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[[Image:Wythoffian construction diagram.png|480px|thumb|Example Wythoff construction triangles with the 7 generator points. Lines to the active mirrors are colored red, yellow, and blue with the 3 nodes opposite them as associated by the Wythoff symbol.]]
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[[Image:Wythoff construction-pqr.png|400px|thumb|The eight forms for the Wythoff constructions from a general triangle (p q r).]]
 
In [[geometry]], the '''Wythoff symbol''' was first used by [[Coxeter]], Longuet-Higgins and Miller in their enumeration of the [[uniform polyhedra]]. It represents a construction by way of [[Wythoff construction|Wythoff's construction]] applied to [[Schwarz triangle]]s. 
 
A Schwarz triangle is a triangle that, with its own reflections in its edges, covers the sphere or the plane a finite number of times. The usual representation for the triangle is three numbers – integers or fractions – such that π/x is the angle at one vertex. For example, the triangle '''(2 3 4)''' represents the symmetry of a [[cube]], while '''(5/2 5/2 5/2)''' is the face of an [[icosahedron]].
 
Wythoff's construction in three dimensions consists of choosing a point in the triangle whose distance from each of the sides, if nonzero, is equal, and dropping perpendiculars to each of the edges.
 
Each edge of the triangle is named for the opposite angle; thus an edge opposite a right angle is designated '2'.  The symbol then corresponds to a representation of '''off | on'''. Each of the numbers ''p'' in the symbol becomes a polygon ''pn'', where n is the number of other edges that appear before the bar. So in '''3 | 4 2''' the vertex – a point, being here a degenerate polygon with 3×0 sides – lies on the π/3 corner of the triangle, and the altitude from that corner can be considered as forming half of the boundary between a [[square (geometry)|square]] (having 4×1 sides) and a [[digon]] (having 2×1 sides) of zero area.
 
The special case of the [[snub (geometry)|snub]] figures is done by using the symbol '''| p q r''', which would normally put the vertex at the centre of the sphere. The faces of a snub alternate as '''p 3 q 3 r 3'''. This gives an [[antiprism]] when q=r=2.
 
Each symbol represents one [[uniform polyhedron]] or tiling, although the same tiling/polyhedron can have different ''Wythoff symbols'' from different symmetry generators. For example, the regular [[cube]] can be represented by '''3 | 4 2''' with [[Octahedral symmetry|O<sub>h</sub> symmetry]], and '''2 4 | 2''' as a square [[Prism (geometry)|prism]] with 2 colors and [[Dihedral symmetry|D<sub>4h</sub> symmetry]], as well as '''2 2 2 |''' with 3 colors and D<sub>2h</sub> symmetry.
 
It can be applied with a slight extension to all uniform polyhedra, but the construction methods do not lead to all uniform tilings in euclidean or hyperbolic space.
 
== Summary table ==
There are seven generator points with each set of p,q,r (and a few special forms):
{| class="wikitable"
!colspan=4|General
!colspan=5|Right triangle (r=2)
|-
!Description
!Wythoff<BR>symbol
![[Vertex configuration|Vertex<BR>configuration]]
![[Coxeter-Dynkin diagram|Coxeter<BR>diagram]]<BR>{{CDD|pqr}}
!Wythoff<BR>symbol
!Vertex<BR>configuration
!colspan=2|[[Schläfli symbol|Schläfli<BR>symbol]]
!Coxeter<BR>diagram<BR>{{CDD|node|p|node|q|node}}
|- align=center
|rowspan=3|[[Regular polyhedron|regular]] and<BR>[[Quasiregular polyhedron|quasiregular]]
| q &#124; p r
| ''(p.r)<sup>q</sup>''
|{{CDD|3|node_1|p|node|q|node|r}}
| q &#124; p 2
| ''p<sup>q</sup>''
|colspan=2| {p,q}
|{{CDD|node_1|p|node|q|node}}
|- align=center
| p &#124; q r
| ''(q.r)<sup>p</sup>''
|{{CDD|3|node|p|node|q|node_1|r}}
| p &#124; q 2
| ''q''<sup>p</sup>
|colspan=2|{q,p}
|{{CDD|node|p|node|q|node_1}}
|- align=center
| r &#124; p q
|''(q.p)<sup>r</sup>''
|{{CDD|3|node|p|node_1|q|node|r}}
| 2 &#124; p q
|''(q.p)²
|r{p,q}||t<sub>1</sub>{p,q}
|{{CDD|node|p|node_1|q|node}}
|- align=center
|rowspan=3|[[Truncation (geometry)|truncated]] and<BR>[[Expansion (geometry)|expanded]]
| q r &#124; p
|''q.2p.r.2p''
|{{CDD|3|node_1|p|node_1|q|node|r}}
| q 2 &#124; p
|''q.2p.2p
|t{p,q}|| t<sub>0,1</sub>{p,q}
|{{CDD|node_1|p|node_1|q|node}}
|- align=center
| p r &#124; q
| ''p.2q.r.2q''
|{{CDD|3|node|p|node_1|q|node_1|r}}
| p 2 &#124; q
| ''p.&nbsp;2q.2q''
|t{q,p}|| t<sub>0,1</sub>{q,p}
|{{CDD|node|p|node_1|q|node_1}}
|- align=center
| p q &#124; r
|''2r.q.2r.p''
|{{CDD|3|node_1|p|node|q|node_1|r}}
| p q &#124; 2
|''4.q.4.p''
| rr{p,q}|| t<sub>0,2</sub>{p,q}
|{{CDD|node_1|p|node|q|node_1}}
|- align=center
|rowspan=2| [[Zonohedron|even-faced]]
| p q r &#124;
| ''2r.2q.2p''
|{{CDD|3|node_1|p|node_1|q|node_1|r}}
| p q 2 &#124;
| ''4.2q.2p''
| tr{p,q}||t<sub>0,1,2</sub>{p,q}
|{{CDD|node_1|p|node_1|q|node_1}}
|- align=center
| p q (r s) &#124;
| ''2p.2q.-2p.-2q''
| -
| p 2 (r s) &#124;
| ''2p.4.-2p.<sup>4</sup>/<sub>3</sub>''
|colspan=2|
| -
|- align=center
|rowspan=2| [[Snub (geometry)|snub]]
| &#124; p q r
| ''3.r.3.q.3.p''
|{{CDD|3|node_h|p|node_h|q|node_h|r}}
| &#124; p q 2
| ''3.3.q.3.p''
|colspan=2| sr{p,q}
|{{CDD|node_h|p|node_h|q|node_h}}
|- align=center
| &#124; p q r s
| ''(4.p.4.q.4.r.4.s)/2''
| -
| -
| -
|colspan=2|
| -
|}
 
There are three special cases:
* '''p q (r s) &#124;''' – This is a mixture of '''p q r &#124;''' and '''p q s &#124;'''.
* '''&#124; p q r''' – Snub forms (alternated) are give this otherwise unused symbol.
* '''&#124; p q r s''' – A unique snub form for [[Great dirhombicosidodecahedron|U75]] that isn't Wythoff-constructible.
 
