Uniform star polyhedron

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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 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[edit]

See Prismatic uniform polyhedron.

Tetrahedral symmetry[edit]

(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 (​32 3 2), and one general triangle (​32 3 3). The general triangle (​32 3 3) generates the octahemioctahedron which is given further on with its full octahedral symmetry.

Vertex arrangement
(Convex hull)
Nonconvex forms
Tetrahedron.png
Tetrahedron
 
Rectified tetrahedron.png
Rectified tetrahedron
Octahedron
Tetrahemihexahedron.png
4.​32.4.3
32 3 | 2
Truncated tetrahedron.png
Truncated tetrahedron
 
Cantellated tetrahedron.png
Cantellated tetrahedron
(Cuboctahedron)
 
Uniform polyhedron-33-t012.png
Omnitruncated tetrahedron
(Truncated octahedron)
 
Uniform polyhedron-33-s012.png
Snub tetrahedron
(Icosahedron)
 

Octahedral symmetry[edit]

(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 (​32 4 2), and (​43 3 2), and two general triangles: (​43 4 3), (​32 4 4).

Vertex arrangement
(Convex hull)
Nonconvex forms
Hexahedron.png
Cube
 
Octahedron.png
Octahedron
 
Cuboctahedron.png
Cuboctahedron
Cubohemioctahedron.png
6.​43.6.4
43 4 | 3
Octahemioctahedron.png
6.​32.6.3
32 3 | 3
Truncated hexahedron.png
Truncated cube
Great rhombihexahedron.png
4.​83.​43.​85
2 ​43 (​3242) |
Great cubicuboctahedron.png
83.3.​83.4
3 4 | ​43
Uniform great rhombicuboctahedron.png
4.​32.4.4
32 4 | 2
Truncated octahedron.png
Truncated octahedron
 
Small rhombicuboctahedron.png
Rhombicuboctahedron
Small rhombihexahedron.png
4.8.​43.8
2 4 (​3242) |
Small cubicuboctahedron.png
8.​32.8.4
32 4 | 4
Stellated truncated hexahedron.png
83.​83.3
2 3 | ​43
Great truncated cuboctahedron convex hull.png
Nonuniform
truncated cuboctahedron
Great truncated cuboctahedron.png
4.6.​83
2 3 ​43 |
Cubitruncated cuboctahedron convex hull.png
Nonuniform
truncated cuboctahedron
Cubitruncated cuboctahedron.png
83.6.8
3 4 ​43 |
Snub hexahedron.png
Snub cube
 

Icosahedral symmetry[edit]

(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.png
Icosahedron
Great dodecahedron.png
{5,​52}
Small stellated dodecahedron.png
{​52,5}
Great icosahedron.png
{3,​52}
Nonuniform truncated icosahedron.png
Nonuniform
truncated icosahedron
Great truncated dodecahedron.png
10.10.​52
2 ​52 | 5
Great dodecicosidodecahedron.png
3.​103.​52.​107
52 3 | ​53
Uniform great rhombicosidodecahedron.png
3.4.​53.4
53 3 | 2
Great rhombidodecahedron.png
4.​103.​43.​107
2 ​53 (​3254) |
Rhombidodecadodecahedron convex hull.png
Nonuniform
truncated icosahedron
Rhombidodecadodecahedron.png
4.​52.4.5
52 5 | 2
Icosidodecadodecahedron.png
5.6.​53.6
53 5 | 3
Rhombicosahedron.png
4.6.​43.​65
2 3 (​5452) |
Small snub icosicosidodecahedron convex hull.png
Nonuniform
truncated icosahedron
Small snub icosicosidodecahedron.png
35.​52
| ​52 3 3
Icosidodecahedron.png
Icosidodecahedron
Small icosihemidodecahedron.png
3.10.​32.10
32 3 | 5
Small dodecahemidodecahedron.png
5.10.​54.10
54 5 | 5
Great icosidodecahedron.png
3.​52.3.​52
2 | 3 ​52
Great dodecahemidodecahedron.png
52.​103.​53.​103
5352 | ​53
Great icosihemidodecahedron.png
3.​103.​32.​103
3 3 | ​53
Dodecadodecahedron.png
5.​52.5.​52
2 | 5 ​52
Small dodecahemicosahedron.png
6.​52.6.​53
5352 | 3
Great dodecahemicosahedron.png
5.6.​54.6
54 5 | 3
Truncated dodecahedron.png
Nonuniform

