Uniform polyhedron
2537223
226152643
2008-07-17T01:39:43Z
Tomruen
63601
/* Nonconvex forms listed by symmetry groups and vertex arrangements */
A '''[[Uniform polytope|uniform]] polyhedron''' is a [[polyhedron]] which has [[regular polygon]]s as faces and is transitive on its [[vertex (geometry)|vertices]] (i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are [[Congruence (geometry)|congruent]], and the polyhedron has a high degree of reflectional and rotational [[symmetry]].
Uniform polyhedra may be [[Regular polyhedron|regular]], [[Quasiregular polyhedron|quasi-regular]] or [[Semiregular polyhedron|semi-regular]]. The faces and vertices need not be [[convex set|convex]], so many of the Uniform polyhedra are also [[Star polyhedron|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 [[Prismatic uniform polyhedron|prisms and antiprisms]] (including star forms)
* 5 [[Platonic solid]]s - regular convex polyhedra
* 4 [[Kepler-Poinsot polyhedra]] - regular nonconvex polyhedra
* 13 [[Archimedean solid]]s - [[Quasiregular polyhedron|quasiregular]] and [[Semiregular polyhedron|semiregular]] convex polyhedra
* 14 [[List of uniform polyhedra#Nonconvex forms with convex faces|nonconvex polyhedra with convex faces]]
* 39 [[List of uniform polyhedra#Nonconvex forms with nonconvex faces|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.
==History==
* The [[Platonic solid]]s date back to the classical Greeks and were studied by [[Plato]], [[Theaetetus (mathematician)|Theaetetus]] and [[Euclid]].
*[[Johannes Kepler]] (1571-1630) was the first to publish the complete list of [[Archimedean solid]]s after the original work of [[Archimedes]] was lost.
* [[Johannes Kepler|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 [[Harold Scott MacDonald Coxeter|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, [[List of Wenninger polyhedron models|''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 (mathematician)]].
* 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 computer construction of the uniform polyhedra and duals via their Kaleidoscopic construction via 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.
===Indexing===
There are four major published indexing efforts from the works above. To distinguish them, they are given by indexing different letters, '''C''' for [[Coxeter]] 1954 first enumeration figures, '''W''' for the 1974 book Polyhedron models by Wenninger, '''K''' for the 1993 Kaleido solution, and '''U''' for the Maeder solution used by Mathematica and extensively reproduced elsewhere.
# ['''C'''] 1954: This paper listed the uniform polyhedra by figures in the paper from 15-92. Starting with 15-32 for the convex forms, 33-35 for 3 infinite prismatic sets, and ending with 36-92 for the nonconvex forms.
# ['''W'''] 1974: Wenninger's book ''Polyhedron model'' numbered figures 1-119: 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'''] 1993 Kaleido: The 80 figures given in the Kaleido solution were grouped by symmetry, numbered 1-80: 1-5 as representatives for the infinite families of prismatic forms with [[Dihedral symmetry in three dimensions|dihedral symmetry]], 6-9 with [[tetrahedral symmetry]], 10-26 with [[Octahedral symmetry]], 46-80 with [[icosahedral symmetry]].
# ['''U'''] 1993 Mathematica: This listing followed the Kaleido one, but moved the 5 prismatic forms to last, shifting the nonprismatic forms back 5, and now 1-75.
== Dihedral symmetry ==
There are two infinite sets of uniform polyhedra with [[dihedral symmetry]]: the [[Prism (geometry)|prisms]] and [[antiprism]]s. These sets both include forms with [[star polygon]]s.
:''Main article: [[Prismatic uniform polyhedron]]''
== Convex forms and fundamental vertex arrangements ==<!-- This section is linked from [[Archimedean solid]] -->
The convex uniform polyhedra can be named by [[Wythoff construction]] operations upon a parent form.
The convex uniform polyhedra are usually classified in three groups:
* [[Platonic solid]]s - 5 regular polyhedra: [[Tetrahedron]], [[cube]], [[octahedron]], [[dodecahedron]], [[icosahedron]].
* [[Archimedean solid]]s - 13 semiregular polyhedra
* [[Prism (geometry)|Prisms]] and [[antiprism]]s - Two infinite families starting as triangles.
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''.
Along with the prisms and their [[dihedral symmetry]], the spherical Wythoff construction process adds two ''regular'' classes which become degenerate as polyhedra - the ''[[Dihedron|dihedra]]'' and [[Hosohedron|hosohedra]], the first having only two faces, and the second only two vertices. The truncation of the regular ''hosohedra'' creates the prisms.