== Description ==
 
The numbers ''p,q,r'' describe the fundamental triangle of the symmetry group: at its vertices, the generating mirrors meet in angles of π/''p'', π/''q'', π/''r''. On the sphere there are 3 main symmetry types: (3 3 2), (4 3 2), (5 3 2), and one infinite family (p 2 2), for any ''p''.  (All simple families have one right angle and so r=2.)
 
The position of the vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle. The generator point can either be on or off each mirror, activated or not. This distinction creates 8 (2³) possible forms, neglecting one where the generator point is on all the mirrors.
 
In this notation the mirrors are labeled by the reflection-order of the opposite vertex. The p,q,r values are listed '''before''' the bar if the corresponding mirror is active.
 
The one ''impossible'' symbol '''&#124; p q r''' implies the generator point is on all mirrors, which is only possible if the triangle is degenerate, reduced to a point. This unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, but odd-numbered reflected images are ignored. The resulting figure has rotational symmetry only.
 
This symbol is functionally similar to the more general [[Coxeter-Dynkin diagram]], in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. (An arc representing a right angle is omitted.)  A node is circled if the generator point is not on the mirror.
 
== Symmetry triangles ==
 
There are 4 symmetry classes of reflection on the [[sphere]], and two in the [[Euclidean plane]]. A few of the [[List of regular polytopes#Hyperbolic tilings|infinitely many]] such patterns in the [[Hyperbolic space|hyperbolic plane]] are also listed.  (Increasing any of the numbers defining a hyperbolic or Euclidean tiling makes another hyperbolic tiling.)
 
'''Point groups:'''
* (p 2 2) [[Dihedral symmetry in three dimensions|dihedral symmetry]], ''p'' = 2, 3, 4... (order 4''p'')
* (3 3 2) [[tetrahedral symmetry]] (order 24)
* (4 3 2) [[octahedral symmetry]] (order 48)
* (5 3 2) [[icosahedral symmetry]] (order 120)
'''Euclidean (affine) groups:'''
* (4 4 2) [[Square tiling|*442 symmetry]]: 45°-45°-90° triangle
* (6 3 2) *[[632 symmetry]]: 30°-60°-90° triangle
* (3 3 3) *[[triangular tiling|333 symmetry]] (60°-60°-60° plane)
'''Hyperbolic groups:'''
* (7 3 2) *[[732 symmetry]]
* (8 3 2) *[[832 symmetry]]
* (4 3 3) *[[433 symmetry]]
* (4 4 3) *[[443 symmetry]]
* (4 4 4) *[[444 symmetry]]
* (5 4 2) *[[542 symmetry]]
* (6 4 2) *[[642 symmetry]]
 
{| class="wikitable"
!colspan=5|Dihedral spherical
!colspan=3|Spherical
|-
!D<sub>2h</sub>
!D<sub>3h</sub>
!D<sub>4h</sub>
!D<sub>5h</sub>
!D<sub>6h</sub>
!T<sub>d</sub>
!O<sub>h</sub>
!I<sub>h</sub>
|-
!*222
!*322
!*422
!*522
!*622
!*332
!*432
!*532
|- align=center
|[[Image:Spherical square bipyramid2.png|100px]]<BR>(2 2 2)
|[[Image:Spherical hexagonal bipyramid2.png|100px]]<BR>(3 2 2)
|[[Image:Spherical octagonal bipyramid2.png|100px]]<BR>(4 2 2)
|[[Image:Spherical decagonal bipyramid2.png|100px]]<BR>(5 2 2)
|[[Image:Spherical dodecagonal bipyramid2.png|100px]]<BR>(6 2 2)
|[[Image:Tetrahedral reflection domains.png|100px]]<BR>(3 3 2)
|[[Image:Octahedral reflection domains.png|100px]]<BR>(4 3 2)
|[[Image:Icosahedral reflection domains.png|100px]]<BR>(5 3 2)
|}
 
The above symmetry groups only includes the integer solutions on the sphere. The list of [[Schwarz triangle]]s includes rational numbers, and determine the full set of solutions of [[nonconvex uniform polyhedron|nonconvex uniform polyhedra]].
 
{| class="wikitable"
|+ Euclidean plane
|-
!p4m
!p3m
!p6m
|-
!*442
!*333
!*632
|-
|[[Image:Tile V488 bicolor.svg|200px]]<BR>(4 4 2)
|[[Image:Tile 3,6.svg|200px]]<BR>(3 3 3)
|[[Image:Tile V46b.svg|200px]]<BR>(6 3 2)
|}
 
{| class="wikitable"
|+ Hyperbolic plane
|-
!*732
!*542
!*433
|-
|[[Image:Order-3 heptakis heptagonal tiling.png|200px]]<BR>(7 3 2)
|[[Image:Order-4 bisected pentagonal tiling.png|200px]]<BR>(5 4 2)
|[[Image:Uniform dual tiling 433-t012.png|200px]]<BR>(4 3 3)
|}
 
''In the tilings above, each triangle is a fundamental domain, colored by even and odd reflections.''
 
== Summary spherical, Euclidean and hyperbolic tilings ==
 
''Selected tilings created by the Wythoff construction are given below.''
 
