View source for Uniform polytope

{{Short description|Isogonal polytope with uniform facets}}
{|class=wikitable style="float:right; margin-left:8px; width:350px"
|+ Convex uniform polytopes
|-
!2D
!3D
|- valign=top
|[[File:Truncated triangle.svg|160px]]<br>[[Truncation (geometry)|Truncated]] [[triangle]] or uniform [[hexagon]], with [[Coxeter diagram]] {{CDD|node_1|3|node_1}}.
|[[File:Truncated octahedron.png|160px]]<br>[[Truncated octahedron]], {{CDD|node_1|3|node_1|4|node}}
|- valign=top
!4D
!5D
|- valign=top
|[[File:Schlegel half-solid truncated 16-cell.png|160px]]<br>[[Truncated 16-cell]], {{CDD|node_1|3|node_1|3|node|4|node}}
|[[File:5-cube t34 B4.svg|160px]]<br>[[Truncated 5-orthoplex]], {{CDD|node_1|3|node_1|3|node|3|node|4|node}}
|}

In [[geometry]], a '''uniform polytope''' of dimension three or higher is a [[vertex-transitive]] [[polytope]] bounded by uniform [[Facet (mathematics)|facets]]. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the same must also be true within each lower-dimensional face of the polytope. In two dimensions (and for two-dimensional faces of higher-dimensional polytopes) a stronger definition is used: only the [[regular polygon]]s are considered as uniform, disallowing polygons that alternate between two different lengths of edges.

This is a generalization of the older category of [[semiregular polytope|''semiregular'' polytopes]], but also includes the [[regular polytope]]s. Further, star regular faces and [[vertex figure]]s ([[star polygon]]s) are allowed, which greatly expand the possible solutions.  A strict definition requires uniform polytopes to be finite, while a more expansive definition allows [[uniform honeycomb]]s (2-dimensional [[uniform tiling|tilings]] and higher dimensional [[Honeycomb (geometry)|honeycombs]]) of [[Euclidean space|Euclidean]] and [[hyperbolic space]] to be considered polytopes as well.

== Operations ==
Nearly every uniform polytope can be generated by a [[Wythoff construction]], and represented by a [[Coxeter diagram]]. Notable exceptions include the [[great dirhombicosidodecahedron]] in three dimensions and the [[grand antiprism]] in four dimensions.

Equivalently, the Wythoffian polytopes can be generated by applying basic operations to the regular polytopes in that dimension. This approach was first used by [[Johannes Kepler]], and is the basis of the [[Conway polyhedron notation]].

=== Rectification operators ===
Regular n-polytopes have ''n'' orders of [[Rectification (geometry)|rectification]]. The zeroth rectification is the original form. The (''n''−1)-th rectification is the [[dual polytope|dual]]. A '''rectification''' reduces edges to vertices, a '''birectification''' reduces faces to vertices, a '''trirectification''' reduces cells to vertices, a '''quadirectification''' reduces 4-faces to vertices, a '''quintirectification''' reduced 5-faces to vertices, and so on.

An extended [[Schläfli symbol]] can be used for representing rectified forms, with a single subscript: 
* ''k''-th rectification = '''t<sub>k</sub>'''{p<sub>1</sub>, p<sub>2</sub>, ..., p<sub>n−1</sub>} = ''k'''''r'''.

=== Truncation operators ===
Truncation operations that can be applied to regular ''n''-polytopes in any combination.  The resulting Coxeter diagram has two ringed nodes, and the operation is named for the distance between them.  '''Truncation''' cuts vertices, '''cantellation''' cuts edges, '''runcination''' cuts faces, '''sterication''' cut cells. Each higher operation also cuts lower ones too, so a cantellation also truncates vertices.

# '''t<sub>0,1</sub>''' or '''t''': [[Truncation (geometry)|'''Truncation''']] - applied to [[polygon]]s and higher. A truncation removes vertices, and inserts a new facet in place of each former vertex. Faces are truncated, doubling their edges. (The term, coined by [[Kepler]], comes from Latin ''truncare'' 'to cut off'.) 
#: [[File:Cube truncation sequence.svg|480px]]
#* There are higher truncations also: '''[[bitruncation]]''' '''t<sub>1,2</sub>''' or '''2t''', '''tritruncation''' '''t<sub>2,3</sub>''' or '''3t''', '''quadritruncation''' '''t<sub>3,4</sub>''' or '''4t''', '''quintitruncation''' '''t<sub>4,5</sub>''' or '''5t''', etc.
# '''t<sub>0,2</sub>''' or '''rr''': [[Cantellation (geometry)|'''Cantellation''']] - applied to [[Uniform polyhedron|polyhedra]] and higher. It can be seen as rectifying its [[rectification (geometry)|rectification]]. A cantellation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically [[Expansion (geometry)|expanded]] copies of themselves. (The term, coined by Johnson, is derived from the verb ''cant'', like ''[[bevel]]'', meaning to cut with a slanted face.) 
#: [[File:Cube cantellation sequence.svg|480px]]
#* There are higher cantellations also: '''bicantellation''' '''t<sub>1,3</sub>''' or '''r2r''', '''tricantellation''' '''t<sub>2,4</sub>''' or '''r3r''', '''quadricantellation''' '''t<sub>3,5</sub>''' or '''r4r''', etc.
#* '''t<sub>0,1,2</sub>''' or '''tr''': '''Cantitruncation''' - applied to [[Uniform polyhedron|polyhedra]] and higher. It can be seen as truncating its [[rectification (geometry)|rectification]]. A cantitruncation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically [[Expansion (geometry)|expanded]] copies of themselves. (The composite term combines cantellation and truncation) 
#** There are higher cantellations also: '''bicantitruncation''' '''t<sub>1,2,3</sub>''' or '''t2r''', '''tricantitruncation''' '''t<sub>2,3,4</sub>''' or '''t3r''', '''quadricantitruncation''' '''t<sub>3,4,5</sub>''' or '''t4r''', etc.
# '''t<sub>0,3</sub>''': '''[[Runcination]]''' - applied to [[Uniform 4-polytope]] and higher. Runcination truncates vertices, edges, and faces, replacing them each with new facets. 4-faces are replaced by topologically expanded copies of themselves. (The term, coined by Johnson, is derived from Latin ''runcina'' 'carpenter's [[Plane (tool)|plane]]'.) 
#* There are higher runcinations also: '''biruncination''' '''t<sub>1,4</sub>''', '''triruncination''' '''t<sub>2,5</sub>''', etc.
# '''t<sub>0,4</sub>''' or '''2r2r''': '''Sterication''' - applied to [[Uniform 5-polytope]]s and higher. It can be seen as birectifying its birectification. Sterication truncates vertices, edges, faces, and cells, replacing each with new facets. 5-faces are replaced by topologically expanded copies of themselves. (The term, coined by Johnson, is derived from Greek ''stereos'' 'solid'.) 
#* There are higher sterications also: '''bisterication''' '''t<sub>1,5</sub>''' or '''2r3r''', '''tristerication''' '''t<sub>2,6</sub>''' or '''2r4r''', etc.
#* '''t<sub>0,2,4</sub>''' or '''2t2r''': '''Stericantellation''' - applied to [[Uniform 5-polytope]]s and higher. It can be seen as bitruncating its birectification. 
#** There are higher sterications also: '''bistericantellation''' '''t<sub>1,3,5</sub>''' or '''2t3r''', '''tristericantellation''' '''t<sub>2,4,6</sub>''' or '''2t4r''', etc.
# '''t<sub>0,5</sub>''': '''Pentellation''' - applied to [[Uniform 6-polytope]]s and higher. Pentellation truncates vertices, edges, faces, cells, and 4-faces, replacing each with new facets. 6-faces are replaced by topologically expanded copies of themselves. (Pentellation is derived from Greek ''[[penta-|pente]]'' 'five'.) 
#* There are also higher pentellations: '''bipentellation''' '''t<sub>1,6</sub>''', '''tripentellation''' '''t<sub>2,7</sub>''', etc.
# '''t<sub>0,6</sub>''' or '''3r3r''': '''Hexication''' - applied to [[Uniform 7-polytope]]s and higher.  It can be seen as trirectifying its trirectification. Hexication truncates vertices, edges, faces, cells, 4-faces, and 5-faces, replacing each with new facets. 7-faces are replaced by topologically expanded copies of themselves. (Hexication is derived from Greek ''[[hexa-|hex]]'' 'six'.) 
#* There are higher hexications also: '''bihexication''': '''t<sub>1,7</sub>''' or '''3r4r''', '''trihexication''': '''t<sub>2,8</sub>''' or '''3r5r''', etc.
#* '''t<sub>0,3,6</sub>''' or '''3t3r''': '''Hexiruncinated''' - applied to [[Uniform 7-polytope]]s and higher.  It can be seen as tritruncating its trirectification. 
#** There are also higher hexiruncinations: '''bihexiruncinated''': '''t<sub>1,4,7</sub>''' or '''3t4r''', '''trihexiruncinated''': '''t<sub>2,5,8</sub>''' or '''3t5r''', etc.
# '''t<sub>0,7</sub>''': '''Heptellation''' - applied to [[Uniform 8-polytope]]s and higher. Heptellation truncates vertices, edges, faces, cells, 4-faces, 5-faces, and 6-faces, replacing each with new facets. 8-faces are replaced by topologically expanded copies of themselves. (Heptellation is derived from Greek ''[[hepta-|hepta]]'' 'seven'.) 
#* There are higher heptellations also: '''biheptellation''' '''t<sub>1,8</sub>''', '''triheptellation''' '''t<sub>2,9</sub>''', etc.
# '''t<sub>0,8</sub>''' or '''4r4r''': '''Octellation''' - applied to [[Uniform 9-polytope]]s and higher.
# '''t<sub>0,9</sub>''': '''Ennecation''' - applied to [[Uniform 10-polytope]]s and higher.
In addition combinations of truncations can be performed which also generate new uniform polytopes. For example, a ''runcitruncation'' is a ''runcination'' and ''truncation'' applied together.

