Uniform polyhedron

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A uniform polyhedron is a polyhedron which has regular polygons as faces and is transitive on its vertices (i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry.

Uniform polyhedra may be regular, quasi-regular or semi-regular. The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra.

Excluding the infinite sets, there are 75 uniform polyhedra (or 76 if edges are allowed to coincide).

Categories include:

Infinite sets of uniform prisms and antiprisms (including star forms)
5 Platonic solids - regular convex polyhedra
4 Kepler-Poinsot polyhedra - regular nonconvex polyhedra
13 Archimedean solids - quasiregular and semiregular convex polyhedra
14 nonconvex polyhedra with convex faces
39 nonconvex polyhedra with nonconvex faces
1 polyhedron found by John Skilling with pairs of edges that coincide, called Great disnub dirhombidodecahedron (Skilling's figure).

They can also be grouped by their symmetry group, which is done below.

[edit] History

The Platonic solids date back to the classical Greeks and were studied by Plato, Theaetetus and Euclid.
Johannes Kepler (1571-1630) was the first to publish the complete list of Archimedean solids after the original work of Archimedes was lost.
Kepler (1619) discovered two of the regular Kepler-Poinsot polyhedra and Louis Poinsot (1809) discovered the other two.
Of the remaining 66, Albert Badoureau (1881) discovered 37. Edmund Hess (1878) discovered 2 more and Pitsch (1881) independently discovered 18, of which 15 had not previously been discovered.
The famous geometer Donald Coxeter discovered the remaining twelve in collaboration with J.C.P. Miller (1930-1932) but did not publish. M.S. and H.C. Longuet-Higgins and independently discovered 11 of these.
In 1954 H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller published the list of uniform polyhedra.
In 1970 S. P. Sopov proved their conjecture that the list was complete.
In 1974, Magnus Wenninger published his book Polyhedron models, which is the first published list of all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by Norman Johnson.
In 1975, John Skilling independently proved the completeness, and showed that if the definition of uniform polyhedron is relaxed to allow edges to coincide then there is just one extra possibility.
In 1993, Zvi Har'El produced a complete kaleidoscopic construction of the uniform polyhedra and duals with a computer program called Kaleido, and summarized in a paper Uniform Solution for Uniform Polyhedra, counting figures 1-80.
Also in 1993, R. Mäder ported this Kaleido solution to Mathematica with a slightly different indexing system.

[edit] Indexing

Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters:

[C] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.
[W] Wenninger, 1974, has 119 figures: 1-5 for the Platonic solids, 6-18 for the Archimedean solids, 19-66 for stellated forms including the 4 regular nonconvex polyhedra, and ended with 67-119 for the nonconvex uniform polyhedra.
[K] Kaleido, 1993: The 80 figures were grouped by symmetry: 1-5 as representatives of the infinite families of prismatic forms with dihedral symmetry, 6-9 with tetrahedral symmetry, 10-26 with Octahedral symmetry, 46-80 with icosahedral symmetry.
[U] Mathematica, 1993, follows the Kaleido series with the 5 prismatic forms moved to last, so that the nonprismatic forms become 1–75.

[edit] Nonconvex uniform polyhedra

The 57 nonprismatic nonconvex forms are compiled by Wythoff constructions within Schwarz triangles.

Main article: Nonconvex uniform polyhedron

[edit] Convex forms by Wythoff construction

The convex uniform polyhedra can be named by Wythoff construction operations and can be named in relation to the regular form.

In more detail the convex uniform polyhedron are given below by their Wythoff construction within each symmetry group.

Within the Wythoff construction, there are repetitions created by lower symmetry forms. The cube is a regular polyhedron, and a square prism. The octahedron is a regular polyhedron, and a triangular antiprism. The octahedron is also a rectified tetrahedron. Many polyhedra are repeated from different construction sources and are colored differently.

The Wythoff construction applies equally to uniform polyhedra and uniform tilings on the surface of a sphere, so images of both are given. The spherical tilings including the set of hosohedrons and dihedrons which are degenerate polyhedra.

