of uniform star prisms and uniform star antiprisms.
  Just as star polygons correspond to circular polygons with overlapping tiles, star polyhedra that do not pass through the center have polytope density greater than 1, and correspond to spherical polyhedra with overlapping tiles; there are 47 nonprismatic such uniform star polyhedra. The remaining 10 nonprismatic uniform star polyhedra, those that pass through the center, are the hemipolyhedra as well as Miller's monster, and do not have well-defined densities.
  The nonconvex forms are constructed from Schwarz triangles.
  All the uniform polyhedra are listed below by their symmetry groups and subgrouped by their vertex arrangements.
context_section_description: 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 (​³⁄₂ 4 2), and (​⁴⁄₃ 3 2), and two general triangles: (​⁴⁄₃ 4 3), (​³⁄₂ 4 4).
caption: (4 3 2) triangles on sphere
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