Uniform polyhedron Information & Uniform polyhedron Links at HealthHaven.com

A display of uniform polyhedra at the Science Museum in London

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).

Convex
- 5 Platonic solids - regular convex polyhedra
- 13 Archimedean solids - quasiregular and semiregular convex polyhedra
Star
- 4 Kepler-Poinsot polyhedra - regular nonconvex polyhedra
- 53 uniform star polyhedra
- 1 star polyhedron found by John Skilling with pairs of edges that coincide, called great disnub dirhombidodecahedron (Skilling's figure).

There are also two infinite sets of uniform prisms and antiprisms, including convex and star forms.

[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.

Regular star polyhedra:

Kepler (1619) discovered two of the regular Kepler-Poinsot polyhedra and Louis Poinsot (1809) discovered the other two.

Other 53 nonregular star polyhedra:

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 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.
Coxeter, Longuet-Higgins & Miller (1954) published the list of uniform polyhedra.
Sopov (1970) proved their conjecture that the list was complete.
In 1974, Magnus Wenninger published his book Polyhedron models, which lists all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by Norman Johnson.
Skilling (1975) 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.
In 2002 Peter W. Messer discovered a minimal set of closed-form expressions for determining the main combinatorial and metrical quantities of any uniform polyhedron (and its dual) given only its Wythoff symbol.^[1]

[edit] Uniform star polyhedra

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

Main article: uniform star 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 (tr. 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 (tr. 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	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Face counts by position			Element counts
#	Name	Picture	Tiling	Vertex figure	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	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Face counts by position			Element counts
#	Name	Picture	Tiling	Vertex figure	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 120 triangles, visible in the faces of the disdyakis triacontahedron and alternately colored triangles on a sphere:

#	Name	Picture	Tiling	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Face counts by position			Element counts
#	Name	Picture	Tiling	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Pos. 2 [5] (12)	Pos. 1 [ ]x[ ] (30)	Pos. 0 [3] (20)	Faces	Edges	Vertices
12	Dodecahedron				{5,3}	{5}			12	30	20
[6]	Icosahedron				{3,5}			{3}	20	30	12
13	Rectified dodecahedron Rectified icosahedron Icosidodecahedron				t₁{5,3}	{5}		{3}	32	60	30
14	Truncated dodecahedron				t_0,1{5,3}	{10}		{3}	32	90	60
15	Truncated icosahedron				t_0,1{3,5}	{5}		{6}	32	90	60
16	Cantellated dodecahedron Cantellated icosahedron Rhombicosidodecahedron				t_0,2{5,3}	{5}	{4}	{3}	62	120	60
17	Omnitruncated dodecahedron Omnitruncated icosahedron Truncated icosidodecahedron				t_0,1,2{5,3}	{10}	{4}	{6}	62	180	120
18	Snub dodecahedron Snub icosahedron				s{5,3}	{5}	2 {3}	{3}	92	150	60

[edit] (p 2 2) Prismatic [p,2], I₂(p) family (D_ph Dihedral symmetry)

Main article: Prismatic uniform polyhedron

The dihedral symmetry of the sphere generates two infinite sets of uniform polyhedra, prisms and antiprisms, and two more infinite set of degenerate polygons, the hosohedrons and dihedrons which exists as tilings on the sphere.

The dihedral symmetry is represented by a fundamental triangle (p 2 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group I₂(p) or [n,2], as well as a prismatic Coxeter-Dynkin diagram: .

Below are the first five dihedral symmetries: D₂ ... D₆. The dihedral symmetry D_p has order 4n, represented the faces of a bipyramid, and on the sphere as an equator line on the longitude, and n equally-spaced lines of longitude.

[edit] (2 2 2) dihedral symmetry

There are 8 fundamental triangles, visible in the faces of the square bipyramid (Octahedron) and alternately colored triangles on a sphere:

#	Name	Picture	Tiling	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Face counts by position			Element counts
#	Name	Picture	Tiling	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Pos. 2 [2] (2)	Pos. 1 [ ]x[ ] (2)	Pos. 0 [ ]x[ ] (2)	Faces	Edges	Vertices
D₂ H₂	digonal dihedron digonal hosohedron				{2,2}	{2}			2	2	2
D₄	truncated digonal dihedron (Same as square dihedron)				t{2,2}={4,2}	{4}			2	4	4
P₄ [7]	omnitruncated digonal dihedron (Same as cube)				t_0,1,2{2,2}	{4}	{4}	{4}	6	12	8
A₂ [1]	snub digonal dihedron (Same as tetrahedron)				s{2,2}		{3}		4	6	4

[edit] (3 2 2) D_3hdihedral symmetry

There are 12 fundamental triangles, visible in the faces of the hexagonal bipyramid and alternately colored triangles on a sphere:

#	Name	Picture	Tiling	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Face counts by position			Element counts
#	Name	Picture	Tiling	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Pos. 2 [3] (2)	Pos. 1 [ ]x[ ] (3)	Pos. 0 [ ]x[ ] (3)	Faces	Edges	Vertices
D₃	Trigonal dihedron				{3,2}	{3}			2	3	3
H₃	Trigonal hosohedron				{2,3}			{2}	3	3	2
D₆	Truncated trigonal dihedron (Same as hexagonal dihedron)				t{3,2}	{6}			2	6	6
P₃	Truncated trigonal hosohedron (Triangular prism)				t{2,3}	{3}	{4}		5	9	6
P₆	Omnitruncated trigonal dihedron (Hexagonal prism)				t{2,3}	{6}	{4}	{4}	8	18	12
A₃ [2]	Snub trigonal dihedron (Same as Triangular antiprism) (Same as octahedron)				s{2,3}	{3}	{3}		8	12	6

