Uniform star polyhedron

File:Uniform-Polyhedra-at-the-Science-Museum.jpg

A display of uniform polyhedra at the Science Museum in London

File:Small snub icosicosidodecahedron.png

The small snub icosicosidodecahedron is a uniform star polyhedron, with vertex figure 3₅.Template:Sfrac

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, 14 quasiregular ones, and 39 semiregular ones.

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

Just as (nondegenerate) star polygons (which have polygon density greater than 1) 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.

Regular polyhedra are labeled by their Schläfli symbol. Other nonregular uniform polyhedra are listed with their vertex configuration.

An additional figure, the pseudo great rhombicuboctahedron, is usually not included as a truly uniform star polytope, despite consisting of regular faces and having the same vertices.

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

File:Disdyakis 6 spherical.png

(3 3 2) triangles on sphere

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 3 2), and one general triangle (3⁄2 3 3). The general triangle (3⁄2 3 3) generates the octahemioctahedron which is given further on with its full octahedral symmetry.

Vertex arrangement (Convex hull)	Nonconvex forms
File:Tetrahedron.png Tetrahedron
File:Rectified tetrahedron.png Rectified tetrahedron Octahedron	File:Tetrahemihexahedron.png 4.3⁄2.4.3 3⁄2 3 | 2
File:Truncated tetrahedron.png Truncated tetrahedron
File:Cantellated tetrahedron.png Cantellated tetrahedron (Cuboctahedron)
File:Uniform polyhedron-33-t012.png Omnitruncated tetrahedron (Truncated octahedron)
File:Uniform polyhedron-33-s012.png Snub tetrahedron (Icosahedron)

Octahedral symmetry

File:Disdyakis 12 spherical.png

(4 3 2) triangles on sphere

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).

Vertex arrangement (Convex hull)	Nonconvex forms
File:Hexahedron.png Cube
File:Octahedron.png Octahedron
File:Cuboctahedron.png Cuboctahedron	File:Cubohemioctahedron.png 6.4⁄3.6.4 4⁄3 4 | 3	File:Octahemioctahedron.png 6.3⁄2.6.3 3⁄2 3 | 3
File:Truncated hexahedron.png Truncated cube	File:Great rhombihexahedron.png 4.8⁄3.4⁄3.8⁄5 2 4⁄3 (3⁄2 4⁄2) |	File:Great cubicuboctahedron.png 8⁄3.3.8⁄3.4 3 4 | 4⁄3	File:Uniform great rhombicuboctahedron.png 4.3⁄2.4.4 3⁄2 4 | 2
File:Truncated octahedron.png Truncated octahedron
File:Small rhombicuboctahedron.png Rhombicuboctahedron	File:Small rhombihexahedron.png 4.8.4⁄3.8⁄7 2 4 (3⁄2 4⁄2) |	File:Small cubicuboctahedron.png 8.3⁄2.8.4 3⁄2 4 | 4	File:Stellated truncated hexahedron.png 8⁄3.8⁄3.3 2 3 | 4⁄3
File:Great truncated cuboctahedron convex hull.png Nonuniform truncated cuboctahedron	File:Great truncated cuboctahedron.png 4.6.8⁄3 2 3 4⁄3 |
File:Cubitruncated cuboctahedron convex hull.png Nonuniform truncated cuboctahedron	File:Cubitruncated cuboctahedron.png 8⁄3.6.8 3 4 4⁄3 |
File:Snub hexahedron.png Snub cube

Icosahedral symmetry

File:Disdyakis 30 spherical.png

(5 3 2) triangles on sphere

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.

Vertex arrangement (Convex hull)	Nonconvex forms
File:Icosahedron.png Icosahedron	File:Great dodecahedron.png {5,5⁄2}	File:Small stellated dodecahedron.png {5⁄2,5}	File:Great icosahedron.png {3,5⁄2}
File:Nonuniform truncated icosahedron.png Nonuniform truncated icosahedron	File:Great truncated dodecahedron.png 10.10.5⁄2 2 5⁄2 | 5	File:Great dodecicosidodecahedron.png 3.10⁄3.5⁄2.10⁄7 5⁄2 3 | 5⁄3	File:Uniform great rhombicosidodecahedron.png 3.4.5⁄3.4 5⁄3 3 | 2	File:Great rhombidodecahedron.png 4.10⁄3.4⁄3.10⁄7 2 5⁄3 (3⁄2 5⁄4) |
File:Rhombidodecadodecahedron convex hull.png Nonuniform truncated icosahedron	File:Rhombidodecadodecahedron.png 4.5⁄2.4.5 5⁄2 5 | 2	File:Icosidodecadodecahedron.png 5.6.5⁄3.6 5⁄3 5 | 3	File:Rhombicosahedron.png 4.6.4⁄3.6⁄5 2 3 (5⁄4 5⁄2) |
File:Small snub icosicosidodecahedron convex hull.png Nonuniform truncated icosahedron	File:Small snub icosicosidodecahedron.png 35.5⁄2 | 5⁄2 3 3
File:Icosidodecahedron.png Icosidodecahedron	File:Small icosihemidodecahedron.png 3.10.3⁄2.10 3⁄2 3 | 5	File:Small dodecahemidodecahedron.png 5.10.5⁄4.10 5⁄4 5 | 5	File:Great icosidodecahedron.png 3.5⁄2.3.5⁄2 2 | 3 5⁄2	File:Great dodecahemidodecahedron.png 5⁄2.10⁄3.5⁄3.10⁄3 5⁄3 5⁄2 | 5⁄3	File:Great icosihemidodecahedron.png 3.10⁄3.3⁄2.10⁄3 3 3 | 5⁄3	File:Dodecadodecahedron.png 5.5⁄2.5.5⁄2 2 | 5 5⁄2	File:Small dodecahemicosahedron.png 6.5⁄2.6.5⁄3 5⁄3 5⁄2 | 3	File:Great dodecahemicosahedron.png 5.6.5⁄4.6 5⁄4 5 | 3
File:Truncated dodecahedron.png truncated dodecahedron	File:Great ditrigonal dodecicosidodecahedron.png 3.10⁄3.5.10⁄3 3 5 | 5⁄3	File:Great icosicosidodecahedron.png 5.6.3⁄2.6 3⁄2 5 | 3	File:Great dodecicosahedron.png 6.10⁄3.6⁄5.10⁄7 3 5⁄3 (3⁄2 5⁄2) |
File:Small retrosnub icosicosidodecahedron convex hull.png Nonuniform truncated dodecahedron	File:Small retrosnub icosicosidodecahedron.png (35.5⁄3)/2 | 3⁄2 3⁄2 5⁄2
File:Dodecahedron.png Dodecahedron	File:Great stellated dodecahedron.png {5⁄2,3}	File:Small ditrigonal icosidodecahedron.png (3.5⁄2)3 3 | 5⁄2 3	File:Ditrigonal dodecadodecahedron.png (5.5⁄3)3 3 | 5⁄3 5	File:Great ditrigonal icosidodecahedron.png ((3.5)3)/2 3⁄2 | 3 5
File:Small rhombicosidodecahedron.png Rhombicosidodecahedron	File:Small dodecicosidodecahedron.png 5.10.3⁄2.10 3⁄2 5 | 5	File:Small rhombidodecahedron.png 4.10.4⁄3.10⁄9 2 5 (3⁄2 5⁄2) |	File:Small stellated truncated dodecahedron.png 5.10⁄3.10⁄3 2 5 | 5⁄3
File:Truncated great icosahedron convex hull.png Nonuniform rhombicosidodecahedron	File:Great truncated icosahedron.png 6.6.5⁄2 2 5⁄2 | 3
File:Nonuniform-rhombicosidodecahedron.png Nonuniform rhombicosidodecahedron	File:Small icosicosidodecahedron.png 6.5⁄2.6.3 5⁄2 3 | 3	File:Small ditrigonal dodecicosidodecahedron.png 3.10.5⁄3.10 5⁄3 3 | 5	File:Small dodecicosahedron.png 6.10.6⁄5.10⁄9 3 5 (3⁄2 5⁄4) |	File:Great stellated truncated dodecahedron.png 3.10⁄3.10⁄3 2 3 | 5⁄3
File:Nonuniform2-rhombicosidodecahedron.png Nonuniform rhombicosidodecahedron	File:Great dirhombicosidodecahedron.png 4.5⁄3.4.3.4.5⁄2.4.3⁄2 | 3⁄2 5⁄3 3 5⁄2	File:Great snub dodecicosidodecahedron.png 3.3.3.5⁄2.3.5⁄3 | 5⁄3 5⁄2 3	File:Great disnub dirhombidodecahedron.png Skilling's figure (see below)
File:Icositruncated dodecadodecahedron convex hull.png Nonuniform truncated icosidodecahedron	File:Icositruncated dodecadodecahedron.png 6.10.10⁄3 3 5 5⁄3 |
File:Truncated dodecadodecahedron convex hull.png Nonuniform truncated icosidodecahedron	File:Truncated dodecadodecahedron.png 4.10⁄9.10⁄3 2 5 5⁄3 |
File:Great truncated icosidodecahedron convex hull.png Nonuniform truncated icosidodecahedron	File:Great truncated icosidodecahedron.png 4.6.10⁄3 2 3 5⁄3 |
File:Snub dodecahedron ccw.png Nonuniform snub dodecahedron	File:Snub dodecadodecahedron.png 3.3.5⁄2.3.5 | 2 5⁄2 5	File:Snub icosidodecadodecahedron.png 3.3.3.5.3.5⁄3 | 5⁄3 3 5	File:Great snub icosidodecahedron.png 34.5⁄2 | 2 5⁄2 3	File:Great inverted snub icosidodecahedron.png 34.5⁄3 | 5⁄3 2 3	File:Inverted snub dodecadodecahedron.png 3.3.5.3.5⁄3 | 5⁄3 2 5	File:Great retrosnub icosidodecahedron.png (34.5⁄2)/2 | 3⁄2 5⁄3 2

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 I_h symmetry.

File:Great disnub dirhombidodecahedron.png

See also

• Star polygon
• List of uniform polyhedra
• List of uniform polyhedra by Schwarz triangle

References

• Brückner, M. Vielecke und vielflache. Theorie und geschichte.. Leipzig, Germany: Teubner, 1900. [1]
• Har'El, Z. Uniform Solution for Uniform Polyhedra., Geometriae Dedicata 47, 57-110, 1993. Zvi Har’El, Kaleido software, Images, dual images
• Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993. [2]
• Messer, Peter W. Closed-Form Expressions for Uniform Polyhedra and Their Duals., Discrete & Computational Geometry 27:353-375 (2002).
• Template:KlitzingPolytopes