Uniform star polyhedron

A display of uniform polyhedra at the Science Museum in London

The small snub icosicosidodecahedron is a uniform star polyhedron, with vertex figure 3⁵.⁵/₂

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 (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 48 nonprismatic such uniform star polyhedra. The remaining 9 nonprismatic uniform star polyhedra, those that pass through the center, are the hemipolyhedra, and do not correspond to spherical polyhedra, as the center cannot be projected uniquely onto the sphere.

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

(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
Tetrahedron
Rectified tetrahedron Octahedron	(4.3/2.4.3) 3/2 3 | 2
Truncated tetrahedron
Cantellated tetrahedron (Cuboctahedron)
Omnitruncated tetrahedron (Truncated octahedron)
Snub tetrahedron (Icosahedron)

Octahedral symmetry

(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
Cube
Octahedron
Cuboctahedron	(6.4/3.6.4) 4/3 4 | 3	(6.3/2.6.3) 3/2 3 | 3
Truncated cube	(4.8/3.4/3.8/5) 2 4/3 (3/2 4/2) |	(8/3.3.8/3.4) 3 4 | 4/3	(4.3/2.4.4) 3/2 4 | 2
Truncated octahedron
Rhombicuboctahedron	(4.8.4/3.8) 2 4 (3/2 4/2) |	(8.3/2.8.4) 3/2 4 | 4	(8/3.8/3.3) 2 3 | 4/3
Nonuniform truncated cuboctahedron	(4.6.8/3) 2 3 4/3 |
Nonuniform truncated cuboctahedron	(8/3.6.8) 3 4 4/3 |
Snub cube

Icosahedral symmetry

(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
Icosahedron	{5,5/2}	{5/2,5}	{3,5/2}
Nonuniform truncated icosahedron 2 5 |3	U37 2 5/2 | 5	U61 5/2 3 | 5/3	U67 5/3 3 | 2	U73 2 5/3 (3/2 5/4) |
Nonuniform truncated icosahedron 2 5 |3	U38 5/2 5 | 2	U44 5/3 5 | 3	U56 2 3 (5/4 5/2) |
Nonuniform truncated icosahedron 2 5 |3	U32 | 5/2 3 3
Icosidodecahedron 2 | 3 5	U49 3/2 3 | 5	U51 5/4 5 | 5	U54 2 | 3 5/2	U70 5/3 5/2 | 5/3	U71 3 3 | 5/3	U36 2 | 5 5/2	U62 5/3 5/2 | 3	U65 5/4 5 | 3
Truncated dodecahedron 2 3 | 5	U42	U48	U63
Nonuniform truncated dodecahedron	U72
Dodecahedron	{5/2,3}	U30	U41	U47
Rhombicosidodecahedron	U33	U39	U58
Beveled Dodecahedron	U55
Nonuniform rhombicosidodecahedron	U31	U43	U50	U66
Nonuniform rhombicosidodecahedron	U75	U64	Skilling's figure (see below)
Nonuniform truncated icosidodecahedron	U45
Nonuniform truncated icosidodecahedron	U59
Nonuniform truncated icosidodecahedron	U68
Nonuniform snub dodecahedron	U40	U46	U57	U69	U60	U74

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.

See also

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

Notes

References

• Coxeter, H. S. M. (May 13, 1954). "Uniform Polyhedra". Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 246 (916): 401–450. doi:10.1098/rsta.1954.0003. 
• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. OCLC 1738087. 
• Brückner, M. Vielecke und vielflache. Theorie und geschichte.. Leipzig, Germany: Teubner, 1900.
• Sopov, S. P. (1970), "A proof of the completeness on the list of elementary homogeneous polyhedra", Ukrainskiui Geometricheskiui Sbornik (8): 139–156, MR 0326550 
• Skilling, J. (1975), "The complete set of uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 278: 111–135, doi:10.1098/rsta.1975.0022, ISSN 0080-4614, JSTOR 74475, MR 0365333 
• 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.
• Messer, Peter W. Closed-Form Expressions for Uniform Polyhedra and Their Duals., Discrete & Computational Geometry 27:353-375 (2002).
• Klitzing, Richard. "3D uniform polyhedra". 

External links

• Weisstein, Eric W. "Uniform Polyhedron". MathWorld.