=== Spherical tilings (''r'' = 2) ===
{| class="wikitable"
|-
!(p q 2)
!Parent
!Truncated
!Rectified
!Bitruncated
!Birectified<BR>(dual)
!Cantellated
!Omnitruncated<BR>(<small>Cantitruncated</small>)
!Snub
|-
!Wythoff<BR>symbol
! q &#124; p 2
! 2 q &#124; p
! 2 &#124; p q
! 2 p &#124; q
! p &#124; q 2
! p q &#124; 2
! p q 2 &#124;
! &#124; p q 2
|-
!rowspan=3|[[Schläfli symbol|Schläfli<BR>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>
|-
!{p,q}
!t{p,q}
!r{p,q}
!t{q,p}
!{q,p}
!rr{p,q}
!tr{p,q}
!rowspan=2|sr{p,q}
|-
!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}
|-
![[Coxeter diagram|Coxeter<BR>diagram]]
!{{CDD|node_1|p|node|q|node}}
!{{CDD|node_1|p|node_1|q|node}}
!{{CDD|node|p|node_1|q|node}}
!{{CDD|node|p|node_1|q|node_1}}
!{{CDD|node|p|node|q|node_1}}
!{{CDD|node_1|p|node|q|node_1}}
!{{CDD|node_1|p|node_1|q|node_1}}
!{{CDD|node_h|p|node_h|q|node_h}}
|-
![[Vertex configuration|Vertex figure]]
!p<sup>q</sup>
!q.2p.2p
!(p.q)<sup>2</sup>
!p.2q.2q
!q<sup>p</sup>
!p.4.q.4
!4.2p.2q
!3.3.p.3.q
|-
![[Image:Tetrahedral reflection domains.png|72px]]<BR>[[Tetrahedral symmetry|(3 3 2)]]
|[[Image:Uniform tiling 332-t0-1-.png|64px]]<BR>[[Tetrahedron|{3,3}]]
|[[File:Uniform tiling 332-t01-1-.png|64px]]<BR>[[Truncated tetrahedron|(3.6.6)]]
|[[Image:Uniform tiling 332-t1-1-.png|64px]]<BR>[[Octahedron|(3.3a.3.3a)]]
|[[Image:Uniform tiling 332-t12.png|64px]]<BR>[[Truncated tetrahedron|(3.6.6)]]
|[[Image:Uniform tiling 332-t2.png|64px]]<BR>[[Tetrahedron|{3,3}]]
| [[Image:Uniform tiling 332-t02.png|64px]]<BR>[[Cuboctahedron|(3a.4.3b.4)]]
|[[Image:Uniform tiling 332-t012.png|64px]]<BR>[[Truncated octahedron|(4.6a.6b)]]
|[[File:Spherical snub tetrahedron.png|64px]]<BR>[[Icosahedron|(3.3.3a.3.3b)]]
|-
![[Image:Octahedral reflection domains.png|73px]]<BR>[[Octahedral symmetry|(4 3 2)]]
|[[Image:Uniform tiling 432-t0.png|64px]]<BR>[[Cube|{4,3}]]
|[[Image:Uniform tiling 432-t01.png|64px]]<BR>[[Truncated cube|(3.8.8)]]
|[[Image:Uniform tiling 432-t1.png|64px]]<BR>[[Cuboctahedron|(3.4.3.4)]]
|[[Image:Uniform tiling 432-t12.png|64px]]<BR>[[Truncated octahedron|(4.6.6)]]
|[[Image:Uniform tiling 432-t2.png|64px]]<BR>[[Octahedron|{3,4}]]
|[[Image:Uniform tiling 432-t02.png|64px]]<BR>[[Rhombicuboctahedron|(3.4.4a.4)]]
|[[Image:Uniform tiling 432-t012.png|64px]]<BR>[[Truncated cuboctahedron|(4.6.8)]]
|[[File:Spherical snub cube.png|65px]]<BR>[[Snub cube|(3.3.3a.3.4)]]
|-
![[Image:Icosahedral reflection domains.png|72px]]<BR>[[Icosahedral symmetry|(5 3 2)]]
|[[Image:Uniform tiling 532-t0.png|64px]]<BR>[[Dodecahedron|{5,3}]]
|[[Image:Uniform tiling 532-t01.png|64px]]<BR>[[Truncated dodecahedron|(3.10.10)]]
|[[Image:Uniform tiling 532-t1.png|64px]]<BR>[[Icosidodecahedron|(3.5.3.5)]]
|[[Image:Uniform tiling 532-t12.png|64px]]<BR>[[Truncated icosahedron|(5.6.6)]]
|[[Image:Uniform tiling 532-t2.png|64px]]<BR>[[Icosahedron|{3,5}]]
|[[Image:Uniform tiling 532-t02.png|64px]]<BR>[[Rhombicosidodecahedron|(3.4.5.4)]]
|[[Image:Uniform tiling 532-t012.png|64px]]<BR>[[Truncated icosidodecahedron|(4.6.10)]]
|[[File:Spherical snub dodecahedron.png|64px]]<BR>[[Snub dodecahedron|(3.3.3a.3.5)]]
|}
==== Some overlapping spherical tilings (''r'' = 2) ====
:''For a more complete list, including cases where ''r'' ≠ 2, see [[List of uniform polyhedra by Schwarz triangle]].''
 
''Tilings are shown as [[polyhedron|polyhedra]].'' Some of the forms are degenerate, given with brackets for [[vertex figure]]s, with overlapping edges or verices.
{| class="wikitable"
|-
!(p q 2)
!Fund.<BR>triangle
!Parent
!Truncated
!Rectified
!Bitruncated
!Birectified<BR>(dual)
!Cantellated
!Omnitruncated<BR>(<small>Cantitruncated</small>)
!Snub
|-
![[Wythoff construction|Wythoff symbol]]
!
! q &#124; p 2
! 2 q &#124; p
! 2 &#124; p q
! 2 p &#124; q
! p &#124; q 2
! p q &#124; 2
! p q 2 &#124;
! &#124; p q 2
|-
!rowspan=3|[[Schläfli symbol]]
!rowspan=3|
!<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>
|-
!{p,q}
!t{p,q}
!r{p,q}
!t{q,p}
!{q,p}
!rr{p,q}
!tr{p,q}
!rowspan=2|sr{p,q}
|-
!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}
|-
![[Coxeter–Dynkin diagram]]
!
!{{CDD|node_1|p|node|q|node}}
!{{CDD|node_1|p|node_1|q|node}}
!{{CDD|node|p|node_1|q|node}}
!{{CDD|node|p|node_1|q|node_1}}
!{{CDD|node|p|node|q|node_1}}
!{{CDD|node_1|p|node|q|node_1}}
!{{CDD|node_1|p|node_1|q|node_1}}
!{{CDD|node_h|p|node_h|q|node_h}}
|-
![[Vertex configuration|Vertex figure]]
!
!p<sup>q</sup>
!(q.2p.2p)
!(p.q.p.q)
!(p.&nbsp;2q.2q)
!q<sup>p</sup>
!(p.&nbsp;4.q.4)
!(4.2p.2q)
!(3.3.p.&nbsp;3.q)
|-
|Icosahedral<BR>(5/2 3 2)
|&nbsp;
|[[Image:Great icosahedron.png|64px]]<BR>[[Great icosahedron|{3,5/2}]]
|[[Image:Great truncated icosahedron.png|64px]]<BR>[[Truncated great icosahedron|(5/2.6.6)]]
|[[Image:Great icosidodecahedron.png|64px]]<BR>[[Great icosidodecahedron|(3.5/2)<sup>2</sup>]]
|[[Image:Icosahedron.png|64px]]<BR>[[Small complex icosidodecahedron|[3.10/2.10/2]]]
|[[Image:Great stellated dodecahedron.png|64px]]<BR>[[Great stellated dodecahedron|{5/2,3}]]
|[[Image:Cantellated great icosahedron.png|64px]]<BR>[[Small complex rhombicosidodecahedron|[3.4.5/2.4]]]
|[[Image:Omnitruncated great icosahedron.png|64px]]<BR>[[Rhombicosahedron|[4.10/2.6]]]
|[[Image:Great snub icosidodecahedron.png|64px]]<BR>[[Great snub icosidodecahedron|(3.3.3.3.5/2)]]
|-
|Icosahedral<BR>(5 5/2 2)
|&nbsp;
|[[Image:Great dodecahedron.png|64px]]<BR>[[Great dodecahedron|{5,5/2}]]
|[[Image:Great truncated dodecahedron.png|64px]]<BR>[[Truncated great dodecahedron|(5/2.10.10)]]
|[[Image:Dodecadodecahedron.png|64px]]<BR>[[Dodecadodecahedron|(5/2.5)<sup>2</sup>]]
|[[Image:dodecahedron.png|64px]]<BR>[[dodecahedron|[5.10/2.10/2]]]
|[[Image:Small stellated dodecahedron.png|64px]]<BR>[[Small stellated dodecahedron|{5/2,5}]]
|[[Image:Cantellated great dodecahedron.png|64px]]<BR>[[Rhombidodecadodecahedron|(5/2.4.5.4)]]
|[[Image:Omnitruncated great dodecahedron.png|64px]]<BR>[[Small rhombidodecahedron|[4.10/2.10]]]
|[[Image:Snub dodecadodecahedron.png|64px]]<BR>[[Snub dodecadodecahedron|(3.3.5/2.3.5)]]
|}
 
==== Dihedral symmetry (''q'' = ''r'' = 2) ====
 
Spherical tilings with [[dihedral symmetry]] exist for all ''p''&nbsp;=&nbsp;2,&nbsp;3,&nbsp;4,&nbsp;... many with [[digon]] faces which become degenerate polyhedra. Two of the eight forms (Rectified and cantellated) are replications and are skipped in the table.
 
