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 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[edit]

See Prismatic uniform polyhedron.

Tetrahedral symmetry[edit]

(3 3 2) triangles on sphere

There is one nonconvex forms, 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.png
Tetrahedron
 
Rectified tetrahedron.png
Rectified tetrahedron
Octahedron
Tetrahemihexahedron.png
(4.3/2.4.3)
3/2 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 (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
Hexahedron.png
Cube
 
Octahedron.png
Octahedron
 
Cuboctahedron.png
Cuboctahedron
Cubohemioctahedron.png
(6.4/3.6.4)
4/3 4 | 3
Octahemioctahedron.png
(6.3/2.6.3)
3/2 3 | 3
Truncated hexahedron.png
Truncated cube
Great rhombihexahedron.png
(4.8/3.4/3.8/5)
2 4/3 (3/2 4/2) |
Great cubicuboctahedron.png
(8/3.3.8/3.4)
3 4 | 4/3
Uniform great rhombicuboctahedron.png
(4.3/2.4.4)
3/2 4 | 2
Truncated octahedron.png
Truncated octahedron
 
Small rhombicuboctahedron.png
Rhombicuboctahedron
Small rhombihexahedron.png
(4.8.4/3.8)
2 4 (3/2 4/2) |
Small cubicuboctahedron.png
(8.3/2.8.4)
3/2 4 | 4
Stellated truncated hexahedron.png
(8/3.8/3.3)
2 3 | 4/3
Great truncated cuboctahedron convex hull.png
Nonuniform
truncated cuboctahedron
Great truncated cuboctahedron.png
(4.6.8/3)
2 3 4/3 |
Cubitruncated cuboctahedron convex hull.png
Nonuniform
truncated cuboctahedron
Cubitruncated cuboctahedron.png
(8/3.6.8)
3 4 4/3 |
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
{5,5/2}
Small stellated dodecahedron.png
{5/2,5}
Great icosahedron.png
{3,5/2}
Nonuniform truncated icosahedron.png
Nonuniform
truncated icosahedron
2 5 |3
Great truncated dodecahedron.png
U37
2 5/2 | 5
Great dodecicosidodecahedron.png
U61
5/2 3 | 5/3
Uniform great rhombicosidodecahedron.png
U67
5/3 3 | 2
Great rhombidodecahedron.png
U73
2 5/3 (3/2 5/4) |
Rhombidodecadodecahedron convex hull.png
Nonuniform
truncated icosahedron
2 5 |3
Rhombidodecadodecahedron.png
U38
5/2 5 | 2
Icosidodecadodecahedron.png
U44
5/3 5 | 3
Rhombicosahedron.png
U56
2 3 (5/4 5/2) |
Small snub icosicosidodecahedron convex hull.png
Nonuniform
truncated icosahedron
2 5 |3
Small snub icosicosidodecahedron.png
U32
| 5/2 3 3
Icosidodecahedron.png
Icosidodecahedron
2 | 3 5
Small icosihemidodecahedron.png
U49
3/2 3 | 5
Small dodecahemidodecahedron.png
U51
5/4 5 | 5
Great icosidodecahedron.png
U54
2 | 3 5/2
Great dodecahemidodecahedron.png
U70
5/3 5/2 | 5/3
Great icosihemidodecahedron.png
U71
3 3 | 5/3
Dodecadodecahedron.png
U36
2 | 5 5/2
Small dodecahemicosahedron.png
U62
5/3 5/2 | 3
Great dodecahemicosahedron.png
U65
5/4 5 | 3
Truncated dodecahedron.png
Truncated dodecahedron
2 3 | 5
Great ditrigonal dodecicosidodecahedron.png
U42
Great icosicosidodecahedron.png
U48
Great dodecicosahedron.png
U63
Small retrosnub icosicosidodecahedron convex hull.png
Nonuniform
truncated dodecahedron
Small retrosnub icosicosidodecahedron.png
U72
Dodecahedron.png
Dodecahedron
Great stellated dodecahedron.png
{5/2,3}
Small ditrigonal icosidodecahedron.png
U30
Ditrigonal dodecadodecahedron.png
U41
Great ditrigonal icosidodecahedron.png
U47
Small rhombicosidodecahedron.png
Rhombicosidodecahedron
Small dodecicosidodecahedron.png
U33
Small rhombidodecahedron.png
U39
Small stellated truncated dodecahedron.png
U58
Truncated great icosahedron convex hull.png
Beveled
Dodecahedron
Great truncated icosahedron.png
U55
Nonuniform-rhombicosidodecahedron.png
Nonuniform
rhombicosidodecahedron
Small icosicosidodecahedron.png
U31
Small ditrigonal dodecicosidodecahedron.png
U43
Small dodecicosahedron.png
U50
Great stellated truncated dodecahedron.png
U66
Nonuniform2-rhombicosidodecahedron.png
Nonuniform
rhombicosidodecahedron
Great dirhombicosidodecahedron.png
U75
Great snub dodecicosidodecahedron.png
U64
Great disnub dirhombidodecahedron.png
Skilling's figure
(see below)
Icositruncated dodecadodecahedron convex hull.png
Nonuniform
truncated icosidodecahedron
Icositruncated dodecadodecahedron.png
U45
Truncated dodecadodecahedron convex hull.png
Nonuniform
truncated icosidodecahedron
Truncated dodecadodecahedron.png
U59
Great truncated icosidodecahedron convex hull.png
Nonuniform
truncated icosidodecahedron
Great truncated icosidodecahedron.png
U68
Snub dodecahedron ccw.png
Nonuniform
snub dodecahedron
Snub dodecadodecahedron.png
U40
Snub icosidodecadodecahedron.png
U46
Great snub icosidodecahedron.png
U57
Great inverted snub icosidodecahedron.png
U69
Inverted snub dodecadodecahedron.png
U60
Great retrosnub icosidodecahedron.png
U74

Skilling's figure[edit]

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.

Great disnub dirhombidodecahedron.png

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:

See also[edit]

Notes[edit]

References[edit]

External links[edit]