If all truncations are applied at once, the operation can be more generally called an [[omnitruncation]].

=== Alternation ===
[[File:Snubcubes in grCO.svg|400px|thumb|An alternation of a [[truncated cuboctahedron]] produces a [[snub cube]].]]
One special operation, called [[Alternation (geometry)|alternation]], removes alternate vertices from a polytope with only even-sided faces.  An alternated omnitruncated polytope is called a ''snub''.

The resulting polytopes always can be constructed, and are not generally reflective, and also do not in general have ''uniform'' polytope solutions.

The set of polytopes formed by alternating the [[hypercube]]s are known as [[demicube]]s. In three dimensions, this produces a [[tetrahedron]]; in four dimensions, this produces a [[16-cell]], or ''demitesseract''.

== Vertex figure ==
Uniform polytopes can be constructed from their [[vertex figure]], the arrangement of edges, faces, cells, etc. around each vertex. Uniform polytopes represented by a [[Coxeter diagram]], marking active mirrors by rings, have reflectional symmetry, and can be simply constructed by recursive reflections of the vertex figure.

A smaller number of nonreflectional uniform polytopes have a single vertex figure but are not repeated by simple reflections.  Most of these can be represented with operations like [[Alternation (geometry)|alternation]] of other uniform polytopes.

Vertex figures for single-ringed Coxeter diagrams can be constructed from the diagram by removing the ringed node, and ringing neighboring nodes. Such vertex figures are themselves vertex-transitive.

Multiringed polytopes can be constructed by a slightly more complicated construction process, and their topology is not a uniform polytope. For example, the vertex figure of a [[truncation (geometry)|truncated]] regular polytope (with 2 rings) is a pyramid. An [[omnitruncation|omnitruncated]] polytope (all nodes ringed) will always have an irregular [[simplex]] as its vertex figure.

== Circumradius ==
Uniform polytopes have equal edge-lengths, and all vertices are an equal distance from the center, called the '''circumradius'''.

Uniform polytopes whose circumradius is equal to the edge length can be used as [[vertex figure]]s for [[uniform honeycomb]]s. For example, the regular [[hexagon]] divides into 6 equilateral triangles and is the vertex figure for the regular [[triangular tiling]]. Also the [[cuboctahedron]] divides into 8 regular tetrahedra and 6 square pyramids (half [[octahedron]]), and it is the vertex figure for the [[alternated cubic honeycomb]].

== Uniform polytopes by dimension ==
It is useful to classify the uniform polytopes by dimension. This is equivalent to the number of nodes on the Coxeter diagram, or the number of hyperplanes in the Wythoffian construction. Because (''n''+1)-dimensional polytopes are tilings of ''n''-dimensional spherical space, tilings of ''n''-dimensional [[Euclidean space|Euclidean]] and [[hyperbolic space]] are also considered to be (''n''+1)-dimensional.  Hence, the tilings of two-dimensional space are grouped with the three-dimensional solids.

=== One dimension ===
The only one-dimensional polytope is the line segment. It corresponds to the Coxeter family A<sub>1</sub>.

=== Two dimensions ===
In two dimensions, there is an infinite family of convex uniform polytopes, the [[regular polygon]]s, the simplest being the equilateral [[triangle]]. Truncated regular polygons become bicolored geometrically [[quasiregular polygon|quasiregular]] polygons of twice as many sides, t{p}={2p}. The first few regular polygons (and quasiregular forms) are displayed below:

{|class="wikitable" style="text-align:center;"
|- bgcolor="#e0e0e0" valign="top"
!Name
![[Equilateral triangle|Triangle]]<br />([[Simplex|2-simplex]])
![[Square (geometry)|Square]]<br />([[Cross-polytope|2-orthoplex]])<br />([[Hypercube|2-cube]])
![[Pentagon]]
![[Hexagon]]
![[Heptagon]]
![[Octagon]]
![[Enneagon]]
![[Decagon]]
![[Hendecagon]]
|- bgcolor="#ffe0e0"
![[Schläfli symbol|Schläfli]]
|{3}
|{4}<br>t{2}
|{5}
|{6}<br>t{3}
|{7}
|{8}<br>t{4}
|{9}
|{10}<br>t{5}
|{11}
|- valign=top
![[Coxeter diagram|Coxeter<br>diagram]]
|{{CDD|node_1|3|node}}
|{{CDD|node_1|4|node}}<br>{{CDD|node_1|2|node_1}}
|{{CDD|node_1|5|node}}
|{{CDD|node_1|6|node}}<br>{{CDD|node_1|3|node_1}}
|{{CDD|node_1|7|node}}
|{{CDD|node_1|8|node}}<br>{{CDD|node_1|4|node_1}}
|{{CDD|node_1|9|node}}
|{{CDD|node_1|10|node}}<br>{{CDD|node_1|5|node_1}}
|{{CDD|node_1|11|node}}
|- valign=top
!Image
|[[File:Regular triangle.svg|75px]]
|[[File:Regular quadrilateral.svg|75px]]<br>[[File:Truncated polygon 4.svg|75px]]
|[[File:Regular pentagon.svg|75px]]
|[[File:Regular hexagon.svg|75px]]<br>[[File:Truncated polygon 6.svg|75px]]
|[[File:Regular heptagon.svg|75px]]
|[[File:Regular octagon.svg|75px]]<br>[[File:Truncated polygon 8.svg|75px]]
|[[File:Regular nonagon.svg|75px]]
|[[File:Regular decagon.svg|75px]]<br>[[File:Truncated polygon 10.svg|75px]]
|[[File:Regular hendecagon.svg|75px]]
|- bgcolor="#e0e0e0" valign="top"
!Name
![[Dodecagon]]
![[Tridecagon]]
![[Tetradecagon]]
![[Pentadecagon]]
![[Hexadecagon]]
![[Heptadecagon]]
![[Octadecagon]]
![[Enneadecagon]]
![[Icosagon]]
|- bgcolor="#ffe0e0"
![[Schläfli symbol|Schläfli]]
|{12}<br>t{6}
|{13}
|{14}<br>t{7}
|{15}
|{16}<br>t{8}
|{17}
|{18}<br>t{9}
|{19}
|{20}<br>t{10}
|- valign=top
![[Coxeter diagram|Coxeter<br>diagram]]
|{{CDD|node_1|12|node}}<br>{{CDD|node_1|6|node_1}}
|{{CDD|node_1|13|node}}
|{{CDD|node_1|14|node}}<br>{{CDD|node_1|7|node_1}}
|{{CDD|node_1|15|node}}
|{{CDD|node_1|16|node}}<br>{{CDD|node_1|8|node_1}}
|{{CDD|node_1|17|node}}
|{{CDD|node_1|18|node}}<br>{{CDD|node_1|9|node_1}}
|{{CDD|node_1|19|node}}
|{{CDD|node_1|20|node}}<br>{{CDD|node_1|10|node_1}}
|- valign=top
!Image
|[[File:Regular dodecagon.svg|75px]]<br>[[File:Truncated polygon 12.svg|75px]]
|[[File:Regular tridecagon.svg|75px]]
|[[File:Regular tetradecagon.svg|75px]]<br>[[File:Truncated polygon 14.svg|75px]]
|[[File:Regular pentadecagon.svg|75px]]
|[[File:Regular hexadecagon.svg|75px]]<br>[[File:Truncated polygon 16.svg|75px]]
|[[File:Regular heptadecagon.svg|75px]]
|[[File:Regular octadecagon.svg|75px]]<br>[[File:Truncated polygon 18.svg|75px]]
|[[File:Regular enneadecagon.svg|75px]]
|[[File:Regular icosagon.svg|75px]]<br>[[File:Truncated polygon 20.svg|75px]]
|}

There is also an infinite set of [[star polygon]]s (one for each [[rational number]] greater than 2), but these are non-convex. The simplest example is the [[pentagram]], which corresponds to the rational number 5/2. Regular star polygons, {p/q}, can be truncated into semiregular star polygons, t{p/q}=t{2p/q}, but become double-coverings if ''q'' is even. A truncation can also be made with a reverse orientation polygon t{p/(p−q)}={2p/(p−q)}, for example t{5/3}={10/3}.