{| class="wikitable"
|-
!(p 2 2)<BR>Fundamental<BR>domain
!Parent
!Truncated
!Bitruncated<BR>(truncated dual)
!Birectified<BR>(dual)
!Omnitruncated<BR>(<small>Cantitruncated</small>)
!Snub
|-
!rowspan=3|Extended<BR>[[Schläfli symbol]]
!<math>\begin{Bmatrix} p , 2 \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p , 2 \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} 2 , p \end{Bmatrix}</math>
!<math>\begin{Bmatrix} 2 , p \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p \\ 2 \end{Bmatrix}</math>
!<math>s\begin{Bmatrix} p \\ 2 \end{Bmatrix}</math>
|-
!{p,2}
!t{p,2}
!t{2,p}
!{2,p}
!tr{p,2}
!rowspan=2|sr{p,2}
|-
!t<sub>0</sub>{p,2}
!t<sub>0,1</sub>{p,2}
!t<sub>1,2</sub>{p,2}
!t<sub>2</sub>{p,2}
!t<sub>0,1,2</sub>{p,2}
|-
!Wythoff symbol
! 2 &#124; p 2
! 2 2 &#124; p
! 2 p &#124; 2
! p &#124; 2 2
! p 2 2 &#124;
! &#124; p 2 2
|-
![[Coxeter–Dynkin diagram]]
!{{CDD|node_1|p|node|2|node}}
!{{CDD|node_1|p|node_1|2|node}}
!{{CDD|node|p|node_1|2|node_1}}
!{{CDD|node|p|node|2|node_1}}
!{{CDD|node_1|p|node_1|2|node_1}}
!{{CDD|node_h|p|node_h|2|node_h}}
|-
![[Vertex configuration|Vertex figure]]
!p²
!(2.2p.2p)
!(4.4.p)
!2<sup>p</sup>
!(4.2p.4)
!(3.3.p.&nbsp;3)
|- align=center
| [[Image:Spherical square bipyramid2.png|64px]]<BR>(2 2 2)<BR>[[Octahedron|V2.2.2]]
| [[Image:Sphere symmetry group cs.png|64px]]<BR>[[Dihedron|{2,2}]]
| [[Dihedron|2.4.4]]
| [[Dihedron|4.4.2]]
| [[Image:Sphere symmetry group cs.png|64px]]<BR>[[Hosohedron|{2,2}]]
| [[Image:Spherical square prism2.png|64px]]<BR>[[Cube|4.4.4]]
| [[Image:Spherical digonal antiprism.png|64px]]<BR>[[Tetrahedron|3.3.3.2]]
|- align=center
| [[Image:Spherical hexagonal bipyramid2.png|64px]]<BR>(3 2 2)<BR>[[Hexagonal bipyramid|V3.2.2]]
| [[Image:Trigonal dihedron.png|64px]]<BR>[[Dihedron|{3,2}]]
| [[Image:Hexagonal dihedron.png|64px]]<BR>[[Dihedron|2.6.6]]
| [[Image:Spherical triangular prism.png|64px]]<BR>[[Triangular prism|4.4.3]]
| [[Image:Triangular hosohedron.png|64px]]<BR>[[Hosohedron|{2,3}]]
| [[Image:Spherical hexagonal prism2.png|64px]]<BR>[[Hexagonal prism|4.4.6]]
| [[Image:Spherical trigonal antiprism.png|64px]]<BR>[[Octahedron|3.3.3.3]]
|- align=center
| [[Image:Spherical octagonal bipyramid2.png|64px]]<BR>(4 2 2)<BR>[[Octagonal bipyramid|V4.2.2]]
| [[Dihedron|{4,2}]]
| [[Dihedron|2.8.8]]
| [[Image:Spherical square prism.png|64px]]<BR>4.4.4
| [[Image:Spherical square hosohedron.png|64px]]<BR>[[Hosohedron|{2,4}]]
| [[Image:Spherical octagonal prism2.png|64px]]<BR>4.4.8
| [[Image:Spherical square antiprism.png|64px]]<BR>3.3.3.4
|- align=center
| [[Image:Spherical decagonal bipyramid2.png|64px]]<BR>(5 2 2)<BR>[[Decagonal bipyramid|V5.2.2]]
| [[Dihedron|{5,2}]]
| [[Dihedron|2.10.10]]
| [[Image:Spherical pentagonal prism.png|64px]]<BR>[[Pentagonal prism|4.4.5]]
| [[Image:Spherical pentagonal hosohedron.png|64px]]<BR>[[Hosohedron|{2,5}]]
| [[Image:Spherical decagonal prism2.png|64px]]<BR>[[Decagonal prism|4.4.10]]
| [[Image:Spherical pentagonal antiprism.png|64px]]<BR>[[Pentagonal antiprism|3.3.3.5]]
|- align=center
| [[Image:Spherical dodecagonal bipyramid2.png|64px]]<BR>(6 2 2)<BR>[[Dodecagonal bipyramid|V6.2.2]]
| [[Image:Hexagonal dihedron.png|64px]]<BR>[[Dihedron|{6,2}]]
| [[Dihedron|2.12.12]]
| [[Image:Spherical hexagonal prism.png|64px]]<BR>[[Hexagonal prism|4.4.6]]
| [[Image:Spherical hexagonal hosohedron.png|64px]]<BR>[[Hosohedron|{2,6}]]
| [[Image:Spherical dodecagonal prism2.png|64px]]<BR>[[Dodecagonal prism|4.4.12]]
| [[Image:Spherical hexagonal antiprism.png|64px]]<BR>[[Hexagonal antiprism|3.3.3.6]]
|-
|colspan=10|...
|}
 
=== Euclidean and hyperbolic tilings (''r'' = 2) ===
 
''Some representative hyperbolic tilings are given, and shown as a [[Poincaré disk model|Poincaré disk]] projection.''
 