Each of these convex forms define set of ''vertices'' that can be identified for the nonconvex forms in the next section.
{| class="wikitable"
|-
!
!Parent
!Truncated
!Rectified
!Bitruncated<BR>(truncated dual)
!Birectified<BR>(dual)
!Cantellated
!Omnitruncated<BR>(<small>Cantitruncated</small>)
!Snub
|-
!rowspan=2|Extended<BR>[[Schläfli symbol]]
!<math>\begin{Bmatrix} p , q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p , q \end{Bmatrix}</math>
!<math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} q , p \end{Bmatrix}</math>
!<math>\begin{Bmatrix} q , p \end{Bmatrix}</math>
!<math>r\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>s\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
|-
!t<sub>0</sub>{p,q}
!t<sub>0,1</sub>{p,q}
!t<sub>1</sub>{p,q}
!t<sub>1,2</sub>{p,q}
!t<sub>2</sub>{p,q}
!t<sub>0,2</sub>{p,q}
!t<sub>0,1,2</sub>{p,q}
!s{p,q}
|-
![[Wythoff symbol]]<BR>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
|-
!rowspan=5|[[Coxeter-Dynkin diagram]]<BR>(variations)
|-
![[Image:CDW ring.png]][[Image:CDW p.png]][[Image:CDW dot.png]][[Image:CDW q.png]][[Image:CDW dot.png]]
![[Image:CDW ring.png]][[Image:CDW p.png]][[Image:CDW ring.png]][[Image:CDW q.png]][[Image:CDW dot.png]]
![[Image:CDW dot.png]][[Image:CDW p.png]][[Image:CDW ring.png]][[Image:CDW q.png]][[Image:CDW dot.png]]
![[Image:CDW dot.png]][[Image:CDW p.png]][[Image:CDW ring.png]][[Image:CDW q.png]][[Image:CDW ring.png]]
![[Image:CDW dot.png]][[Image:CDW p.png]][[Image:CDW dot.png]][[Image:CDW q.png]][[Image:CDW ring.png]]
![[Image:CDW ring.png]][[Image:CDW p.png]][[Image:CDW dot.png]][[Image:CDW q.png]][[Image:CDW ring.png]]
![[Image:CDW ring.png]][[Image:CDW p.png]][[Image:CDW ring.png]][[Image:CDW q.png]][[Image:CDW ring.png]]
![[Image:CDW hole.png]][[Image:CDW p.png]][[Image:CDW hole.png]][[Image:CDW q.png]][[Image:CDW hole.png]]
|-
!(o)-p-o-q-o
!(o)-p-(o)-q-o
!o-p-(o)-q-o
!o-p-(o)-q-(o)
!o-p-o-q-(o)
!(o)-p-o-q-(o)
!(o)-p-(o)-q-(o)
!( )-p-( )-q-( )
|-
!xPoQo
!xPxQo
!oPxQo
!oPxQx
!oPoQx
!xPoQx
!xPxQx
!sPsQs
|-
![p,q]:001
![p,q]:011
![p,q]:010
![p,q]:110
![p,q]:100
![p,q]:101
![p,q]:111
![p,q]:111s
|-
![[Vertex configuration|Vertex figure]]
!p<sup>q</sup>
!(q.2p.2p)
!(p.q.p.q)
!(p.2q.2q)
!q<sup>p</sup>
!(p.4.q.4)
!(4.2p.2q)
!(3.3.p.3.q)
|-
|[[Tetrahedral symmetry|Tetrahedral]]<BR>3-3-2
|[[Image:Uniform polyhedron-33-t0.png|64px]]<BR>[[Tetrahedron|{3,3}]]
|[[Image:Uniform polyhedron-33-t01.png|64px]]<BR>[[Truncated tetrahedron|(3.6.6)]]
|[[Image:Uniform polyhedron-33-t1.png|64px]]<BR>[[Octahedron|(3.3.3.3)]]
|[[Image:Uniform polyhedron-33-t12.png|64px]]<BR>[[Truncated tetrahedron|(3.6.6)]]
|[[Image:Uniform polyhedron-33-t2.png|64px]]<BR>[[Tetrahedron|{3,3}]]
| [[Image:Uniform polyhedron-33-t02.png|64px]]<BR>[[Cuboctahedron|(3.4.3.4)]]
|[[Image:Uniform polyhedron-33-t012.png|64px]]<BR>[[Truncated octahedron|(4.6.6)]]