{|class="wikitable" style="text-align:center;"
|- bgcolor="#e0e0e0"
!Name
|[[Pentagram]]
| colspan="2" | [[Heptagram]]s
|[[Octagram]]
| colspan="2" | [[Enneagram (geometry)|Enneagrams]]
|[[Decagram (geometry)|Decagram]]
|...[[star polygon|n-agrams]]
|- bgcolor="#ffe0e0"
![[Schläfli symbol|Schläfli]]
|{5/2}
|{7/2}
|{7/3}
|{8/3}<br>t{4/3}
|{9/2}
|{9/4}
|{10/3}<br>t{5/3}
|{''p/q''}
|-
![[Coxeter diagram|Coxeter<br>diagram]]
|{{CDD|node_1|5|rat|d2|node}}
|{{CDD|node_1|7|rat|d2|node}}
|{{CDD|node_1|7|rat|d3|node}}
|{{CDD|node_1|8|rat|d3|node}}<br>{{CDD|node_1|4|rat|d3|node_1}}
|{{CDD|node_1|9|rat|d2|node}}
|{{CDD|node_1|9|rat|d4|node}}
|{{CDD|node_1|10|rat|d3|node}}<br>{{CDD|node_1|5|rat|d3|node_1}}
|{{CDD|node_1|p|rat|dq|node}}
|- valign=top
!Image
|[[File:Star polygon 5-2.svg|75px]]
|[[File:Star polygon 7-2.svg|75px]]
|[[File:Star polygon 7-3.svg|75px]]
|[[File:Star polygon 8-3.svg|75px]]<br>[[File:Regular polygon truncation 4 3.svg|75px]]
|[[File:Star polygon 9-2.svg|75px]]
|[[File:Star polygon 9-4.svg|75px]]
|[[File:Star polygon 10-3.svg|75px]]<br>[[File:Regular star truncation 5-3 1.svg|75px]]
|&nbsp;
|}

Regular polygons, represented by [[Schläfli symbol]] {p} for a p-gon. Regular polygons are self-dual, so the rectification produces the same polygon. The uniform truncation operation doubles the sides to {2p}. The snub operation, alternating the truncation, restores the original polygon {p}. Thus all uniform polygons are also regular. The following operations can be performed on regular polygons to derive the uniform polygons, which are also regular polygons:

{|class="wikitable" valign=center
!rowspan=2|Operation
!rowspan=2 colspan=2|Extended<br>Schläfli<br>Symbols
!rowspan=2|Regular<br>result
!rowspan=2|[[Coxeter diagram|Coxeter<br>diagram]]
!colspan=2|Position
!rowspan=2|[[Coxeter notation|Symmetry]]
|- BGCOLOR="#f0e0e0" align=center
! (1)
! (0)
|- BGCOLOR="#f0e0e0" align=center
| '''Parent'''
| {p}|| t<sub>0</sub>{p}
| {p}
|{{CDD|node_1|p|node}}
| {}
| --
| [p]<br>(order 2p)
|- BGCOLOR="#e0e0f0" align=center
| '''Rectified'''<br>(Dual)
| r{p}||t<sub>1</sub>{p}
| {p}
|{{CDD|node|p|node_1}}
| --
| {}
| [p]<br>(order 2p)
|- BGCOLOR="#e0f0e0" align=center
| '''Truncated'''
| t{p}||t<sub>0,1</sub>{p}
| {2p}
| {{CDD|node_1|p|node_1}}
| {}
| {}
| [<nowiki/>[p]]=[2p]<br>(order 4p)
|- BGCOLOR="#e0f0e0" align=center
| Half
| colspan=2|h{2p}
| {p}
|{{CDD|node_h|2x|p|node}}
| --
| --
| [1<sup>+</sup>,2p]=[p]<br>(order 2p)
|- BGCOLOR="#e0f0e0" align=center
| '''Snub'''
|colspan=2| s{p}
| {p}
|{{CDD|node_h|p|node_h}}
| --
| --
| [<nowiki/>[p]]<sup>+</sup>=[p]<br>(order 2p)
|}

=== Three dimensions ===
{{Main article|Uniform polyhedron}}

In three dimensions, the situation gets more interesting. There are five convex regular polyhedra, known as the [[Platonic solid]]s:

{|class="wikitable" style="text-align:center;"
|- valign="top"
!Name
![[Schläfli symbol|Schläfli]]<br />{p,q}
![[Coxeter diagram|Diagram]]<br />{{CDD|node_1|p|node|q|node}}
!Image<br>(transparent)
!Image<br>(solid)
!Image<br>(sphere)
![[Face (geometry)|Faces]]<br />{p}
![[Edge (geometry)|Edges]]
![[Vertex (geometry)|Vertices]]<br />{q}
![[Symmetry group|Symmetry]]
![[Dual polyhedron|Dual]]
|- bgcolor="#e0e0e0"
|[[Tetrahedron]]<br />([[Simplex|3-simplex]])<br />(Pyramid)
|{3,3}
|{{CDD|node_1|3|node|3|node}}
|[[File:Tetrahedron.svg|75px]]
|[[File:Tetrahedron.png|75px]]
|[[File:Uniform tiling 332-t0-1-.svg|75px]]
|4<br />{3}
|6
|4<br />{3}
|T<sub>d</sub>
|(self)
|- bgcolor="#ffe0e0"
|[[Cube]] <br />([[hypercube|3-cube]])<br />(Hexahedron)
|{4,3}
|{{CDD|node_1|4|node|3|node}}
|[[File:Hexahedron.svg|75px]]
|[[File:Hexahedron.png|75px]]
|[[File:Uniform tiling 432-t0.png|75px]]
|6<br />{4}
|12
|8<br />{3}
|O<sub>h</sub>
|Octahedron
|- bgcolor="#e0e0ff"
|[[Octahedron]]<br />([[Cross-polytope|3-orthoplex]])
|{3,4}
|{{CDD|node_1|3|node|4|node}}
|[[File:Octahedron.svg|75px]]
|[[File:Octahedron.png|75px]]
|[[File:Uniform tiling 432-t2.png|75px]]
|8<br />{3}
|12
|6<br />{4}
|O<sub>h</sub>
|Cube
|- bgcolor="#ffe0e0"
|[[Dodecahedron]]
|{5,3}
|{{CDD|node_1|5|node|3|node}}
|[[File:POV-Ray-Dodecahedron.svg|75px]]
|[[File:Dodecahedron.png|75px]]
|[[File:Uniform tiling 532-t0.png|75px]]
|12<br />{5}
|30
|20<br />{3}2
|I<sub>h</sub>
|Icosahedron
|- bgcolor="#e0e0ff"
|[[Icosahedron]]
|{3,5}
|{{CDD|node_1|3|node|5|node}}
|[[File:Icosahedron.svg|75px]]
|[[File:Icosahedron.png|75px]]
|[[File:Uniform tiling 532-t2.png|75px]]
|20<br />{3}
|30
|12<br />{5}
|I<sub>h</sub>
|Dodecahedron
|}

In addition to these, there are also 13 semiregular polyhedra, or [[Archimedean solid]]s, which can be obtained via [[Wythoff construction]]s, or by performing operations such as [[truncation (geometry)|truncation]] on the Platonic solids, as demonstrated in the following table:

{|class="wikitable"
|-
!
!Parent
!Truncated
!Rectified
!Bitruncated<br>(tr. dual)
!Birectified<br>(dual)
!Cantellated
!Omnitruncated<br>({{small|Cantitruncated}})
!Snub
|-
|[[Tetrahedral symmetry|Tetrahedral]]<br>3-3-2
|[[File:Uniform polyhedron-33-t0.png|64px]]<br>[[Tetrahedron|{3,3}]]
|[[File:Uniform polyhedron-33-t01.png|64px]]<br>[[Truncated tetrahedron|(3.6.6)]]
|[[File:Uniform polyhedron-33-t1.svg|64px]]<br>[[Octahedron|(3.3.3.3)]]
|[[File:Uniform polyhedron-33-t12.png|64px]]<br>[[Truncated tetrahedron|(3.6.6)]]
|[[File:Uniform polyhedron-33-t2.png|64px]]<br>[[Tetrahedron|{3,3}]]
| [[File:Uniform polyhedron-33-t02.svg|64px]]<br>[[Cuboctahedron|(3.4.3.4)]]
|[[File:Uniform polyhedron-33-t012.png|64px]]<br>[[Truncated octahedron|(4.6.6)]]
|[[File:Uniform polyhedron-33-s012.svg|64px]]<br>[[Icosahedron|(3.3.3.3.3)]]
|-
|[[Octahedral symmetry|Octahedral]]<br>4-3-2
|[[File:Uniform polyhedron-43-t0.svg|64px]]<br>[[Cube|{4,3}]]
|[[File:Uniform polyhedron-43-t01.svg|64px]]<br>[[Truncated cube|(3.8.8)]]
|[[File:Uniform polyhedron-43-t1.svg|64px]]<br>[[Cuboctahedron|(3.4.3.4)]]
|[[File:Uniform polyhedron-43-t12.svg|64px]]<br>[[Truncated octahedron|(4.6.6)]]
|[[File:Uniform polyhedron-43-t2.svg|64px]]<br>[[Octahedron|{3,4}]]
|[[File:Uniform polyhedron-43-t02.png|64px]]<br>[[Small rhombicuboctahedron|(3.4.4.4)]]
|[[File:Uniform polyhedron-43-t012.png|64px]]<br>[[Truncated cuboctahedron|(4.6.8)]]
|[[File:Uniform polyhedron-43-s012.png|64px]]<br>[[Snub cube|(3.3.3.3.4)]]
|-
|[[Icosahedral symmetry|Icosahedral]]<br>5-3-2
|[[File:Uniform polyhedron-53-t0.svg|64px]]<br>[[Dodecahedron|{5,3}]]
|[[File:Uniform polyhedron-53-t01.svg|64px]]<br>[[Truncated dodecahedron|(3.10.10)]]
|[[File:Uniform polyhedron-53-t1.svg|64px]]<br>[[Icosidodecahedron|(3.5.3.5)]]
|[[File:Uniform polyhedron-53-t12.svg|64px]]<br>[[Truncated icosahedron|(5.6.6)]]
|[[File:Uniform polyhedron-53-t2.svg|64px]]<br>[[Icosahedron|{3,5}]]
|[[File:Uniform polyhedron-53-t02.png|64px]]<br>[[Rhombicosidodecahedron|(3.4.5.4)]]
|[[File:Uniform polyhedron-53-t012.png|64px]]<br>[[Truncated icosidodecahedron|(4.6.10)]]
|[[File:Uniform polyhedron-53-s012.png|64px]]<br>[[Snub dodecahedron|(3.3.3.3.5)]]
|}

There is also the infinite set of [[Prism (geometry)|prisms]], one for each regular polygon, and a corresponding set of [[antiprism]]s.

{|class="wikitable"
!#
!Name
!Picture
!Tiling
![[Vertex figure|Vertex<br>figure]]
![[Coxeter diagram|Diagram]]<br>and [[Schläfli symbol#Extended for uniform 4-polytope and 3-space honeycombs|Schläfli]]<br>symbols
|- BGCOLOR="#e0f0e0"
!P<sub>2p</sub>
|[[Prism (geometry)|Prism]]
|[[File:Dodecagonal prism.png|60px]]
|[[File:Spherical truncated hexagonal prism.png|60px]]
|[[File:Dodecagonal prism vf.png|60px]]
|align=center|{{CDD|node_1|p|node_1|2|node_1}}<br>tr{2,p}
|- BGCOLOR="#d0f0f0"
!A<sub>p</sub>
|[[Antiprism]]
|[[File:Hexagonal antiprism.png|60px]]
|[[File:Spherical hexagonal antiprism.svg|60px]]
|[[File:Hexagonal antiprism vertfig.png|60px]]
|align=center|{{CDD|node_h|p|node_h|2x|node_h}}<br>sr{2,p}
|}

The uniform star polyhedra include a further 4 regular star polyhedra, the [[Kepler-Poinsot polyhedra]], and 53 semiregular star polyhedra. There are also two infinite sets, the star prisms (one for each star polygon) and star antiprisms (one for each rational number greater than 3/2).

==== Constructions ====
The Wythoffian uniform polyhedra and tilings can be defined by their [[Wythoff symbol]], which specifies the [[fundamental region]] of the object. An extension of [[Schläfli symbol|Schläfli]] notation, also used by [[Harold Scott MacDonald Coxeter|Coxeter]], applies to all dimensions; it consists of the letter 't', followed by a series of subscripted numbers corresponding to the ringed nodes of the [[Coxeter diagram]], and followed by the Schläfli symbol of the regular seed polytope. For example, the [[truncated octahedron]] is represented by the notation: t<sub>0,1</sub>{3,4}.

{|class="wikitable" valign=center
!rowspan=2|Operation
!rowspan=2 colspan=3|Schläfli<br>Symbol
!rowspan=2 colspan=2|[[Coxeter diagram|Coxeter<br>diagram]]
!rowspan=2|[[Wythoff construction|Wythoff<br>symbol]]
!colspan=7|Position: {{CDD|node_n0|p|node_n1|q|node_n2}}
|- BGCOLOR="#f0e0e0" align=center
! {{CDD|node_n0|p|node_n1|2|node_x}}|| {{CDD|node_n0|2|node_x|2|node_n2}}|| {{CDD|node_x|2|node_n1|q|node_n2}}|| {{CDD|node_x|2|node_x|2|node_n2}}|| {{CDD|node_x|2|node_n1|2|node_x}}|| {{CDD|node_n0|2|node_x|2|node_x}}
|- BGCOLOR="#f0e0e0" align=center
| '''Parent'''
| <math>\begin{Bmatrix} p , q \end{Bmatrix}</math>
| {p,q}
| t<sub>0</sub>{p,q}
|colspan=2|{{CDD|node_1|p|node|q|node}}
| q {{pipe}} 2 p|| {p}|| { }|| --|| --|| --|| { }
|-BGCOLOR="#e0e0f0" align=center
| Birectified<br>(or '''[[Dual polyhedron|dual]]''')
| <math>\begin{Bmatrix} q , p \end{Bmatrix}</math>
| {q,p}
| t<sub>2</sub>{p,q}
|colspan=2|{{CDD|node|p|node|q|node_1}}
|| p {{pipe}} 2 q|| --|| { }|| {q}|| { }|| --|| --
|- BGCOLOR="#f0e0e0" align=center
| [[Truncation (geometry)|Truncated]]
| <math>t\begin{Bmatrix} p , q \end{Bmatrix}</math>
| t{p,q}
| t<sub>0,1</sub>{p,q}
|colspan=2|{{CDD|node_1|p|node_1|q|node}}
| 2 q {{pipe}} p|| {2p}|| { }|| {q}|| --|| { }|| { }
|- BGCOLOR="#e0e0f0" align=center
| [[Bitruncation (geometry)|Bitruncated]]<br>(or truncated dual)
| <math>t\begin{Bmatrix} q , p \end{Bmatrix}</math>
| t{q,p}
| t<sub>1,2</sub>{p,q}
|colspan=2|{{CDD|node|p|node_1|q|node_1}}
| 2 p {{pipe}} q|| {p}|| { }|| {2q}|| { }|| { }
| --
|- BGCOLOR="#e0f0e0" align=center
| '''[[Rectification (geometry)|Rectified]]'''
|<math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
| r{p,q}
| t<sub>1</sub>{p,q}
|{{CDD|node_1|split1-pq|nodes}}||{{CDD|node|p|node_1|q|node}}
| 2 {{pipe}} p q|| {p}|| --|| {q}|| --|| { }|| --
|- BGCOLOR="#e0f0e0" align=center
|[[Cantellation (geometry)|Cantellated]]<br>(or [[Expansion (geometry)|expanded]])
|<math>r\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
| rr{p,q}
| t<sub>0,2</sub>{p,q}
|{{CDD|node|split1-pq|nodes_11}}||{{CDD|node_1|p|node|q|node_1}}
|| p q {{pipe}} 2|| {p}|| { }×{ }|| {q}|| { }|| --|| { }
|- BGCOLOR="#e0f0e0" align=center
| Cantitruncated<br>(or [[Omnitruncation|Omnitruncated]]) 
|<math>t\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
| tr{p,q}
| t<sub>0,1,2</sub>{p,q}
|{{CDD|node_1|split1-pq|nodes_11}}||{{CDD|node_1|p|node_1|q|node_1}}
|| 2 p q {{pipe}}|| {2p}|| { }×{}|| {2q}|| { }|| { }|| { }
|}

{|class="wikitable" valign=center
!rowspan=2|Operation
!rowspan=2 colspan=3|Schläfli<br>Symbol
!rowspan=2 colspan=2|[[Coxeter diagram|Coxeter<br>diagram]]
!rowspan=2|[[Wythoff construction|Wythoff<br>symbol]]
!colspan=7|Position: {{CDD|node_n0|p|node_n1|q|node_n2}}
|- BGCOLOR="#f0e0e0" align=center
! {{CDD|node_n0|p|node_n1|2|node_x}}|| {{CDD|node_n0|2|node_x|2|node_n2}}|| {{CDD|node_x|2|node_n1|q|node_n2}}|| {{CDD|node_x|2|node_x|2|node_n2}}|| {{CDD|node_x|2|node_n1|2|node_x}}|| {{CDD|node_n0|2|node_x|2|node_x}}
|- BGCOLOR="#e0f0e0" align=center
|[[Snub (geometry)|Snub rectified]]
|<math>s\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
|colspan=2|sr{p,q}
||{{CDD|node_h|split1-pq|nodes_hh}}||{{CDD|node_h|p|node_h|q|node_h}}
|| {{pipe}} 2 p q|| {p}|| {3}<br>{3}|| {q}|| --|| --|| --
|- BGCOLOR="#f0e0e0" align=center
| [[Snub (geometry)|Snub]]
| <math>s\begin{Bmatrix} p , q \end{Bmatrix}</math>
| s{p,2q}
| ht<sub>0,1</sub>{p,q}
|colspan=2|{{CDD|node_h|p|node_h|2x|q|node}}
| || s{2p}|| {3}|| {q}|| --|| {3}||
|}

{|
|[[File:Polyhedron truncation example3.png|320px]]
|[[File:Wythoffian construction diagram.svg|320px]]<br>Generating triangles
|}

=== Four dimensions ===
{{Main article|Uniform 4-polytope}}

In four dimensions, there are 6 [[convex regular 4-polytope]]s, 17 prisms on the Platonic and Archimedean solids (excluding the cube-prism, which has already been counted as the [[tesseract]]), and two infinite sets: the prisms on the convex antiprisms, and the [[duoprism]]s. There are also 41 convex semiregular 4-polytope, including the [[non-Wythoffian]] [[grand antiprism]] and the [[snub 24-cell]]. Both of these special 4-polytope are composed of subgroups of the vertices of the [[600-cell]].

The four-dimensional uniform star polytopes have not all been enumerated. The ones that have include the 10 regular star (Schläfli-Hess) 4-polytopes and 57 prisms on the uniform star polyhedra, as well as three infinite families: the prisms on the star antiprisms, the duoprisms formed by [[Cartesian product|multiplying]] two star polygons, and the duoprisms formed by multiplying an ordinary polygon with a star polygon. There is an unknown number of 4-polytope that do not fit into the above categories; over one thousand have been discovered so far.

[[File:Polychoron tetrahedral domain.png|240px|thumb|Example tetrahedron in [[cubic honeycomb]] cell.<br>''There are 3 right dihedral angles (2 intersecting perpendicular mirrors):<br>Edges 1 to 2, 0 to 2, and 1 to 3.'']]
[[File:Polychoron truncation chart.png|240px|thumb|Summary chart of truncation operations]]
Every regular polytope can be seen as the images of a [[fundamental region]] in a small number of mirrors.  In a 4-dimensional polytope (or 3-dimensional cubic honeycomb) the fundamental region is bounded by four mirrors.  A mirror in 4-space is a three-dimensional [[hyperplane]], but it is more convenient for our purposes to consider only its two-dimensional intersection with the three-dimensional surface of the [[hypersphere]]; thus the mirrors form an irregular [[tetrahedron]].

Each of the sixteen [[List of regular polytopes#Four-dimensional regular polytopes|regular 4-polytopes]] is generated by one of four symmetry groups, as follows:
* group [3,3,3]: the [[5-cell]] {3,3,3}, which is self-dual;
* group [3,3,4]: [[16-cell]] {3,3,4} and its dual [[tesseract]] {4,3,3};
* group [3,4,3]: the [[24-cell]] {3,4,3}, self-dual;
* group [3,3,5]: [[600-cell]] {3,3,5}, its dual [[120-cell]] {5,3,3}, and their ten regular stellations.
* group [3<sup>1,1,1</sup>]: contains only repeated members of the [3,3,4] family.

(The groups are named in [[Coxeter notation]].)

Eight of the [[convex uniform honeycomb]]s in Euclidean 3-space are analogously generated from the [[cubic honeycomb]] {4,3,4}, by applying the same operations used to generate the Wythoffian uniform 4-polytopes.

For a given symmetry simplex, a generating point may be placed on any of the four vertices, 6 edges, 4 faces, or the interior volume.  On each of these 15 elements there is a point whose images, reflected in the four mirrors, are the vertices of a uniform 4-polytope.

The extended Schläfli symbols are made by a '''t''' followed by inclusion of one to four subscripts 0,1,2,3.  If there is one subscript, the generating point is on a corner of the fundamental region, i.e. a point where three mirrors meet.  These corners are notated as
* '''0''': vertex of the parent 4-polytope (center of the dual's cell)
* '''1''': center of the parent's edge (center of the dual's face)
* '''2''': center of the parent's face (center of the dual's edge)
* '''3''': center of the parent's cell (vertex of the dual)

(For the two self-dual 4-polytopes, "dual" means a similar 4-polytope in dual position.)  Two or more subscripts mean that the generating point is between the corners indicated.

==== Constructive summary ====
The 15 constructive forms by family are summarized below. The self-dual families are listed in one column, and others as two columns with shared entries on the symmetric [[Coxeter diagram]]s. The final 10th row lists the snub 24-cell constructions. This includes all nonprismatic uniform 4-polytopes, except for the [[non-Wythoffian]] [[grand antiprism]], which has no Coxeter family.