{| class="wikitable"
|-
!(p q 2)
!Fund.<BR>triangles
!Parent
!Truncated
!Rectified
!Bitruncated
!Birectified<BR>(dual)
!Cantellated
!Omnitruncated<BR>(<small>Cantitruncated</small>)
!Snub
|-
![[Wythoff construction|Wythoff symbol]]
!
! q &#124; p 2
! 2 q &#124; p
! 2 &#124; p q
! 2 p &#124; q
! p &#124; q 2
! p q &#124; 2
! p q 2 &#124;
! &#124; p q 2
|-
!rowspan=3|[[Schläfli symbol]]
!rowspan=3|
!<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>
|-
!{p,q}
!t{p,q}
!r{p,q}
!t{q,p}
!{q,p}
!rr{p,q}
!tr{p,q}
!rowspan=2|sr{p,q}
|-
!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}
|-
![[Coxeter–Dynkin diagram]]
!
!{{CDD|node_1|p|node|q|node}}
!{{CDD|node_1|p|node_1|q|node}}
!{{CDD|node|p|node_1|q|node}}
!{{CDD|node|p|node_1|q|node_1}}
!{{CDD|node|p|node|q|node_1}}
!{{CDD|node_1|p|node|q|node_1}}
!{{CDD|node_1|p|node_1|q|node_1}}
!{{CDD|node_h|p|node_h|q|node_h}}
|-
![[Vertex configuration|Vertex figure]]
!
!p<sup>q</sup>
!(q.2p.2p)
!(p.q.p.q)
!(p.&nbsp;2q.2q)
!q<sup>p</sup>
!(p.&nbsp;4.q.4)
!(4.2p.2q)
!(3.3.p.&nbsp;3.q)
|- align=center
|[[Hexagonal tiling]]<BR>(6 3 2)
|[[Image:Tile V46b.svg|64px]] <br> [[Bisected hexagonal tiling|V4.6.12]]
|[[Image:Uniform tiling 63-t0.png|64px]]<BR>[[Hexagonal tiling|{6,3}]]
|[[Image:Uniform tiling 63-t01.png|64px]]<BR>[[Truncated hexagonal tiling|3.12.12]]
|[[Image:Uniform tiling 63-t1.png|64px]]<BR>[[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 63-t12.png|64px]]<BR>[[Hexagonal tiling|6.6.6]]
|[[Image:Uniform tiling 63-t2.png|64px]]<BR>[[Triangular tiling|{3,6}]]
|[[Image:Uniform tiling 63-t02.png|64px]]<BR>[[Rhombitrihexagonal tiling|3.4.6.4]]
|[[Image:Uniform tiling 63-t012.png|64px]]<BR>[[Truncated trihexagonal tiling|4.6.12]]
|[[Image:Uniform tiling 63-snub.png|65px]]<BR>[[Snub hexagonal tiling|3.3.3.3.6]]
|- align=center
|(Hyperbolic plane)<BR>(7 3 2)
|[[File:Hyperbolic_domains_732.png|72px]]<BR>[[Order 3-7 kisrhombille|V4.6.14]]
|[[Image:Uniform tiling 73-t0.png|64px]]<BR>[[Heptagonal tiling|{7,3}]]
|[[Image:Uniform tiling 73-t01.png|64px]]<BR>[[Truncated heptagonal tiling|3.14.14]]
|[[Image:Uniform tiling 73-t1.png|64px]]<BR>[[Triheptagonal tiling|3.7.3.7]]
|[[Image:Uniform tiling 73-t12.png|64px]]<BR>[[Truncated order-7 triangular tiling|7.6.6]]
|[[Image:Uniform tiling 73-t2.png|64px]]<BR>[[Order-7 triangular tiling|{3,7}]]
|[[Image:Uniform tiling 73-t02.png|64px]]<BR>[[Rhombitriheptagonal tiling|3.4.7.4]]
|[[Image:Uniform tiling 73-t012.png|64px]]<BR>[[Truncated triheptagonal tiling|4.6.14]]
|[[Image:Uniform tiling 73-snub.png|65px]]<BR>[[Snub heptagonal tiling|3.3.3.3.7]]
|- align=center
|(Hyperbolic plane)<BR>(8 3 2)
|[[File:Hyperbolic_domains_832.png|72px]]<BR>[[Order 3-8 kisrhombille|V4.6.16]]
|[[Image:Uniform tiling 83-t0.png|64px]]<BR>[[Octagonal tiling|{8,3}]]
|[[Image:Uniform tiling 83-t01.png|64px]]<BR>[[Truncated octagonal tiling|3.16.16]]
|[[Image:Uniform tiling 83-t1.png|64px]]<BR>[[Trioctagonal tiling|3.8.3.8]]
|[[Image:Uniform tiling 83-t12.png|64px]]<BR>[[Truncated order-8 triangular tiling|8.6.6]]
|[[Image:Uniform tiling 83-t2.png|64px]]<BR>[[Order-8 triangular tiling|{3,8}]]
|[[Image:Uniform tiling 83-t02.png|64px]]<BR>[[Rhombitrioctagonal tiling|3.4.8.4]]
|[[Image:Uniform tiling 83-t012.png|64px]]<BR>[[Truncated trioctagonal tiling|4.6.16]]
|[[Image:Uniform tiling 83-snub.png|65px]]<BR>[[Snub octagonal tiling|3.3.3.3.8]]
|- align=center
|[[Square tiling]]<BR>(4 4 2)
|[[Image:Tiling Dual Semiregular V4-8-8 Tetrakis Square-2-color-zoom.svg|64px]] <br> [[Tetrakis square tiling|V4.8.8]]
|[[Image:Uniform tiling 44-t0.png|64px]]<BR>[[Square tiling|{4,4}]]
|[[Image:Uniform tiling 44-t01.png|64px]]<BR>[[Truncated square tiling|4.8.8]]
|[[Image:Uniform tiling 44-t1.png|64px]]<BR>[[Square tiling|4.4a.4.4a]]
|[[Image:Uniform tiling 44-t12.png|64px]]<BR>[[Truncated square tiling|4.8.8]]
|[[Image:Uniform tiling 44-t2.png|64px]]<BR>[[Square tiling|{4,4}]]
|[[Image:Uniform tiling 44-t02.png|64px]]<BR>[[Square tiling|4.4a.4b.4a]]
|[[Image:Uniform tiling 44-t012.png|64px]]<BR>[[Truncated square tiling|4.8.8]]
|[[Image:Uniform tiling 44-snub.png|64px]]<BR>[[Snub square tiling|3.3.4a.3.4b]]
|- align=center
|(Hyperbolic plane)<BR>(5 4 2)
|[[File:Hyperbolic_domains_542.png|72px]]<BR>V4.8.10
|[[Image:Uniform tiling 54-t0.png|64px]]<BR>[[Order-4 pentagonal tiling|{5,4}]]
|[[Image:Uniform tiling 54-t01.png|64px]]<BR>[[Truncated order-4 pentagonal tiling|4.10.10]]
|[[Image:Uniform tiling 54-t1.png|64px]]<BR>[[Tetrapentagonal tiling|4.5.4.5]]
|[[Image:Uniform tiling 54-t12.png|64px]]<BR>[[Truncated order-5 square tiling|5.8.8]]
|[[Image:Uniform tiling 54-t2.png|64px]]<BR>[[Order-5 square tiling|{4,5}]]