truncated dodecahedron

Great ditrigonal dodecicosidodecahedron.png
3.​103.5.​103
3 5 | ​53
Great icosicosidodecahedron.png
5.6.​32.6
32 5 | 3
Great dodecicosahedron.png
6.​103.​65.​107
3 ​53 (​3252) |
Small retrosnub icosicosidodecahedron convex hull.png
Nonuniform
truncated dodecahedron
Small retrosnub icosicosidodecahedron.png
(35.​53)/2
| ​323252
Dodecahedron.png
Dodecahedron
Great stellated dodecahedron.png
{​52,3}
Small ditrigonal icosidodecahedron.png
(3.​52)3
3 | ​52 3
Ditrigonal dodecadodecahedron.png
(5.​53)3
3 | ​53 5
Great ditrigonal icosidodecahedron.png
(3.53)/2

32 | 3 5

Small rhombicosidodecahedron.png
Rhombicosidodecahedron
Small dodecicosidodecahedron.png
5.10.​32.10
32 5 | 5
Small rhombidodecahedron.png
4.10.​43.​109
2 5 (​3252) |
Small stellated truncated dodecahedron.png
5.​103.​103
2 5 | ​53
Truncated great icosahedron convex hull.png
Nonuniform
rhombicosidodecahedron
Great truncated icosahedron.png
6.6.​52
2 ​52 | 3
Nonuniform-rhombicosidodecahedron.png
Nonuniform
rhombicosidodecahedron
Small icosicosidodecahedron.png
6.​52.6.3
52 3 | 3
Small ditrigonal dodecicosidodecahedron.png
3.10.​53.10
53 3 | 5
Small dodecicosahedron.png
6.10.​65.​109
3 5 (​3254) |
Great stellated truncated dodecahedron.png
3.​103.​103
2 3 | ​53
Nonuniform2-rhombicosidodecahedron.png
Nonuniform
rhombicosidodecahedron
Great dirhombicosidodecahedron.png
4.​53.4.3.4.​52.4.​32
| ​3253 3 ​52
Great snub dodecicosidodecahedron.png
3.3.3.​52.3.​53
| ​5352 3
Great disnub dirhombidodecahedron.png
Skilling's figure
(see below)
Icositruncated dodecadodecahedron convex hull.png
Nonuniform
truncated icosidodecahedron
Icositruncated dodecadodecahedron.png
6.10.​103
3 5 ​53 |
Truncated dodecadodecahedron convex hull.png
Nonuniform
truncated icosidodecahedron
Truncated dodecadodecahedron.png
4.​109.​103
2 5 ​53 |
Great truncated icosidodecahedron convex hull.png
Nonuniform
truncated icosidodecahedron
Great truncated icosidodecahedron.png
4.6.​103
2 3 ​53 |
Snub dodecahedron ccw.png
Nonuniform
snub dodecahedron
Snub dodecadodecahedron.png
3.3.​52.3.5
| 2 ​52 5
Snub icosidodecadodecahedron.png
3.3.3.5.3.​53
| ​53 3 5
Great snub icosidodecahedron.png
34.​52
| 2 ​52 3
Great inverted snub icosidodecahedron.png
34.​53
| ​53 2 3
Inverted snub dodecadodecahedron.png
3.3.5.3.​53
| ​53 2 5
Great retrosnub icosidodecahedron.png
(34.​52)/2
| ​3253 2


Degenerate cases[edit]

Coxeter identified a number of degenerate star polyhedra by the Wythoff construction method, which contain overlapping edges or vertices. These degenerate forms include:

Skilling's figure[edit]

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 Ih symmetry.

Great disnub dirhombidodecahedron.png

See also[edit]

References[edit]

  • 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. [1]
  • 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. [2]
  • 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[edit]