|[[Image:Uniform polyhedron-33-s012.png|64px]]<BR>[[Icosahedron|(3.3.3.3.3)]]
|-
|[[Octahedral symmetry|Octahedral]]<BR>4-3-2
|[[Image:Uniform polyhedron-43-t0.png|64px]]<BR>[[Cube|{4,3}]]
|[[Image:Uniform polyhedron-43-t01.png|64px]]<BR>[[Truncated cube|(3.8.8)]]
|[[Image:Uniform polyhedron-43-t1.png|64px]]<BR>[[Cuboctahedron|(3.4.3.4)]]
|[[Image:Uniform polyhedron-43-t12.png|64px]]<BR>[[Truncated octahedron|(4.6.6)]]
|[[Image:Uniform polyhedron-43-t2.png|64px]]<BR>[[Octahedron|{3,4}]]
|[[Image:Uniform polyhedron-43-t02.png|64px]]<BR>[[Small rhombicuboctahedron|(3.4.4.4)]]
|[[Image:Uniform polyhedron-43-t012.png|64px]]<BR>[[Great rhombicuboctahedron|(4.6.8)]]
|[[Image:Uniform polyhedron-43-s012.png|64px]]<BR>[[Snub cube|(3.3.3.3.4)]]
|-
|[[Icosahedral symmetry|Icosahedral]]<BR>5-3-2
|[[Image:Uniform polyhedron-53-t0.png|64px]]<BR>[[Dodecahedron|{5,3}]]
|[[Image:Uniform polyhedron-53-t01.png|64px]]<BR>[[Truncated dodecahedron|(3.10.10)]]
|[[Image:Uniform polyhedron-53-t1.png|64px]]<BR>[[Icosidodecahedron|(3.5.3.5)]]
|[[Image:Uniform polyhedron-53-t12.png|64px]]<BR>[[Truncated icosahedron|(5.6.6)]]
|[[Image:Uniform polyhedron-53-t2.png|64px]]<BR>[[Icosahedron|{3,5}]]
|[[Image:Uniform polyhedron-53-t02.png|64px]]<BR>[[Rhombicosidodecahedron|(3.4.5.4)]]
|[[Image:Uniform polyhedron-53-t012.png|64px]]<BR>[[Truncated icosidodecahedron|(4.6.10)]]
|[[Image:Uniform polyhedron-53-s012.png|64px]]<BR>[[Snub dodecahedron|(3.3.3.3.5)]]
|-
|[[Dihedral symmetry|Dihedral]]<BR>p-2-2<BR>Example p=5
| [[Dihedron|{5,2}]]
| 2.10.10
| 2.5.2.5
| [[Image:pentagonal prism.png|64px]]<BR>4.4.5
| [[Hosohedron|{2,5}]]
| 2.4.5.4
| [[Image:decagonal prism.png|64px]]<BR>4.4.10
| [[Image:pentagonal antiprism.png|64px]]<BR>3.3.3.5
|}
====Definition of operations====
{|width=640|
|[[Image:Wythoffian construction diagram.png|320px]]
|[[Image:Polyhedron truncation example3.png|320px]]<BR>Example forms from the [[cube]] and [[octahedron]]
|}
{|width=720 class="wikitable"
!Operation
!colspan=2|Extended<BR>[[Schläfli symbol|Schläfli<BR>symbols]]
![[Coxeter-Dynkin diagram|Coxeter-<BR>Dynkin<BR>diagram]]
!Description
|-
! Parent
|width=70| t<sub>0</sub>{p,q}
| <math>\begin{Bmatrix} p , q \end{Bmatrix}</math>
|[[Image:Dynkins-100.png]]
| Any regular polyhedron or tiling
|-
! [[Rectification (geometry)|Rectified]]
| t<sub>1</sub>{p,q}
| <math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
|[[Image:Dynkins-010.png]]
|The edges are fully truncated into single points. The polyhedron now has the combined faces of the parent and dual.
|-
!Birectified<BR>Also [[Dual polyhedron|Dual]]
| t<sub>2</sub>{p,q}
| <math>\begin{Bmatrix} q , p \end{Bmatrix}</math>
|[[Image:Dynkins-001.png]]
|[[Image:Dual Cube-Octahedron.svg|100px|right]]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}.