|[[Image:Uniform tiling 54-t02.png|64px]]<BR>[[Rhombitetrapentagonal tiling|4.4.5.4]]
|[[Image:Uniform tiling 54-t012.png|64px]]<BR>[[Truncated tetrapentagonal tiling|4.8.10]]
|[[Image:Uniform tiling 54-snub.png|64px]]<BR>[[Snub tetrapentagonal tiling|3.3.4.3.5]]
|- align=center
|(Hyperbolic plane)<BR>(6 4 2)
|[[File:Hyperbolic_domains_642.png|72px]]<BR>V4.8.12
|[[Image:Uniform tiling 64-t0.png|64px]]<BR>[[Order-4 hexagonal tiling|{6,4}]]
|[[Image:Uniform tiling 64-t01.png|64px]]<BR>[[Truncated order-4 hexagonal tiling|4.12.12]]
|[[Image:Uniform tiling 64-t1.png|64px]]<BR>[[Tetrahexagonal tiling|4.6.4.6]]
|[[Image:Uniform tiling 64-t12.png|64px]]<BR>[[Truncated order-6 square tiling|6.8.8]]
|[[Image:Uniform tiling 64-t2.png|64px]]<BR>[[Order-6 square tiling|{4,6}]]
|[[Image:Uniform tiling 64-t02.png|64px]]<BR>[[Rhombitetrahexagonal tiling|4.4.6.4]]
|[[Image:Uniform tiling 64-t012.png|64px]]<BR>[[Truncated tetrahexagonal tiling|4.8.12]]
|[[Image:Uniform tiling 64-snub.png|64px]]<BR>[[Snub tetrahexagonal tiling|3.3.4.3.6]]
|- align=center
|(Hyperbolic plane)<BR>(7 4 2)
|[[File:Hyperbolic_domains_742.png|72px]]<BR>V4.8.14
|[[Image:Uniform tiling 74-t0.png|64px]]<BR>[[Order-4 heptagonal tiling|{7,4}]]
|[[Image:Uniform tiling 74-t01.png|64px]]<BR>[[Truncated order-4 heptagonal tiling|4.14.14]]
|[[Image:Uniform tiling 74-t1.png|64px]]<BR>[[Tetraheptagonal tiling|4.7.4.7]]
|[[Image:Uniform tiling 74-t12.png|64px]]<BR>[[Truncated order-7 square tiling|7.8.8]]
|[[Image:Uniform tiling 74-t2.png|64px]]<BR>[[Order-7 square tiling|{4,7}]]
|[[Image:Uniform tiling 74-t02.png|64px]]<BR>[[Rhombitetraheptagonal tiling|4.4.7.4]]
|[[Image:Uniform tiling 74-t012.png|64px]]<BR>[[Truncated tetraheptagonal tiling|4.8.14]]
|[[Image:Uniform tiling 74-snub.png|64px]]<BR>[[Snub tetraheptagonal tiling|3.3.4.3.7]]
|- align=center
|(Hyperbolic plane)<BR>(8 4 2)
|[[File:Hyperbolic_domains_842.png|72px]]<BR>V4.8.16
|[[Image:Uniform tiling 84-t0.png|64px]]<BR>[[Order-4 octagonal tiling|{8,4}]]
|[[Image:Uniform tiling 84-t01.png|64px]]<BR>[[Truncated order-4 octagonal tiling|4.16.16]]
|[[Image:Uniform tiling 84-t1.png|64px]]<BR>[[Tetraoctagonal tiling|4.8.4.8]]
|[[Image:Uniform tiling 84-t12.png|64px]]<BR>[[Octagonal tiling|8.8.8]]
|[[Image:Uniform tiling 84-t2.png|64px]]<BR>[[Order-8 square tiling|{4,8}]]
|[[Image:Uniform tiling 84-t02.png|64px]]<BR>[[Rhombitetraoctagonal tiling|4.4.8.4]]
|[[Image:Uniform tiling 84-t012.png|64px]]<BR>[[Truncated tetraoctagonal tiling|4.8.16]]
|[[Image:Uniform tiling 84-snub.png|64px]]<BR>[[Snub tetraoctagonal tiling|3.3.4.3.8]]
|- align=center
|(Hyperbolic plane)<BR>(5 5 2)
|[[File:Hyperbolic_domains_552.png|72px]]<BR>V4.10.10
|[[Image:Uniform tiling 552-t0.png|64px]]<BR>[[Order-5 pentagonal tiling|{5,5}]]
|[[Image:Uniform tiling 552-t01.png|64px]]<BR>[[Truncated order-5 pentagonal tiling|5.10.10]]
|[[Image:Uniform tiling 552-t1.png|64px]]<BR>[[Order-4 pentagonal tiling|5.5.5.5]]
|[[Image:Uniform tiling 552-t12.png|64px]]<BR>[[Truncated order-5 pentagonal tiling|5.10.10]]
|[[Image:Uniform tiling 552-t2.png|64px]]<BR>[[Order-5 pentagonal tiling|{5,5}]]
|[[Image:Uniform tiling 552-t02.png|64px]]<BR>[[Tetrapentagonal tiling|5.4.5.4]]
|[[Image:Uniform tiling 552-t012.png|64px]]<BR>[[Truncated order-4 pentagonal tiling|4.10.10]]
|[[Image:Uniform tiling 552-snub.png|64px]]<BR>[[Snub pentapentagonal tiling|3.3.5.3.5]]
|- align=center
|(Hyperbolic plane)<BR>(6 5 2)
|[[File:Hyperbolic_domains_652.png|72px]]<BR>V4.10.12
|[[File:H2 tiling 256-1.png|64px]]<BR>[[Order-5 hexagonal tiling|{6,5}]]
|[[File:H2 tiling 256-3.png|64px]]<BR>[[Truncated order-5 hexagonal tiling|5.12.12]]
|[[File:H2 tiling 256-2.png|64px]]<BR>[[Pentahexagonal tiling|5.6.5.6]]
|[[File:H2 tiling 256-6.png|64px]]<BR>[[Truncated order-6 pentagonal tiling|6.10.10]]
|[[File:H2 tiling 256-4.png|64px]]<BR>[[Order-6 pentagonal tiling|{5,6}]]
|[[File:H2 tiling 256-5.png|64px]]<BR>[[Rhombipentahexagonal tiling|5.4.6.4]]
|[[File:H2 tiling 256-7.png|64px]]<BR>[[Truncated pentahexagonal tiling|4.10.12]]
|[[File:Uniform tiling 65-snub.png|64px]]<BR>[[Snub pentahexagonal tiling|3.3.5.3.6]]
|- align=center
|(Hyperbolic plane)<BR>(7 5 2)
|[[File:Hyperbolic_domains_752.png|72px]]<BR>V4.10.14
|[[File:H2 tiling 257-1.png|64px]]<BR>[[Order-5 heptagonal tiling|{7,5}]]
|[[File:H2 tiling 257-3.png|64px]]<BR>[[Truncated order-5 heptagonal tiling|5.14.14]]
|[[File:H2 tiling 257-2.png|64px]]<BR>[[Pentaheptagonal tiling|5.7.5.7]]
|[[File:H2 tiling 257-6.png|64px]]<BR>[[Truncated order-7 pentagonal tiling|7.10.10]]
|[[File:H2 tiling 257-4.png|64px]]<BR>[[Order-7 pentagonal tiling|{5,7}]]
|[[File:H2 tiling 257-5.png|64px]]<BR>[[Rhombipentaheptagonal tiling|5.4.7.4]]