These symmetry groups are formed from the reflectional point groups in three dimensions, each represented by a fundamental triangle (p q r), where p>1, q>1, r>1 and 1/p+1/q+1/r<1.

Tetrahedral symmetry (3 3 2) - order 24
Octahedral symmetry (4 3 2) - order 48
Icosahedral symmetry (5 3 2) - order 60
Dihedral symmetry (n 2 2), for all n=3,4,5,... - order 4n

The remaining nonreflective forms are constructed by alternation operations applied to the polyhedra with an even number of sides.

Along with the prisms and their dihedral symmetry, the spherical Wythoff construction process adds two regular classes which become degenerate as polyhedra - the dihedra and hosohedra, the first having only two faces, and the second only two vertices. The truncation of the regular hosohedra creates the prisms.

Below the convex uniform polyhedra are indexed 1-18 for the nonprismatic forms as they are presented in the tables by symmetry form. Repeated forms are in brackets.

For the infinite set of prismatic forms, they are indexed in four families:

Hosohedrons H_2... (Only as spherical tilings)
Dihedrons D_2... (Only as spherical tilings)
Prisms P_3... (Truncated hosohedrons)
Antiprisms A_3... (Snub prisms)

[edit] Summary tables

	Parent	Truncated	Rectified	Bitruncated (truncated dual)	Birectified (dual)	Cantellated	Omnitruncated (Cantitruncated)	Snub
Extended Schläfli symbol	$\begin{Bmatrix} p , q \end{Bmatrix}$	$t\begin{Bmatrix} p , q \end{Bmatrix}$	$\begin{Bmatrix} p \\ q \end{Bmatrix}$	$t\begin{Bmatrix} q , p \end{Bmatrix}$	$\begin{Bmatrix} q , p \end{Bmatrix}$	$r\begin{Bmatrix} p \\ q \end{Bmatrix}$	$t\begin{Bmatrix} p \\ q \end{Bmatrix}$	$s\begin{Bmatrix} p \\ q \end{Bmatrix}$
Extended Schläfli symbol	t₀{p,q}	t_0,1{p,q}	t₁{p,q}	t_1,2{p,q}	t₂{p,q}	t_0,2{p,q}	t_0,1,2{p,q}	s{p,q}
Wythoff symbol p-q-2	q \| p 2	2 q \| p	2 \| p q	2 p \| q	p \| q 2	p q \| 2	p q 2 \|	\| p q 2
Coxeter-Dynkin diagram
Vertex figure	p^q	(q.2p.2p)	(p.q.p.q)	(p.2q.2q)	q^p	(p.4.q.4)	(4.2p.2q)	(3.3.p.3.q)
Tetrahedral 3-3-2	{3,3}	(3.6.6)	(3.3.3.3)	(3.6.6)	{3,3}	(3.4.3.4)	(4.6.6)	(3.3.3.3.3)
Octahedral 4-3-2	{4,3}	(3.8.8)	(3.4.3.4)	(4.6.6)	{3,4}	(3.4.4.4)	(4.6.8)	(3.3.3.3.4)
Icosahedral 5-3-2	{5,3}	(3.10.10)	(3.5.3.5)	(5.6.6)	{3,5}	(3.4.5.4)	(4.6.10)	(3.3.3.3.5)

And a sampling of Dihedral symmetries:

(p 2 2)	Parent	Truncated	Rectified	Bitruncated (truncated dual)	Birectified (dual)	Cantellated	Omnitruncated (Cantitruncated)	Snub
Extended Schläfli symbol	$\begin{Bmatrix} p , 2 \end{Bmatrix}$	$t\begin{Bmatrix} p , 2 \end{Bmatrix}$	$\begin{Bmatrix} p \\ 2 \end{Bmatrix}$	$t\begin{Bmatrix} 2 , p \end{Bmatrix}$	$\begin{Bmatrix} 2 , p \end{Bmatrix}$	$r\begin{Bmatrix} p \\ 2 \end{Bmatrix}$	$t\begin{Bmatrix} p \\ 2 \end{Bmatrix}$	$s\begin{Bmatrix} p \\ 2 \end{Bmatrix}$
Extended Schläfli symbol	t₀{p,2}	t_0,1{p,2}	t₁{p,2}	t_1,2{p,2}	t₂{p,2}	t_0,2{p,2}	t_0,1,2{p,2}	s{p,2}
Wythoff symbol	2 \| p 2	2 2 \| p	2 \| p 2	2 p \| 2	p \| 2 2	p 2 \| 2	p 2 2 \|	\| p 2 2
Coxeter-Dynkin diagram
Vertex figure	p²	(2.2p.2p)	(p.2.p.2)	(p.4.4)	2^p	(p.4.2.4)	(4.2p.4)	(3.3.p.3.2)
Dihedral (2 2 2)	{2,2}	2.4.4	2.2.2.2	4.4.2	{2,2}	2.4.2.4	4.4.4	3.3.3.2
Dihedral (3 2 2)	{3,2}	2.6.6	2.3.2.3	4.4.3	{2,3}	2.4.3.4	4.4.6	3.3.3.3
Dihedral (4 2 2)	{4,2}	2.8.8	2.4.2.4	4.4.4	{2,4}	2.4.4.4	4.4.8	3.3.3.4
Dihedral (5 2 2)	{5,2}	2.10.10	2.5.2.5	4.4.5	{2,5}	2.4.5.4	4.4.10	3.3.3.5
Dihedral (6 2 2)	{6,2}	2.12.12	2.6.2.6	4.4.6	{2,6}	2.4.6.4	4.4.12	3.3.3.6

[edit] Wythoff construction operators

Example forms from the cube and octahedron

Operation	Extended Schläfli symbols		Description
Parent	t₀{p,q}	$\begin{Bmatrix} p , q \end{Bmatrix}$	Any regular polyhedron or tiling
Rectified	t₁{p,q}	$\begin{Bmatrix} p \\ q \end{Bmatrix}$	The edges are fully truncated into single points. The polyhedron now has the combined faces of the parent and dual.
Birectified Also Dual	t₂{p,q}	$\begin{Bmatrix} q , p \end{Bmatrix}$	The birectified (dual) is a further truncation so that the original faces are reduced to points. New faces are formed under each parent vertex. The number of edges is unchanged and are rotated 90 degrees. The dual of the regular polyhedron {p, q} is also a regular polyhedron {q, p}.
Truncated	t_0,1{p,q}	$t\begin{Bmatrix} p , q \end{Bmatrix}$	Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated polyhedron. The polyhedron has its original faces doubled in sides, and contains the faces of the dual.
Bitruncated	t_1,2{p,q}	$t\begin{Bmatrix} q , p \end{Bmatrix}$	Same as truncated dual.
Cantellated (or rhombated) (Also expanded)	t_0,2{p,q}	$r\begin{Bmatrix} p \\ q \end{Bmatrix}$	In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.
Omnitruncated (or cantitruncated) (or rhombitruncated)	t_0,1,2{p,q}	$t\begin{Bmatrix} p \\ q \end{Bmatrix}$	The truncation and cantellation operations are applied together to create an omnitruncated form which has the parent's faces doubled in sides, the dual's faces doubled in sides, and squares where the original edges existed.
Snub	s{p,q}	$s\begin{Bmatrix} p \\ q \end{Bmatrix}$	The snub takes the omnitruncated form and rectifies alternate vertices. (This operation is only possible for polyhedra with all even-sided faces.) All the original faces end up with half as many sides, and the squares degenerate into edges. Since the omnitruncated forms have 3 faces/vertex, new triangles are formed. Usually these alternated faceting forms are slightly deformed thereafter in order to end again as uniform polyhedra. The possibility of the latter variation depends on the degree of freedom.

[edit] (3 3 2) T_d Tetrahedral symmetry

The tetrahedral symmetry of the sphere generates 5 uniform polyhedra, and a 6th form by a snub operation.