{|class="wikitable" width=420
!A<sub>4</sub>
!colspan=2|BC<sub>4</sub>
!D<sub>4</sub>
!F<sub>4</sub>
!colspan=2|H<sub>4</sub>
|-
![3,3,3]<br>{{CDD|node|3|node|3|node|3|node}}
!colspan=2|[4,3,3]<br>{{CDD|node|4|node|3|node|3|node}}
![3,3<sup>1,1</sup>]<br>{{CDD|nodea|3a|branch|3a|nodea}}
![3,4,3]<br>{{CDD|node|3|node|4|node|3|node}}
!colspan=2|[5,3,3]<br>{{CDD|node|5|node|3|node|3|node}}
|- align=center valign=top
|[[5-cell]]<br>[[File:Schlegel wireframe 5-cell.png|60px]]<br>{{CDD|node_1|3|node|3|node|3|node}}<br>{3,3,3}
|[[16-cell]]<br>[[File:Schlegel wireframe 16-cell.png|60px]]<br>{{CDD|node|4|node|3|node|3|node_1}}<br>{3,3,4}
|[[tesseract]]<br>[[File:Schlegel wireframe 8-cell.png|60px]]<br>{{CDD|node_1|4|node|3|node|3|node}}<br>{4,3,3}
|[[demitesseract]]<br>[[File:Schlegel wireframe 16-cell.png|60px]]<br>{{CDD|nodea_1|3a|branch|3a|nodea}}<br>{3,3<sup>1,1</sup>}
|[[24-cell]]<br>[[File:Schlegel wireframe 24-cell.png|60px]]<br>{{CDD|node_1|3|node|4|node|3|node}}<br>{3,4,3}
![[600-cell]]<br>[[File:Schlegel wireframe 600-cell vertex-centered.png|60px]]<br>{{CDD|node|5|node|3|node|3|node_1}}<br>{3,3,5}
![[120-cell]]<br>[[File:Schlegel wireframe 120-cell.png|60px]]<br>{{CDD|node_1|5|node|3|node|3|node}}<br>{5,3,3}
|- align=center valign=top
|[[rectified 5-cell]]<br>[[File:Schlegel half-solid rectified 5-cell.png|60px]]<br>{{CDD|node|3|node_1|3|node|3|node}}<br>r{3,3,3}
|[[24-cell|rectified 16-cell]]<br>[[File:Schlegel half-solid rectified 16-cell.png|60px]]<br>{{CDD|node|4|node|3|node_1|3|node}}<br>r{3,3,4}
|[[rectified tesseract]]<br>[[File:Schlegel half-solid rectified 8-cell.png|60px]]<br>{{CDD|node|4|node_1|3|node|3|node}}<br>r{4,3,3}
|[[rectified demitesseract]]<br>[[File:Schlegel wireframe 24-cell.png|60px]]<br>{{CDD|nodea|3a|branch_10|3a|nodea}}<br>r{3,3<sup>1,1</sup>}
|[[rectified 24-cell]]<br>[[File:Schlegel half-solid cantellated 16-cell.png|60px]]<br>{{CDD|node|3|node_1|4|node|3|node}}<br>r{3,4,3}
![[rectified 600-cell]]<br>[[File:Rectified 600-cell schlegel halfsolid.png|60px]]<br>{{CDD|node|5|node|3|node_1|3|node}}<br>r{3,3,5}
![[rectified 120-cell]]<br>[[File:Rectified 120-cell schlegel halfsolid.png|60px]]<br>{{CDD|node|5|node_1|3|node|3|node}}<br>r{5,3,3}
|- align=center valign=top
|[[truncated 5-cell]]<br>[[File:Schlegel half-solid truncated pentachoron.png|60px]]<br>{{CDD|node_1|3|node_1|3|node|3|node}}<br>t{3,3,3}
|[[truncated 16-cell]]<br>[[File:Schlegel half-solid truncated 16-cell.png|60px]]<br>{{CDD|node|4|node|3|node_1|3|node_1}}<br>t{3,3,4}
|[[truncated tesseract]]<br>[[File:Schlegel half-solid truncated tesseract.png|60px]]<br>{{CDD|node_1|4|node_1|3|node|3|node}}<br>t{4,3,3}
|[[truncated demitesseract]]<br>[[File:Schlegel half-solid truncated 16-cell.png|60px]]<br>{{CDD|nodea_1|3a|branch_10|3a|nodea}}<br>t{3,3<sup>1,1</sup>}
|[[truncated 24-cell]]<br>[[File:Schlegel half-solid truncated 24-cell.png|60px]]<br>{{CDD|node_1|3|node_1|4|node|3|node}}<br>t{3,4,3}
![[truncated 600-cell]]<br>[[File:Schlegel half-solid truncated 600-cell.png|60px]]<br>{{CDD|node|5|node|3|node_1|3|node_1}}<br>t{3,3,5}
![[truncated 120-cell]]<br>[[File:Schlegel half-solid truncated 120-cell.png|60px]]<br>{{CDD|node_1|5|node_1|3|node|3|node}}<br>t{5,3,3}
|- align=center valign=top
|[[cantellated 5-cell]]<br>[[File:Schlegel half-solid cantellated 5-cell.png|60px]]<br>{{CDD|node_1|3|node|3|node_1|3|node}}<br>rr{3,3,3}
|[[rectified 24-cell|cantellated 16-cell]]<br>[[File:Schlegel half-solid cantellated 16-cell.png|60px]]<br>{{CDD|node|4|node_1|3|node|3|node_1}}<br>rr{3,3,4}
|[[cantellated tesseract]]<br>[[File:Schlegel half-solid cantellated 8-cell.png|60px]]<br>{{CDD|node_1|4|node|3|node_1|3|node}}<br>rr{4,3,3}
|[[cantellated demitesseract]]<br>[[File:Schlegel half-solid rectified 8-cell.png|60px]]<br>{{CDD|nodea_1|3a|branch|3a|nodea_1}}<br>2r{3,3<sup>1,1</sup>}
|[[cantellated 24-cell]]<br>[[File:Cantel 24cell1.png|60px]]<br>{{CDD|node_1|3|node|4|node_1|3|node}}<br>rr{3,4,3}
![[cantellated 600-cell]]<br>[[File:Cantellated 600 cell center.png|60px]]<br>{{CDD|node|5|node_1|3|node|3|node_1}}<br>rr{3,3,5}
![[cantellated 120-cell]]<br>[[File:Cantellated 120 cell center.png|51px]]<br>{{CDD|node_1|5|node|3|node_1|3|node}}<br>rr{5,3,3}
|- align=center valign=top
|[[runcinated 5-cell]]<br>[[File:Schlegel half-solid runcinated 5-cell.png|60px]]<br>{{CDD|node_1|3|node|3|node|3|node_1}}<br>t<sub>0,3</sub>{3,3,3}
|[[runcinated tesseract|runcinated 16-cell]]<br>[[File:Schlegel half-solid runcinated 16-cell.png|60px]]<br>{{CDD|node_1|4|node|3|node|3|node_1}}<br>t<sub>0,3</sub>{3,3,4}
|[[runcinated tesseract]]<br>[[File:Schlegel half-solid runcinated 8-cell.png|60px]]<br>{{CDD|node_1|4|node|3|node|3|node_1}}<br>t<sub>0,3</sub>{4,3,3}
|
|[[runcinated 24-cell]]<br>[[File:Runcinated 24-cell Schlegel halfsolid.png|60px]]<br>{{CDD|node_1|3|node|4|node|3|node_1}}<br>t<sub>0,3</sub>{3,4,3}
|colspan=2|[[runcinated 600-cell]]<br>[[runcinated 120-cell]]<br>[[File:runcinated 120-cell.png|60px]]<br>{{CDD|node_1|5|node|3|node|3|node_1}}<br>t<sub>0,3</sub>{3,3,5}
|- align=center valign=top
|[[bitruncated 5-cell]]<br>[[File:Schlegel half-solid bitruncated 5-cell.png|60px]]<br>{{CDD|node|3|node_1|3|node_1|3|node}}<br>t<sub>1,2</sub>{3,3,3}
|[[bitruncated tesseract|bitruncated 16-cell]]<br>[[File:Schlegel half-solid bitruncated 16-cell.png|60px]]<br>{{CDD|node|4|node_1|3|node_1|3|node}}<br>2t{3,3,4}
|[[bitruncated tesseract]]<br>[[File:Schlegel half-solid bitruncated 8-cell.png|60px]]<br>{{CDD|node|4|node_1|3|node_1|3|node}}<br>2t{4,3,3}
|[[cantitruncated demitesseract]]<br>[[File:Schlegel half-solid bitruncated 16-cell.png|60px]]<br>{{CDD|nodea_1|3a|branch_10|3a|nodea_1}}<br>2t{3,3<sup>1,1</sup>}
|[[bitruncated 24-cell]]<br>[[File:Bitruncated 24-cell Schlegel halfsolid.png|60px]]<br>{{CDD|node|3|node_1|4|node_1|3|node}}<br>2t{3,4,3}
|colspan=2|[[bitruncated 600-cell]]<br>[[bitruncated 120-cell]]<br>[[File:Bitruncated 120-cell schlegel halfsolid.png|60px]]<br>{{CDD|node|5|node_1|3|node_1|3|node}}<br>2t{3,3,5}
|- align=center valign=top
|[[cantitruncated 5-cell]]<br>[[File:Schlegel half-solid cantitruncated 5-cell.png|60px]]<br>{{CDD|node_1|3|node_1|3|node_1|3|node}}<br>tr{3,3,3}
|[[truncated 24-cell|cantitruncated 16-cell]]<br>[[File:Schlegel half-solid cantitruncated 16-cell.png|60px]]<br>{{CDD|node|4|node_1|3|node_1|3|node_1}}<br>tr{3,3,4}
|[[cantitruncated tesseract]]<br>[[File:Schlegel half-solid cantitruncated 8-cell.png|60px]]<br>{{CDD|node_1|4|node_1|3|node_1|3|node}}<br>tr{4,3,3}
|[[omnitruncated demitesseract]]<br>[[File:Schlegel half-solid truncated 24-cell.png|60px]]<br>{{CDD|nodea_1|3a|branch_11|3a|nodea_1}}<br>tr{3,3<sup>1,1</sup>}
|[[cantitruncated 24-cell]]<br>[[File:Cantitruncated 24-cell schlegel halfsolid.png|60px]]<br>{{CDD|node_1|3|node_1|4|node_1|3|node}}<br>tr{3,4,3}
![[cantitruncated 600-cell]]<br>[[File:Cantitruncated 600-cell.png|60px]]<br>{{CDD|node|5|node_1|3|node_1|3|node_1}}<br>tr{3,3,5}
![[cantitruncated 120-cell]]<br>[[File:Cantitruncated 120-cell.png|51px]]<br>{{CDD|node_1|5|node_1|3|node_1|3|node}}<br>tr{5,3,3}
|- align=center valign=top
|[[runcitruncated 5-cell]]<br>[[File:Schlegel half-solid runcitruncated 5-cell.png|60px]] <br>{{CDD|node_1|3|node_1|3|node|3|node_1}}<br>t<sub>0,1,3</sub>{3,3,3}
|[[runcitruncated 16-cell]]<br>[[File:Schlegel half-solid runcitruncated 16-cell.png|60px]]<br>{{CDD|node_1|4|node|3|node_1|3|node_1}}<br>t<sub>0,1,3</sub>{3,3,4}
|[[runcitruncated tesseract]]<br>[[File:Schlegel half-solid runcitruncated 8-cell.png|60px]]<br>{{CDD|node_1|4|node_1|3|node|3|node_1}}<br>t<sub>0,1,3</sub>{4,3,3}
|[[runcicantellated demitesseract]]<br>[[File:Schlegel half-solid cantellated 16-cell.png|60px]]<br>{{CDD|nodea_1|3a|branch_01lr|3a|nodea_1}}<br>rr{3,3<sup>1,1</sup>}
|[[runcitruncated 24-cell]]<br>[[File:Runcitruncated 24-cell.png|60px]]<br>{{CDD|node_1|3|node_1|4|node|3|node_1}}<br>t<sub>0,1,3</sub>{3,4,3}
![[runcitruncated 600-cell]]<br>[[File:Runcitruncated 600-cell.png|60px]]<br>{{CDD|node_1|5|node|3|node_1|3|node_1}}<br>t<sub>0,1,3</sub>{3,3,5}
![[runcitruncated 120-cell]]<br>[[File:Runcitruncated 120-cell.png|60px]]<br>{{CDD|node_1|5|node_1|3|node|3|node_1}}<br>t<sub>0,1,3</sub>{5,3,3}
|- align=center valign=top
|[[omnitruncated 5-cell]]<br>[[File:Schlegel half-solid omnitruncated 5-cell.png|60px]]<br>{{CDD|node_1|3|node_1|3|node_1|3|node_1}}<br>t<sub>0,1,2,3</sub>{3,3,3}
|[[omnitruncated tesseract|omnitruncated 16-cell]]<br>[[File:Schlegel half-solid omnitruncated 16-cell.png|60px]]<br>{{CDD|node_1|4|node_1|3|node_1|3|node_1}}<br>t<sub>0,1,2,3</sub>{3,3,4}
|[[omnitruncated tesseract]]<br>[[File:Schlegel half-solid omnitruncated 8-cell.png|60px]]<br>{{CDD|node_1|4|node_1|3|node_1|3|node_1}}<br>t<sub>0,1,2,3</sub>{3,3,4}
|
|[[omnitruncated 24-cell]]<br>[[File:Omnitruncated 24-cell.png|60px]]<br>{{CDD|node_1|3|node_1|4|node_1|3|node_1}}<br>t<sub>0,1,2,3</sub>{3,4,3}
|colspan=2|[[omnitruncated 120-cell]]<br>[[omnitruncated 600-cell]]<br>[[File:Omnitruncated 120-cell wireframe.png|60px]]<br>{{CDD|node_1|5|node_1|3|node_1|3|node_1}}<br>t<sub>0,1,2,3</sub>{5,3,3}
|- align=center valign=top
|
|[[snub 24-cell|alternated cantitruncated 16-cell]]<br>[[File:Schlegel half-solid alternated cantitruncated 16-cell.png|60px]]<br>{{CDD|node|4|node_h|3|node_h|3|node_h}}<br>sr{3,3,4}
|
|[[snub demitesseract]]<br>[[File:Ortho solid 969-uniform polychoron 343-snub.png|60px]]<br>{{CDD|nodea_h|3a|branch_hh|3a|nodea_h}}<br>sr{3,3<sup>1,1</sup>}
|[[snub 24-cell|Alternated truncated 24-cell]]<br>[[File:Ortho solid 969-uniform polychoron 343-snub.png|60px]]<br>{{CDD|node_h|3|node_h|4|node|3|node}}<br>s{3,4,3}
|
|
|}

==== Truncated forms ====
The following table defines 15 forms of truncation. Each form can have from one to four cell types, located in positions 0,1,2,3 as defined above. The cells are labeled by polyhedral truncation notation.
* An '''n'''-gonal prism is represented as : {n}×{ }.
* The green background is shown on forms that are equivalent to either the parent or the dual.
* The red background shows the truncations of the parent, and blue the truncations of the dual.

