# Nonconvex uniform polyhedron

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Nonconvex uniform polyhedron

In geometry, a nonconvex uniform polyhedron, or uniform star polyhedron, is a self-intersecting uniform polyhedron. Each 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.

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

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.

Regular polyhedra are labeled by their Schläfli symbol. Other nonregular uniform polyhedra are listed with their vertex configuration or their Uniform polyhedron index U(1-80).

Note: For nonconvex forms below an additional descriptor Nonuniform is used when the convex hull vertex arrangement has same topology as one of these, but has nonregular faces. For example an "nonuniform cantellated" form may have rectangles 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 and truncated tetrahedron, and one nonconvex form, the tetrahemihexahedron which have "tetrahedral symmetry". The tetrahedron is self dual.

In addition the octahedron, truncated octahedron, cuboctahedron, and icosahedron 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 Ih symmetry.

*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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