|-
![[Truncation (geometry)|Truncated]]
| t<sub>0,1</sub>{p,q}
| <math>t\begin{Bmatrix} p , q \end{Bmatrix}</math>
|[[Image:Dynkins-110.png]]
|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.<BR>[[Image:Cube truncation sequence.svg|400px]]
|-
!'''Bitruncated'''
| t<sub>1,2</sub>{p,q}
| <math>t\begin{Bmatrix} q , p \end{Bmatrix}</math>
|[[Image:Dynkins-011.png]]
|Same as truncated dual.
|-
! [[Cantellation (geometry)|Cantellated]]<BR>(or rhombated)<BR>(Also [[Expansion (geometry)|expanded]])
| t<sub>0,2</sub>{p,q}
| <math>r\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
|[[Image:Dynkins-101.png]]
|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.<BR>[[Image:Cube cantellation sequence.svg|400px]]
|-
![[Omnitruncation (geometry)|Omnitruncated]]<BR>(or cantitruncated)<BR>(or rhombitruncated)
| t<sub>0,1,2</sub>{p,q}
| <math>t\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
|[[Image:Dynkins-111.png]]
|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 (geometry)|Snub]]
| s{p,q}
| <math>s\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
|[[Image:Dynkins-sss.png]]
|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.<BR>[[Image:Snubcubes in grCO.svg|400px]]
|}
== Nonconvex forms listed by symmetry groups and vertex arrangements ==
The nonconvex forms are constructed from [[Schwarz triangle]]s.
All the uniform polyhedra are listed below by their [[symmetry group]]s 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]] of the 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 [[Square (geometry)|squares]].
=== Tetrahedral symmetry ===
There are 2 convex uniform polyhedra, the [[tetrahedron]] and [[truncated tetrahedron]], and one nonconvex form, the [[tetrahemihexahedron]] which have ''[[tetrahedral symmetry]]''. The [[tetrahedron]] is self dual.
In addition the [[octahedron]], [[truncated octahedron]], [[cuboctahedron]], and [[icosahedron]] have tetrahedral symmetry as well as higher symmetry. They are added for completeness below, although their nonconvex forms with octahedral symmetry are not included here.
{| class="wikitable"
!Vertex group
!Convex
!colspan=2|Nonconvex
|-
|(Tetrahedral)
| [[Image:Tetrahedron.png|64px]]<BR>[[Tetrahedron|{3,3}]]
|-
|Truncated (*)
| [[Image:Truncated tetrahedron.png|64px]]<BR>[[Truncated tetrahedron|(3.6.6)]]
|-
|Rectified (*)
| [[Image:Rectified tetrahedron.png|64px]]<BR>[[Octahedron|{3,4}]]
| [[Image:Tetrahemihexahedron.png|64px]]<BR>[[Tetrahemihexahedron|(4.3/2.4.3)]]
|-
|Cantellated (*)
| [[Image:Cantellated tetrahedron.png|64px]]<BR>[[Cuboctahedron|(3.4.3.4)]]
|-
|Omnitruncated (*)
| [[Image:Omnitruncated tetrahedron.png|64px]]<BR>[[Truncated octahedron|(4.6.6)]]
|-
|Snub (*)
| [[Image:Snub tetrahedron.png|64px]]<BR>[[Icosahedron|{3,5}]]
|}
=== Octahedral symmetry ===
There are 8 convex forms, and 10 nonconvex forms with ''[[octahedral symmetry]]''.
{| class="wikitable"
!Vertex group
!Convex
!colspan=3|Nonconvex
|-
|(Octahedral)
|[[Image:Octahedron.png|64px]]<BR>[[Octahedron|{3,4}]]
|-
|Truncated (*)
| [[Image:Truncated octahedron.png|64px]]<BR>[[Truncated octahedron|(4.6.6)]]
|-
|Rectified (*)
| [[Image:Cuboctahedron.png|64px]]<BR>[[Cuboctahedron|(3.4.3.4)]]
| [[Image:Cubohemioctahedron.png|64px]]<BR>[[Cubohemioctahedron|(6.4/3.6.4)]]
| [[Image:Octahemioctahedron.png|64px]]<BR>[[Octahemioctahedron|(6.3/2.6.3)]]
|-
|Truncated dual (*)
| [[Image:Truncated hexahedron.png|64px]]<BR>[[Truncated cube|(3.8.8)]]
| [[Image:Great rhombihexahedron.png|64px]]<BR>[[Great rhombihexahedron|(4.8/3.4/3.8/5)]]
| [[Image:Great cubicuboctahedron.png|64px]]<BR>[[Great cubicuboctahedron|(8/3.3.8/3.4)]]
| [[Image:Uniform great rhombicuboctahedron.png|64px]]<BR>[[Uniform great rhombicuboctahedron|(4.3/2.4.4)]]
|-
|Dual (*)
|[[Image:Hexahedron.png|64px]]<BR>[[Cube|{4,3}]]
|-
|Cantellated (*)
| [[Image:Small rhombicuboctahedron.png|64px]]<BR>[[Small rhombicuboctahedron|(3.4.4.4)]]
| [[Image:Small rhombihexahedron.png|64px]]<BR>[[Small rhombihexahedron|(4.8.4/3.8)]]
| [[Image:Small cubicuboctahedron.png|64px]]<BR>[[Small cubicuboctahedron|(8.3/2.8.4)]]
| [[Image:Stellated truncated hexahedron.png|64px]]<BR>[[Stellated truncated hexahedron|(8/3.8/3.3)]]
|-
|Omnitruncated (*)
| [[Image:Great rhombicuboctahedron.png|64px]]<BR>[[Great rhombicuboctahedron|(4.6.8)]]
|-
| Nonuniform omnitruncated (*)
| [[Great rhombicuboctahedron|(4.6.8)]]
| [[Image:Great truncated cuboctahedron.png|64px]]<BR>[[Great truncated cuboctahedron|(8/3.4.6)]]
| [[Image:Cubitruncated cuboctahedron.png|64px]]<BR>[[Cubitruncated cuboctahedron|(8/3.6.8)]]
|-
| Snub (*)
| [[Image:snub hexahedron.png|64px]]<BR>[[Snub cube|(3.3.3.3.4)]]
|}
=== Icosahedral symmetry ===<!-- This section is linked from [[Pentagram]] -->
There are 8 convex forms and 46 nonconvex forms with ''[[icosahedral symmetry]]'' (or 47 nonconvex forms if Skilling's figure is included). Some of the nonconvex snub forms have nonuniform chiral symmetry, and some have achiral symmetry.
{| class="wikitable"
!Vertex group
!Convex
!colspan=4|Nonconvex
|-
| (Icosahedral)
| [[Image:Icosahedron.png|64px]]<BR>[[Icosahedron|{3,5}]]
| [[Image:Small stellated dodecahedron.png|64px]]<BR>[[Small stellated dodecahedron|{5/2,5}]]
| [[Image:Great dodecahedron.png|64px]]<BR>[[Great dodecahedron|{5,5/2}]]
| [[Image:Great icosahedron.png|64px]]<BR>[[Great icosahedron|{3,5/2}]]
|-
| Truncated (*)
| [[Image:truncated icosahedron.png|64px]]<BR>[[Truncated icosahedron|(5.6.6)]]
|-
| Nonuniform truncated (*)
| [[Truncated icosahedron|(5.6.6)]]
| [[Image:Great truncated dodecahedron.png|64px]]<BR>[[Truncated great dodecahedron|U37]]
| [[Image:Great dodecicosidodecahedron.png|64px]]<BR>[[Great dodecicosidodecahedron|U61]]
| [[Image:Uniform great rhombicosidodecahedron.png|64px]]<BR>[[Uniform great rhombicosidodecahedron|U67]]
| [[Image:Great rhombidodecahedron.png|64px]]<BR>[[Great rhombidodecahedron|U73]]
| [[Image:Rhombidodecadodecahedron.png|64px]]<BR>[[Rhombidodecadodecahedron|U38]]
| [[Image:Icosidodecadodecahedron.png|64px]]<BR>[[Icosidodecadodecahedron|U44]]
| [[Image:Rhombicosahedron.png|64px]]<BR>[[Rhombicosahedron|U56]]
| [[Image:Small snub icosicosidodecahedron.png|64px]]<BR>[[Small snub icosicosidodecahedron|U32]]
|-
| Rectified (*)
| [[Image:icosidodecahedron.png|64px]]<BR>[[Icosidodecahedron|(3.5.3.5)]]
| [[Image:Small icosihemidodecahedron.png|64px]]<BR>[[Small icosihemidodecahedron|U49]]
| [[Image:Small dodecahemidodecahedron.png|64px]]<BR>[[Small dodecahemidodecahedron|U51]]
| [[Image:Great icosidodecahedron.png|64px]]<BR>[[Great icosidodecahedron|U54]]
| [[Image:Great dodecahemidodecahedron.png|64px]]<BR>[[Great dodecahemidodecahedron|U70]]
| [[Image:Great icosihemidodecahedron.png|64px]]<BR>[[Great icosihemidodecahedron|U71]]
| [[Image:Dodecadodecahedron.png|64px]]<BR>[[Dodecadodecahedron|U36]]
| [[Image:Small dodecahemicosahedron.png|64px]]<BR>[[Small dodecahemicosahedron|U62]]
| [[Image:Great dodecahemicosahedron.png|64px]]<BR>[[Great dodecahemicosahedron|U65]]
|-
| Truncated dual (*)
| [[Image:truncated dodecahedron.png|64px]]<BR>[[Truncated dodecahedron|(3.10.10)]]
| [[Image:Great ditrigonal dodecicosidodecahedron.png|64px]]<BR>[[Great ditrigonal dodecicosidodecahedron|U42]]
| [[Image:Great icosicosidodecahedron.png|64px]]<BR>[[Great icosicosidodecahedron|U48]]
| [[Image:Great dodecicosahedron.png|64px]]<BR>[[Great dodecicosahedron|U63]]
|-
| Nonuniform truncated dual (*)
| [[Truncated dodecahedron|(3.10.10)]]
| [[Image:Small retrosnub icosicosidodecahedron.png|64px]]<BR>[[Small retrosnub icosicosidodecahedron|U72]]
|-
| Dual (*)
| [[Image:Dodecahedron.png|64px]]<BR>[[Dodecahedron|{5,3}]]
| [[Image:Great stellated dodecahedron.png|64px]]<BR>[[Great stellated dodecahedron|{5/2,3}]]
| [[Image:Small ditrigonal icosidodecahedron.png|64px]]<BR>[[Small ditrigonal icosidodecahedron|U30]]
| [[Image:Ditrigonal dodecadodecahedron.png|64px]]<BR>[[Ditrigonal dodecadodecahedron|U41]]
| [[Image:Great ditrigonal icosidodecahedron.png|64px]]<BR>[[Great ditrigonal icosidodecahedron|U47]]
|-
| Cantellated (*)
| [[Image:Small rhombicosidodecahedron.png|64px]]<BR>[[Rhombicosidodecahedron|(3.4.5.4)]]
| [[Image:Small dodecicosidodecahedron.png|64px]]<BR>[[Small dodecicosidodecahedron|U33]]
| [[Image:Small rhombidodecahedron.png|64px]]<BR>[[Small rhombidodecahedron|U39]]
| [[Image:Small stellated truncated dodecahedron.png|64px]]<BR>[[Small stellated truncated dodecahedron|U58]]
|-
| Nonuniform Cantellated (*)
| [[Rhombicosidodecahedron|(3.4.5.4)]]
| [[Image:Small icosicosidodecahedron.png|64px]]<BR>[[Small icosicosidodecahedron|U31]]
| [[Image:Small ditrigonal dodecicosidodecahedron.png|64px]]<BR>[[Small ditrigonal dodecicosidodecahedron|U43]]
| [[Image:Small dodecicosahedron.png|64px]]<BR>[[Small dodecicosahedron|U50]]
| [[Image:Great stellated truncated dodecahedron.png|64px]]<BR>[[Great stellated truncated dodecahedron|U66]]
| [[Image:Great truncated icosahedron.png|64px]]<BR>[[Truncated great icosahedron|U55]]
| [[Image:Great dirhombicosidodecahedron.png|64px]]<BR>[[Great dirhombicosidodecahedron|U75]]
| [[Image:Great snub dodecicosidodecahedron.png|64px]]<BR>[[Great snub dodecicosidodecahedron|U64]]
|-
| Omnitruncated (*)
| [[Image:Great rhombicosidodecahedron.png|64px]]<BR>[[Truncated icosidodecahedron|(4.6.10)]]
|-
| Nonuniform omnitruncated (*)
| [[Truncated icosidodecahedron|(4.6.10)]]
| [[Image:Great truncated icosidodecahedron.png|64px]]<BR>[[Great truncated icosidodecahedron|U68]]
| [[Image:Truncated dodecadodecahedron.png|64px]]<BR>[[Truncated dodecadodecahedron|U59]]
| [[Image:Icositruncated dodecadodecahedron.png|64px]]<BR>[[Icositruncated dodecadodecahedron|U45]]
|-
| Snub (*)
|[[Image:snub dodecahedron ccw.png|64px]]<BR>[[Snub dodecahedron|(3.3.3.3.5)]]
|-
| Nonuniform Snub (*)
| [[Snub dodecahedron|(3.3.3.3.5)]]
| [[Image:Snub dodecadodecahedron.png|64px]]<BR>[[Snub dodecadodecahedron|U40]]
| [[Image:Snub icosidodecadodecahedron.png|64px]]<BR>[[Snub icosidodecadodecahedron|U46]]
| [[Image:Great snub icosidodecahedron.png|64px]]<BR>[[Great snub icosidodecahedron|U57]]
| [[Image:Great inverted snub icosidodecahedron.png|64px]]<BR>[[Great inverted snub icosidodecahedron|U69]]
| [[Image:Inverted snub dodecadodecahedron.png|64px]]<BR>[[Inverted snub dodecadodecahedron|U60]]
| [[Image:Great retrosnub icosidodecahedron.png|64px]]<BR>[[Great retrosnub icosidodecahedron|U74]]
|}
== Skilling's figure ==
[[Image:Great disnub dirhombidodecahedron.png|right|64px]]
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 '''I'''<sub>h</sub> symmetry.