[edit] (4 2 2) D_4hdihedral symmetry

There are 16 fundamental triangles, visible in the faces of the octagonal bipyramid and alternately colored triangles on a sphere:

#	Name	Picture	Tiling	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Face counts by position			Element counts
#	Name	Picture	Tiling	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Pos. 2 [4] (2)	Pos. 1 [ ]x[ ] (4)	Pos. 0 [ ]x[ ] (4)	Faces	Edges	Vertices
D₄	square dihedron				{4,2}	{4}			2	4	4
H₄	square hosohedron				{2,4}			{2}	4	4	2
D₈	Truncated square dihedron (Same as octagonal dihedron)				t{4,2}	{8}			2	8	8
P₄ [7]	Truncated square hosohedron (Cube)				t{2,4}	{4}	{4}		6	12	8
D₈	Omnitruncated square dihedron (Octagonal prism)				t{2,4}	{8}	{4}	{4}	10	24	16
A₄	Snub square dihedron (Square antiprism)				t{2,4}	{4}	{3}		10	16	8

[edit] (5 2 2) D_5h dihedral symmetry

There are 20 fundamental triangles, visible in the faces of the decagonal bipyramid and alternately colored triangles on a sphere:

#	Name	Picture	Tiling	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Face counts by position			Element counts
#	Name	Picture	Tiling	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Pos. 2 [5] (2)	Pos. 1 [ ]x[ ] (5)	Pos. 0 [ ]x[ ] (5)	Faces	Edges	Vertices
D₅	Pentagonal dihedron				{5,2}	{5}			2	5	5
H₅	Pentagonal hosohedron					{2,5}			{2}	5	5	2
D₁₀	Truncated pentagonal dihedron (Same as decagonal dihedron)				t{5,2}	{10}			2	10	10
P₅	Truncated pentagonal hosohedron (Same as pentagonal prism)				t{2,5}	{5}	{4}		7	15	10
P₁₀	Omnitruncated pentagonal dihedron (Decagonal prism)				t{2,5}	{10}	{4}	{4}	12	30	20
A₅	Snub pentagonal dihedron (Pentagonal antiprism)				t{2,5}	{5}	{3}		12	20	10

[edit] (6 2 2) D_6hdihedral symmtry

There are 24 fundamental triangles, visible in the faces of the dodecagonal bipyramid and alternately colored triangles on a sphere.

#	Name	Picture	Tiling	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Face counts by position			Element counts
#	Name	Picture	Tiling	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Pos. 2 [6] (2)	Pos. 1 [ ]x[ ] (6)	Pos. 0 [ ]x[ ] (6)	Faces	Edges	Vertices
D₆	Hexagonal dihedron				{6,2}	{6}			2	6	6
H₆	Hexagonal hosohedron				{2,6}			{2}	6	6	2
D₁₂	Truncated hexagonal dihedron (Same as dodecagonal dihedron)				t{6,2}	{12}			2	12	12
H₆	Truncated hexagonal hosohedron (Same as hexagonal prism)				t{2,6}	{6}	{4}		8	18	12
P₁₂	Omnitruncated hexagonal dihedron (Dodecagonal prism)				t{2,6}	{12}	{4}	{4}	14	36	24
A₆	Snub hexagonal dihedron (Hexagonal antiprism)				t{2,6}	{6}	{3}		14	24	12

[edit] See also

[edit] Notes

^ [http://www.springerlink.com/content/me48wm7823jhdcpe/fulltext.pdf?page=1 Closed-Form Expressions for Uniform Polyhedra and Their Duals, Peter W. Messer, Discrete Comput Geom 27:353–375 (2002)]

[edit] References

Brückner, M. Vielecke und vielflache. Theorie und geschichte.. Leipzig, Germany: Teubner, 1900. [1]
Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences 246: 401–450, MR 0062446, ISSN 0080-4614, http://www.jstor.org/stable/91532
Sopov, S. P. (1970), "A proof of the completeness on the list of elementary homogeneous polyhedra", Ukrainskiui Geometricheskiui Sbornik (8): 139–156, MR 0326550
Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
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, MR 0365333, ISSN 0080-4614, http://www.jstor.org/stable/74475
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).

[edit] External links

Weisstein, Eric W., "Uniform Polyhedron" from MathWorld.
Uniform Solution for Uniform Polyhedra
The Uniform Polyhedra
Virtual Polyhedra Uniform Polyhedra
Stella: Polyhedron Navigator - Software able to generate and print nets for all uniform polyhedra. Used to create many of the images on this page.
Paper models:
- Uniform/Dual Polyhedra
- Paper Models of Uniform (and other) Polyhedra
uniform polyhedra in 3d

[0] [http://www.springerlink.com/content/me48wm7823jhdcpe/fulltext.pdf?page=1 Closed-Form Expressions for Uniform Polyhedra and Their Duals, Peter W. Messer, Discrete Comput Geom 27:353–375 (2002)]

[1]

Contents

[edit] History

[edit] Uniform star polyhedra

[edit] Convex forms by Wythoff construction

[edit] Summary tables

[edit] Wythoff construction operators

[edit] (3 3 2) Td Tetrahedral symmetry

[edit] (4 3 2) Oh Octahedral symmetry

[edit] (5 3 2) Ih Icosahedral symmetry

[edit] (p 2 2) Prismatic [p,2], I2(p) family (Dph Dihedral symmetry)