{|class="wikitable"
!rowspan=2| Operation
!rowspan=2 colspan=2|Schläfli symbol
!rowspan=2|[[Coxeter diagram|Coxeter<br>diagram]]
!colspan=4|Cells by position: {{CDD|node_n0|p|node_n1|q|node_n2|r|node_n3}}
|- BGCOLOR="#f0e0e0" align=center
!(3)<br>{{CDD|node_n0|p|node_n1|q|node_n2|2|node_x}}
!(2)<br>{{CDD|node_n0|p|node_n1|2|node_x|2|node_n3}}
!(1)<br>{{CDD|node_n0|2|node_x|2|node_n2|r|node_n3}}
!(0)<br>{{CDD|node_x|2|node_n1|q|node_n2|r|node_n3}}
|- BGCOLOR="#f0e0e0" align=center
| '''Parent'''
|{p,q,r}||t<sub>0</sub>{p,q,r}
|{{CDD|node_1|p|node|q|node|r|node}}
|{{CDD|node_1|p|node|q|node}}<br>{p,q}
|{{CDD|node_1|p|node|2|node}}<br>--
|{{CDD|node_1|2|node|r|node}}<br>--
|{{CDD|node|q|node|r|node}}<br>--
|- BGCOLOR="#f0e0e0" align=center
|'''[[Rectification (geometry)|Rectified]]'''
|r{p,q,r}||t<sub>1</sub>{p,q,r}
|{{CDD|node|p|node_1|q|node|r|node}}
|{{CDD|node|p|node_1|q|node}}<br>r{p,q}
|{{CDD|node|p|node_1|2|node}}<br>--
|{{CDD|node|2|node|r|node}}<br>--
|{{CDD|node_1|q|node|r|node}}<br>{q,r}
|- BGCOLOR="#e0e0f0" align=center
| Birectified<br>(or rectified dual)
|2r{p,q,r}<br>= r{r,q,p}||t<sub>2</sub>{p,q,r}
|{{CDD|node|p|node|q|node_1|r|node}}
|{{CDD|node|p|node|q|node_1}}<br>{q,p}
|{{CDD|node|p|node|2|node}}<br>--
|{{CDD|node|2|node_1|r|node}}<br>--
|{{CDD|node|q|node_1|r|node}}<br>r{q,r}
|- BGCOLOR="#e0e0f0" align=center
| Trirectifed<br>(or [[Dual polytope|dual]]) 
|3r{p,q,r}<br>= {r,q,p}||t<sub>3</sub>{p,q,r}
|{{CDD|node|p|node|q|node|r|node_1}}
|{{CDD|node|p|node|q|node}}<br>--
|{{CDD|node|p|node|2|node_1}}<br>--
|{{CDD|node|2|node|r|node_1}}<br>--
|{{CDD|node|q|node|r|node_1}}<br>{r,q}
|- BGCOLOR="#f0e0e0" align=center
|[[Truncation (geometry)|Truncated]]
|t{p,q,r}||t<sub>0,1</sub>{p,q,r}
|{{CDD|node_1|p|node_1|q|node|r|node}}
|{{CDD|node_1|p|node_1|q|node}}<br>t{p,q}
|{{CDD|node_1|p|node_1|2|node}}<br>--
|{{CDD|node_1|2|node|r|node}}<br>--
|{{CDD|node_1|q|node|r|node}}<br>{q,r}
|- BGCOLOR="#e0f0e0" align=center
|[[Bitruncation (geometry)|Bitruncated]]
|2t{p,q,r}||2t{p,q,r}
|{{CDD|node|p|node_1|q|node_1|r|node}}
|{{CDD|node|p|node_1|q|node_1}}<br>t{q,p}
|{{CDD|node|p|node_1|2|node}}<br>--
|{{CDD|node|2|node_1|r|node}}<br>--
|{{CDD|node_1|q|node_1|r|node}}<br>t{q,r}
|- BGCOLOR="#e0e0f0" align=center
| Tritruncated<br>(or truncated dual)
|3t{p,q,r}<br>= t{r,q,p}||t<sub>2,3</sub>{p,q,r}
|{{CDD|node|p|node|q|node_1|r|node_1}}
|{{CDD|node|p|node|q|node_1}}<br>{q,p}
|{{CDD|node|p|node|2|node_1}}<br>--
|{{CDD|node|2|node_1|r|node_1}}<br>--
|{{CDD|node|q|node_1|r|node_1}}<br>t{r,q}
|- BGCOLOR="#f0e0e0" align=center
|[[Cantellation (geometry)|Cantellated]]
|rr{p,q,r}||t<sub>0,2</sub>{p,q,r}
|{{CDD|node_1|p|node|q|node_1|r|node}}
|{{CDD|node_1|p|node|q|node_1}}<br>rr{p,q}
|{{CDD|node_1|p|node|2|node}}<br>--
|{{CDD|node_1|2|node_1|r|node}}<br>{ }×{r}
|{{CDD|node|q|node_1|r|node}}<br>r{q,r}
|- BGCOLOR="#e0e0f0" align=center
| Bicantellated<br>(or cantellated dual)
|r2r{p,q,r}<br>= rr{r,q,p}||t<sub>1,3</sub>{p,q,r}
|{{CDD|node|p|node_1|q|node|r|node_1}}
|{{CDD|node|p|node_1|q|node}}<br>r{p,q}
|{{CDD|node|p|node_1|2|node_1}}<br>{p}×{ }
|{{CDD|node|2|node|r|node_1}}<br>--
|{{CDD|node_1|q|node|r|node_1}}<br>rr{q,r}
|- BGCOLOR="#e0f0e0" align=center
|[[Runcination (geometry)|Runcinated]]<br>(or [[Expansion (geometry)|expanded]])
|e{p,q,r}||t<sub>0,3</sub>{p,q,r}
|{{CDD|node_1|p|node|q|node|r|node_1}}
|{{CDD|node_1|p|node|q|node}}<br>{p,q}
|{{CDD|node_1|p|node|2|node_1}}<br>{p}×{ }
|{{CDD|node_1|2|node|r|node_1}}<br>{ }×{r}
|{{CDD|node|q|node|r|node_1}}<br>{r,q}
|- BGCOLOR="#f0e0e0" align=center
| Cantitruncated
|tr{p,q,r}||tr{p,q,r}
|{{CDD|node_1|p|node_1|q|node_1|r|node}}
|{{CDD|node_1|p|node_1|q|node_1}}<br>tr{p,q}
|{{CDD|node_1|p|node_1|2|node}}<br>--
|{{CDD|node_1|2|node_1|r|node}}<br>{ }×{r}
|{{CDD|node_1|q|node_1|r|node}}<br>t{q,r}
|- BGCOLOR="#e0e0f0" align=center
| Bicantitruncated<br>(or cantitruncated dual)
|t2r{p,q,r}<br>= tr{r,q,p}||t<sub>1,2,3</sub>{p,q,r}
|{{CDD|node|p|node_1|q|node_1|r|node_1}}
|{{CDD|node|p|node_1|q|node_1}}<br>t{q,p}
|{{CDD|node|p|node_1|2|node_1}}<br>{p}×{ }
|{{CDD|node|2|node_1|r|node_1}}<br>--
|{{CDD|node_1|q|node_1|r|node_1}}<br>tr{q,r}
|- BGCOLOR="#f0e0e0" align=center
| Runcitruncated
|e<sub>t</sub>{p,q,r}||t<sub>0,1,3</sub>{p,q,r}
|{{CDD|node_1|p|node_1|q|node|r|node_1}}
|{{CDD|node_1|p|node_1|q|node}}<br>t{p,q}
|{{CDD|node_1|p|node_1|2|node_1}}<br>{2p}×{ }
|{{CDD|node_1|2|node|r|node_1}}<br>{ }×{r}
|{{CDD|node_1|q|node|r|node_1}}<br>rr{q,r}
|- BGCOLOR="#e0e0f0" align=center
| Runcicantellated<br>(or runcitruncated dual)
|e<sub>3t</sub>{p,q,r}<br>= e<sub>t</sub>{r,q,p}||t<sub>0,2,3</sub>{p,q,r}
|{{CDD|node_1|p|node|q|node_1|r|node_1}}
|{{CDD|node_1|p|node|q|node_1}}<br>tr{p,q}
|{{CDD|node_1|p|node|2|node_1}}<br>{p}×{ }
|{{CDD|node_1|2|node_1|r|node_1}}<br>{ }×{2r}
|{{CDD|node|q|node_1|r|node_1}}<br>t{r,q}
|- BGCOLOR="#e0f0e0" align=center
| Runcicantitruncated<br>(or [[Omnitruncation|omnitruncated]])
|o{p,q,r}||t<sub>0,1,2,3</sub>{p,q,r}
|{{CDD|node_1|p|node_1|q|node_1|r|node_1}}
|{{CDD|node_1|p|node_1|q|node_1}}<br>tr{p,q}
|{{CDD|node_1|p|node_1|2|node_1}}<br>{2p}×{ }
|{{CDD|node_1|2|node_1|r|node_1}}<br>{ }×{2r}
|{{CDD|node_1|q|node_1|r|node_1}}<br>tr{q,r}
|}