== See also ==
*[[Polyhedron]]
**[[Regular polyhedron]]
**[[Quasiregular polyhedron]]
**[[Semiregular polyhedron]]
*[[List of uniform polyhedra]]
*[[List of Wenninger polyhedron models]]
*[[Polyhedron model]]
*[[List of uniform polyhedra by vertex figure]]
*[[List of uniform polyhedra by Wythoff symbol]]
== References ==
*Brückner, M. ''Vielecke und vielflache. Theorie und geschichte.''. Leipzig, Germany: Teubner, 1900. [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8316.0001.001]
*H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, ''Uniform polyhedra'', '''Phil. Trans.''' 1954, 246 A, 401-50 [http://links.jstor.org/sici?sici=0080-4614%2819540513%29246%3A916%3C401%3AUP%3E2.0.CO%3B2-4]
*S. P. Sopov ''A proof of the completeness on the list of elementary homogeneous polyhedra.'' (Russian) Ukrain. Geometr. Sb. No. 8, (1970), 139-156
* {{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger | title=Polyhedron Models | publisher=Cambridge University Press | year=1974 | id=ISBN 0-521-09859-9 }}
*[[John Skilling]], ''The complete set of uniform polyhedra.'', Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 111-135 [http://links.jstor.org/sici?sici=0080-4614%2819750306%29278%3A1278%3C111%3ATCSOUP%3E2.0.CO%3B2-F]
* Har'El, Z. ''Uniform Solution for Uniform Polyhedra.'', Geometriae Dedicata 47, 57-110, 1993. [http://www.math.technion.ac.il/~rl Zvi Har’El] [http://www.math.technion.ac.il/~rl/docs/uniform.pdf], [http://www.math.technion.ac.il/~rl/kaleido Kaleido software], [http://www.math.technion.ac.il/~rl/kaleido/poly.html Images], [http://www.math.technion.ac.il/~rl/kaleido/dual.html dual images]
* [http://www.mathconsult.ch/showroom/unipoly Mäder, R. E.] ''Uniform Polyhedra.'' Mathematica J. 3, 48-57, 1993. [http://library.wolfram.com/infocenter/Articles/2254]
==External links==
* {{MathWorld | urlname=UniformPolyhedron | title=Uniform Polyhedron}}
*[http://www.math.technion.ac.il/~rl/docs/uniform.pdf Uniform Solution for Uniform Polyhedra]
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[http://www.georgehart.com/virtual-polyhedra/uniform-info.html Virtual Polyhedra] Uniform Polyhedra
*[http://www.software3d.com/Stella.php 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:
**[http://www.software3d.com/Uniform.php Uniform/Dual Polyhedra]
**[http://www.polyedergarten.de/ Paper Models of Uniform (and other) Polyhedra]
* [http://www.tfh-berlin.de/~s17299/applets/polyh2.html uniform polyhedra in 3d]
[[Category:Polyhedra]]
[[Category:Uniform polyhedra]]
[[eo:Uniforma pluredro]]
[[fr:Polyèdre uniforme]]
[[it:Poliedro uniforme]]
[[ja:一様多面体]]