The tetrahedral symmetry is represented by a fundamental triangle with one vertex with two mirrors, and two vertices withr three mirrors, represented by the symbol (3 3 2). It can also be represented by the Coxeter group A₂ or [3,3], as well as a Coxeter-Dynkin diagram: .

There are 24 triangles, visible in the faces of the tetrakis hexahedron and alternately colored triangles on a sphere:

#	Name	Picture	Tiling	Coxeter-Dynkin and Schläfli symbols	Face counts by position			Element counts
#	Name	Picture	Tiling	Coxeter-Dynkin and Schläfli symbols	Pos. 2 [3] (4)	Pos. 1 [ ]x[ ] (6)	Pos. 0 [3] (4)	Faces	Edges	Vertices
1	tetrahedron			{3,3}	{3}			4	6	4
[1]	Birectified tetrahedron (Same as tetrahedron)			t₂{3,3}			{3}	4	6	4
2	rectified tetrahedron (Same as octahedron)			t₁{3,3}	{3}		{3}	8	12	6
3	truncated tetrahedron			t_0,1{3,3}	{6}		{3}	8	18	12
[3]	Bitruncated tetrahedron (Same as truncated tetrahedron)			t_1,2{3,3}	{3}		{6}	8	18	12
4	cantellated tetrahedron (Same as cuboctahedron)			t_0,2{3,3}	{3}	{4}	{3}	14	24	12
5	omnitruncated tetrahedron (Same as truncated octahedron)			t_0,1,2{3,3}	{6}	{4}	{6}	14	36	24
6	Snub tetrahedron (Same as icosahedron)			s{3,3}	{3}	2 {3}	{3}	20	30	12

[edit] (4 3 2) O_h Octahedral symmetry

The octahedral symmetry of the sphere generates 7 uniform polyhedra, and a 3 more by alternation. Four of these forms are repeated from the tetrahedral symmetry table above.

The octaahedral symmetry is represented by a fundamental triangle (4 3 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group B₂ or [4,3], as well as a Coxeter-Dynkin diagram: .

There are 48 triangles, visible in the faces of the disdyakis dodecahedron and alternately colored triangles on a sphere:

#	Name	Picture	Tiling	Coxeter-Dynkin and Schläfli symbols	Face counts by position			Element counts
#	Name	Picture	Tiling	Coxeter-Dynkin and Schläfli symbols	Pos. 2 [4] (8)	Pos. 1 [ ]x[ ] (6)	Pos. 0 [3] (12)	Faces	Edges	Vertices
7	Cube			{4,3}	{4}			6	12	8
[2]	Octahedron			{3,4}			{3}	8	12	6
[4]	rectified cube rectified octahedron (Cuboctahedron)			{4,3}	{4}		{3}	14	24	12
8	Truncated cube			t_0,1{4,3}	{8}		{3}	14	36	24
[5]	Truncated octahedron			t_0,1{3,4}	{4}		{6}	14	36	24
9	Cantellated cube cantellated octahedron Rhombicuboctahedron			t_0,2{4,3}	{8}	{4}	{6}	26	48	24
10	Omnitruncated cube omnitruncated octahedron Truncated cuboctahedron			t_0,1,2{4,3}	{8}	{4}	{6}	26	72	48
[6]	Alternated truncated octahedron (Same as Icosahedron)			h_0,1{3,4}		{3}	{3}	20	30	12
[1]	Alternated cube (Same as tetrahedron)			h{4,3}			¹/₂ {3}	6	12	8
11	Snub cube			s{4,3}	{4}	2 {3}	{3}	38	60	24

[edit] (5 3 2) I_h Icosahedral symmetry

The icosahedral symmetry of the sphere generates 7 uniform polyhedra, and a 1 more by alternation. Only one is repeated from the tetrahedral and octahedral symmetry table above.

The icosahedral symmetry is represented by a fundamental triangle (5 3 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group G₂ or [5,3], as well as a Coxeter-Dynkin diagram: .

There are 60 triangles, visible in the faces of the disdyakis triacontahedron and alternately colored triangles on a sphere: