- Nonconvex uniform polyhedron
In

geometry , a**nonconvex**, oruniform polyhedron **uniform star polyhedron**, is a self-intersectinguniform polyhedron . Each can contain eitherstar polygon faces,star polygon vertex figure s" or both.The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler-Poinsot polyhedra.

There are also two infinite sets of "uniform star prisms" and "uniform star antiprisms".

The nonconvex forms are constructed from

Schwarz triangle s.All the uniform polyhedra are listed below by their

symmetry group s and subgrouped by their vertex arrangements.Regular polyhedra are labeled by their

Schläfli symbol . Other nonregular uniform polyhedra are listed with theirvertex configuration or their Uniform polyhedron index U(1-80).Note: For nonconvex forms below an additional descriptor

**Nonuniform**is used when theconvex hull vertex arrangement has same topology as one of these, but has nonregular faces. For example an "nonuniform cantellated" form may haverectangles created in place of the edges rather than squares.**Dihedral symmetry**See:

Prismatic_uniform_polyhedron **Tetrahedral symmetry**There are 2 convex uniform polyhedra, the

tetrahedron andtruncated tetrahedron , and one nonconvex form, thetetrahemihexahedron which have "tetrahedral symmetry ". Thetetrahedron is self dual.In addition the

octahedron ,truncated octahedron ,cuboctahedron , andicosahedron have tetrahedral symmetry as well as higher symmetry. They are added for completeness below, although their nonconvex forms with octahedral symmetry are not included here.**Skilling's figure**One further nonconvex 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 sometimes but not always counted as a uniform polyhedron. It has**I**_{h}symmetry.**See also***

Star polygon

*list of uniform polyhedra**References*** cite journal

last = Coxeter

first = H. S. M.

authorlink = Harold Scott MacDonald Coxeter

title = Uniform Polyhedra

journal = Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences

volume = 246,

number = 916

date = May 13, 1954

pages = 401–450

doi = 10.1098/rsta.1954.0003

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