|[[File:H2 tiling 257-7.png|64px]]<BR>[[Truncated pentaheptagonal tiling|4.10.14]]
|[[File:Uniform tiling 75-snub.png|64px]]<BR>[[Snub pentaheptagonal tiling|3.3.5.3.7]]
|- align=center
|(Hyperbolic plane)<BR>(8 5 2)
|[[File:Hyperbolic_domains_852.png|72px]]<BR>V4.10.16
|[[File:H2 tiling 258-1.png|64px]]<BR>[[Order-5 octagonal tiling|{8,5}]]
|[[File:H2 tiling 258-3.png|64px]]<BR>[[Truncated order-5 octagonal tiling|5.16.16]]
|[[File:H2 tiling 258-2.png|64px]]<BR>[[Pentaoctagonal tiling|5.8.5.8]]
|[[File:H2 tiling 258-6.png|64px]]<BR>[[Truncated order-8 pentagonal tiling|8.10.10]]
|[[File:H2 tiling 258-4.png|64px]]<BR>[[Order-8 pentagonal tiling|{5,8}]]
|[[File:H2 tiling 258-5.png|64px]]<BR>[[Rhombipentaoctagonal tiling|5.4.8.4]]
|[[File:H2 tiling 258-7.png|64px]]<BR>[[Truncated pentaoctagonal tiling|4.10.16]]
|valign=bottom|[[Snub pentaoctagonal tiling|3.3.5.3.8]]
|- align=center
|(Hyperbolic plane)<BR>(6 6 2)
|[[File:Hyperbolic_domains_662.png|72px]]<BR>V4.12.12
|[[Image:Uniform tiling 66-t2.png|64px]]<BR>[[Order-6 hexagonal tiling|{6,6}]]
|[[Image:Uniform tiling 66-t12.png|64px]]<BR>[[Truncated order-6 hexagonal tiling|6.12.12]]
|[[Image:Uniform tiling 66-t1.png|64px]]<BR>[[Order-4 hexagonal tiling|6.6.6.6]]
|[[Image:Uniform tiling 66-t01.png|64px]]<BR>[[Truncated order-6 hexagonal tiling|6.12.12]]
|[[Image:Uniform tiling 66-t0.png|64px]]<BR>[[Order-6 hexagonal tiling|{6,6}]]
|[[Image:Uniform tiling 66-t02.png|64px]]<BR>[[Tetrahexagonal tiling|6.4.6.4]]
|[[Image:Uniform tiling 66-t012.png|64px]]<BR>[[Truncated order-4 hexagonal tiling|4.12.12]]
|[[Image:Uniform tiling 66-snub.png|64px]]<BR>[[Snub hexahexagonal tiling|3.3.6.3.6]]
|- align=center
|(Hyperbolic plane)<BR>(7 6 2)
|[[File:Hyperbolic_domains_762.png|72px]]<BR>V4.12.14
|[[File:H2 tiling 267-1.png|64px]]<BR>[[Order-6 heptagonal tiling|{7,6}]]
|[[File:H2 tiling 267-3.png|64px]]<BR>[[Truncated order-6 heptagonal tiling|6.14.14]]
|[[File:H2 tiling 267-2.png|64px]]<BR>[[Hexaheptagonal tiling|6.7.6.7]]
|[[File:H2 tiling 267-6.png|64px]]<BR>[[Truncated order-7 hexagonal tiling|7.12.12]]
|[[File:H2 tiling 267-4.png|64px]]<BR>[[Order-7 hexagonal tiling|{6,7}]]
|[[File:H2 tiling 267-5.png|64px]]<BR>[[Rhombihexaheptagonal tiling|6.4.7.4]]
|[[File:H2 tiling 267-7.png|64px]]<BR>[[Truncated hexaheptagonal tiling|4.12.14]]
|valign=bottom|[[Snub hexaheptagonal tiling|3.3.6.3.7]]
|- align=center
|(Hyperbolic plane)<BR>(8 6 2)
|[[File:Hyperbolic_domains_862.png|72px]]<BR>V4.12.16
|[[File:H2 tiling 268-1.png|64px]]<BR>[[Order-6 octagonal tiling|{8,6}]]
|[[File:H2 tiling 268-3.png|64px]]<BR>[[Truncated order-6 octagonal tiling|6.16.16]]
|[[File:H2 tiling 268-2.png|64px]]<BR>[[Hexaoctagonal tiling|6.8.6.8]]
|[[File:H2 tiling 268-6.png|64px]]<BR>[[Truncated order-8 hexagonal tiling|8.12.12]]
|[[File:H2 tiling 268-4.png|64px]]<BR>[[Order-8 hexagonal tiling|{6,8}]]
|[[File:H2 tiling 268-5.png|64px]]<BR>[[Rhombihexaoctagonal tiling|6.4.8.4]]
|[[File:H2 tiling 268-7.png|64px]]<BR>[[Truncated hexaoctagonal tiling|4.12.16]]
|[[File:Uniform tiling 86-snub.png|64px]]<BR>[[Snub hexaoctagonal tiling|3.3.6.3.8]]
|- align=center
|(Hyperbolic plane)<BR>(7 7 2)
|[[File:Hyperbolic_domains_772.png|72px]]<BR>V4.14.14
|[[Image:Uniform tiling 77-t2.png|64px]]<BR>[[Order-7 heptagonal tiling|{7,7}]]
|[[Image:Uniform tiling 77-t12.png|64px]]<BR>[[Truncated order-7 heptagonal tiling|7.14.14]]
|[[Image:Uniform tiling 77-t1.png|64px]]<BR>[[Order-4 heptagonal tiling|7.7.7.7]]
|[[Image:Uniform tiling 77-t01.png|64px]]<BR>[[Truncated order-7 heptagonal tiling|7.14.14]]
|[[Image:Uniform tiling 77-t0.png|64px]]<BR>[[Order-7 heptagonal tiling|{7,7}]]
|[[Image:Uniform tiling 77-t02.png|64px]]<BR>[[Tetraheptagonal tiling|7.4.7.4]]
|[[Image:Uniform tiling 77-t012.png|64px]]<BR>[[Truncated order-4 heptagonal tiling|4.14.14]]
|[[Image:Uniform tiling 77-snub.png|64px]]<BR>[[Snub heptaheptagonal tiling|3.3.7.3.7]]
|- align=center
|(Hyperbolic plane)<BR>(8 7 2)
|[[File:Hyperbolic_domains_872.png|72px]]<BR>V4.14.16
|[[File:H2 tiling 278-1.png|64px]]<BR>[[Order-7 octagonal tiling|{8,7}]]
|[[File:H2 tiling 278-3.png|64px]]<BR>[[Truncated order-7 octagonal tiling|7.16.16]]
|[[File:H2 tiling 278-2.png|64px]]<BR>[[Heptaoctagonal tiling|7.8.7.8]]
|[[File:H2 tiling 278-6.png|64px]]<BR>[[Truncated order-8 heptagonal tiling|8.14.14]]
|[[File:H2 tiling 278-4.png|64px]]<BR>[[Order-8 heptagonal tiling|{7,8}]]
|[[File:H2 tiling 278-5.png|64px]]<BR>[[Rhombiheptaoctagonal tiling|7.4.8.4]]
|[[File:H2 tiling 278-7.png|64px]]<BR>[[Truncated heptaoctagonal tiling|4.14.16]]
|valign=bottom|[[Snub heptaoctagonal tiling|3.3.7.3.8]]
|- align=center
|(Hyperbolic plane)<BR>(8 8 2)