==== Half forms ====
Half constructions exist with ''holes'' rather than ringed nodes. Branches neighboring ''holes'' and inactive nodes must be even-order. Half construction have the vertices of an identically ringed construction.
{|class="wikitable"
!rowspan=2| Operation
!rowspan=2 colspan=2|Schläfli symbol
!rowspan=2|[[Coxeter diagram|Coxeter<br>diagram]]
!colspan=4|Cells by position: {{CDD|node_n0|p|node_n1|q|node_n2|r|node_n3}}
|- BGCOLOR="#f0e0e0" align=center
!(3)<br>{{CDD|node_n0|p|node_n1|q|node_n2|2|node_x}}
!(2)<br>{{CDD|node_n0|p|node_n1|2|node_x|2|node_n3}}
!(1)<br>{{CDD|node_n0|2|node_x|2|node_n2|r|node_n3}}
!(0)<br>{{CDD|node_x|2|node_n1|q|node_n2|r|node_n3}}
|- BGCOLOR="#f0e0e0" align=center
|Half<br>[[Alternation (geometry)|Alternated]]
|h{p,2q,r}||ht<sub>0</sub>{p,2q,r}
|{{CDD|node_h|2x|p|node|q|node|r|node}}
|{{CDD|node_h|2x|p|node|q|node}}<br>h{p,2q}
|{{CDD|node_h|2x|p|node|2|node}}<br>--
|{{CDD|node_h|2|node|r|node}}<br>--
|{{CDD|node|q|node|r|node}}<br>--
|- BGCOLOR="#f0e0e0" align=center
|Alternated rectified
|hr{2p,2q,r}||ht<sub>1</sub>{2p,2q,r}
|{{CDD|node|2x|p|node_h|2x|q|node|r|node}}
|{{CDD|node|2x|p|node_h|2x|q|node}}<br>hr{2p,2q}
|{{CDD|node|2x|p|node_h|2|node}}<br>--
|{{CDD|node|2|node_h|r|node}}<br>--
|{{CDD|node_h|q|node|r|node}}<br>h{2q,r}
|- BGCOLOR="#f0e0e0" align=center
|Snub<br>Alternated truncation
|s{p,2q,r}||ht<sub>0,1</sub>{p,2q,r}
|{{CDD|node_h|p|node_h|2x|q|node|r|node}}
|{{CDD|node_h|p|node_h|2x|q|node}}<br>s{p,2q}
|{{CDD|node_h|p|node_h|2|node}}<br>--
|{{CDD|node_h|2|node|r|node}}<br>--
|{{CDD|node_h|2x|q|node|r|node}}<br>h{2q,r}
|- BGCOLOR="#e0f0e0" align=center
|Bisnub<br>Alternated bitruncation
|2s{2p,q,2r}||ht<sub>1,2</sub>{2p,q,2r}
|{{CDD|node|2x|p|node_h|q|node_h|2x|r|node}}
|{{CDD|node|2x|p|node_h|q|node_h}}<br>s{q,2p}
|{{CDD|node|2x|p|node_h|2|node}}<br>--
|{{CDD|node|2|node_h|2x|r|node}}<br>--
|{{CDD|node_h|q|node_h|2x|r|node}}<br>s{q,2r}
|- BGCOLOR="#f0e0e0" align=center
|Snub rectified<br>Alternated truncated rectified
|sr{p,q,2r}||ht<sub>0,1,2</sub>{p,q,2r}
|{{CDD|node_h|p|node_h|q|node_h|2x|r|node}}
|{{CDD|node_h|p|node_h|q|node_h}}<br>sr{p,q}
|{{CDD|node_h|p|node_h|2|node}}<br>--
|{{CDD|node_h|2x|node_h|2x|r|node}}<br>s{2,2r}
|{{CDD|node_h|q|node_h|2x|r|node}}<br>s{q,2r}
|- BGCOLOR="#e0f0e0" align=center
|Omnisnub<br>Alternated omnitruncation
|os{p,q,r}||ht<sub>0,1,2,3</sub>{p,q,r}
|{{CDD|node_h|p|node_h|q|node_h|r|node_h}}
|{{CDD|node_h|p|node_h|q|node_h}}<br>sr{p,q}
|{{CDD|node_h|p|node_h|2x|node_h}}<br>{p}×{ }
|{{CDD|node_h|2x|node_h|r|node_h}}<br>{ }×{r}
|{{CDD|node_h|q|node_h|r|node_h}}<br>sr{q,r}
|}

=== Five and higher dimensions ===
In five and higher dimensions, there are 3 regular polytopes, the [[hypercube]], [[simplex]] and [[cross-polytope]]. They are generalisations of the three-dimensional cube, tetrahedron and octahedron, respectively. There are no regular star polytopes in these dimensions. Most uniform higher-dimensional polytopes are obtained by modifying the regular polytopes, or by taking the Cartesian product of polytopes of lower dimensions.

In six, seven and eight dimensions, the [[exceptional object|exceptional]] [[simple Lie group]]s, [[E6 (mathematics)|E<sub>6</sub>]], [[E7 (mathematics)|E<sub>7</sub>]] and [[E8 (mathematics)|E<sub>8</sub>]] come into play. By placing rings on a nonzero number of nodes of the [[Coxeter diagram]]s, one can obtain 39 new 6-polytopes, 127 new 7-polytopes and 255 new 8-polytopes. A notable example is the [[4 21 polytope|4<sub>21</sub> polytope]].

== Uniform honeycombs ==
Related to the subject of finite uniform polytopes are uniform honeycombs in Euclidean and hyperbolic spaces. Euclidean uniform honeycombs are generated by [[Coxeter diagram#Affine Coxeter groups|affine Coxeter groups]] and hyperbolic honeycombs are generated by the [[Coxeter diagram#Hyperbolic Coxeter groups|hyperbolic Coxeter groups]]. Two affine Coxeter groups can be multiplied together.

There are two classes of hyperbolic Coxeter groups, compact and paracompact. Uniform honeycombs generated by compact groups have finite facets and vertex figures, and exist in 2 through 4 dimensions. Paracompact groups have affine or hyperbolic subgraphs, and infinite facets or vertex figures, and exist in 2 through 10 dimensions.

== See also ==
* [[Schläfli symbol]]

== References ==
{{reflist}}

* [[Harold Scott MacDonald Coxeter|Coxeter]] ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, {{isbn|978-0-486-40919-1}} (Chapter 3: Wythoff's Construction for Uniform Polytopes)
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** Norman Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 [http://roice3.org/ruen/tup ONLINE]
* [[Alicia Boole Stott|A. Boole Stott]]: ''Geometrical deduction of semiregular from regular polytopes and space fillings'', Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910 [https://dwc.knaw.nl/DL/publications/PU00011492.pdf PDF]
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: 
** H.S.M. Coxeter, [[Michael S. Longuet-Higgins|M.S. Longuet-Higgins]] and J.C.P. Miller: ''Uniform Polyhedra'', Philosophical Transactions of the Royal Society of London, Londne, 1954 [https://www.bendwavy.org/polytile/NWJ/1954_uniform_polyhedra_paper.pdf PDF]
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 
* ''[http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html Kaleidoscopes: Selected Writings of H.S.M. Coxeter]'', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{isbn|978-0-471-01003-6}}
** (Paper 22) H.S.M. Coxeter, [https://doi.org/10.1007%2FBF01181449 Regular and Semi Regular Polytopes I], [Math. Zeit. 46 (1940) 380-407, MR 2,10]
** (Paper 23) H.S.M. Coxeter, [https://doi.org/10.1007%2FBF01161657 Regular and Semi-Regular Polytopes II], [Math. Zeit. 188 (1985) 559-591]
** (Paper 24) H.S.M. Coxeter, [https://doi.org/10.1007%2FBF01161745 Regular and Semi-Regular Polytopes III], [Math. Zeit. 200 (1988) 3-45]
* Marco Möller, ''Vierdimensionale Archimedische Polytope'', Dissertation, Universität Hamburg, Hamburg (2004) {{in lang|de}} [https://ediss.sub.uni-hamburg.de/handle/ediss/689 ONLINE]

== External links ==
* {{GlossaryForHyperspace | anchor=Uniform | title=Uniform polytope }}
* [http://www.polytope.de uniform, convex polytopes in four dimensions:], Marco Möller {{in lang|de}}

{{Polytopes}}

[[Category:Uniform polytopes]]