|[[File:Hyperbolic_domains_882.png|72px]]<BR>V4.16.16
|[[Image:Uniform tiling 88-t2.png|64px]]<BR>[[Order-8 octagonal tiling|{8,8}]]
|[[Image:Uniform tiling 88-t12.png|64px]]<BR>[[Truncated order-8 octagonal tiling|8.16.16]]
|[[Image:Uniform tiling 88-t1.png|64px]]<BR>[[Order-4 octagonal tiling|8.8.8.8]]
|[[Image:Uniform tiling 88-t01.png|64px]]<BR>[[Truncated order-8 octagonal tiling|8.16.16]]
|[[Image:Uniform tiling 88-t0.png|64px]]<BR>[[Order-8 octagonal tiling|{8,8}]]
|[[Image:Uniform tiling 88-t02.png|64px]]<BR>[[Tetraoctagonal tiling|8.4.8.4]]
|[[Image:Uniform tiling 88-t012.png|64px]]<BR>[[Truncated order-4 octagonal tiling|4.16.16]]
|[[Image:Uniform tiling 88-snub.png|64px]]<BR>[[Snub octaoctagonal tiling|3.3.8.3.8]]
|- align=center
|(Hyperbolic plane)<BR>(&infin; 3 2)
|[[File:H2checkers_23i.png|72px]]<BR>V4.6.&infin;
|[[File:H2 tiling 23i-1.png|65px]]<BR>[[Order-3 apeirogonal tiling|{&infin;,3}]]
|[[File:H2 tiling 23i-3.png|65px]]<BR>[[Truncated order-3 apeirogonal tiling|3.&infin;.&infin;]]
|[[File:H2 tiling 23i-2.png|65px]]<BR>[[Triapeirogonal tiling|3.&infin;.3.&infin;]]
|[[File:H2 tiling 23i-6.png|65px]]<BR>[[Truncated infinite-order triangular tiling|&infin;.6.6]]
|[[File:H2 tiling 23i-4.png|65px]]<BR>[[Infinite-order triangular tiling|{3,&infin;}]]
|[[File:H2 tiling 23i-5.png|65px]]<BR>[[Rhombitriapeirogonal tiling|3.4.&infin;.4]]
|[[File:H2 tiling 23i-7.png|65px]]<BR>[[Truncated triapeirogonal tiling|4.6.&infin;]]
|[[File:Uniform tiling i32-snub.png|65px]]<BR>[[Snub triapeirogonal tiling|3.3.3.3.&infin;]]
|- align=center
|(Hyperbolic plane)<BR>(&infin; 4 2)
|[[File:H2checkers_24i.png|72px]]<BR>V4.8.&infin;
|[[File:H2 tiling 24i-1.png|65px]]<BR>[[Order-4 apeirogonal tiling|{&infin;,4}]]
|[[File:H2 tiling 24i-3.png|65px]]<BR>[[Truncated order-4 apeirogonal tiling|4.&infin;.&infin;]]
|[[File:H2 tiling 24i-2.png|65px]]<BR>[[Tetraapeirogonal tiling|4.&infin;.4.&infin;]]
|[[File:H2 tiling 24i-6.png|65px]]<BR>[[Truncated infinite-order square tiling|&infin;.8.8]]
|[[File:H2 tiling 24i-4.png|65px]]<BR>[[Infinite-order square tiling|{4,&infin;}]]
|[[File:H2 tiling 24i-5.png|65px]]<BR>[[Rhombitetraapeirogonal tiling|4.4.&infin;.4]]
|[[File:H2 tiling 24i-7.png|65px]]<BR>[[Truncated tetraapeirogonal tiling|4.8.&infin;]]
|[[File:Uniform tiling i42-snub.png|65px]]<BR>[[Snub tetraapeirogonal tiling|3.3.4.3.&infin;]]
|- align=center
|(Hyperbolic plane)<BR>(&infin; 5 2)
|[[File:H2checkers_25i.png|72px]]<BR>V4.10.&infin;
|[[File:H2 tiling 25i-1.png|65px]]<BR>[[Order-5 apeirogonal tiling|{&infin;,5}]]
|[[File:H2 tiling 25i-3.png|65px]]<BR>[[Truncated order-5 apeirogonal tiling|5.&infin;.&infin;]]
|[[File:H2 tiling 25i-2.png|65px]]<BR>[[Pentaapeirogonal tiling|5.&infin;.5.&infin;]]
|[[File:H2 tiling 25i-6.png|65px]]<BR>[[Truncated infinite-order pentagonal tiling|&infin;.10.10]]
|[[File:H2 tiling 25i-4.png|65px]]<BR>[[Infinite-order pentagonal tiling|{5,&infin;}]]
|[[File:H2 tiling 25i-5.png|65px]]<BR>[[Rhombipentaapeirogonal tiling|5.4.&infin;.4]]
|[[File:H2 tiling 25i-7.png|65px]]<BR>[[Truncated pentaapeirogonal tiling|4.10.&infin;]]
|[[File:Uniform tiling i52-snub.png|65px]]<BR>[[Snub pentaapeirogonal tiling|3.3.5.3.&infin;]]
|- align=center
|(Hyperbolic plane)<BR>(&infin; 6 2)
|[[File:H2checkers_26i.png|72px]]<BR>V4.12.&infin;
|[[File:H2 tiling 26i-1.png|65px]]<BR>[[Order-6 apeirogonal tiling|{&infin;,6}]]
|[[File:H2 tiling 26i-3.png|65px]]<BR>[[Truncated order-6 apeirogonal tiling|6.&infin;.&infin;]]
|[[File:H2 tiling 26i-2.png|65px]]<BR>[[Hexaapeirogonal tiling|6.&infin;.6.&infin;]]
|[[File:H2 tiling 26i-6.png|65px]]<BR>[[Truncated infinite-order hexagonal tiling|&infin;.12.12]]
|[[File:H2 tiling 26i-4.png|65px]]<BR>[[Infinite-order hexagonal tiling|{6,&infin;}]]
|[[File:H2 tiling 26i-5.png|65px]]<BR>[[Rhombihexaapeirogonal tiling|6.4.&infin;.4]]
|[[File:H2 tiling 26i-7.png|65px]]<BR>[[Truncated hexaapeirogonal tiling|4.12.&infin;]]
|[[File:Uniform tiling i62-snub.png|65px]]<BR>[[Snub hexaapeirogonal tiling|3.3.6.3.&infin;]]
|- align=center
|(Hyperbolic plane)<BR>(&infin; 7 2)
|[[File:H2checkers_27i.png|72px]]<BR>V4.14.&infin;
|[[File:H2 tiling 27i-1.png|65px]]<BR>[[Order-7 apeirogonal tiling|{&infin;,7}]]
|[[File:H2 tiling 27i-3.png|65px]]<BR>[[Truncated order-7 apeirogonal tiling|7.&infin;.&infin;]]
|[[File:H2 tiling 27i-2.png|65px]]<BR>[[Heptaapeirogonal tiling|7.&infin;.7.&infin;]]
|[[File:H2 tiling 27i-6.png|65px]]<BR>[[Truncated infinite-order heptagonal tiling|&infin;.14.14]]
|[[File:H2 tiling 27i-4.png|65px]]<BR>[[Infinite-order heptagonal tiling|{7,&infin;}]]
|[[File:H2 tiling 27i-5.png|65px]]<BR>[[Rhombiheptaapeirogonal tiling|7.4.&infin;.4]]
|[[File:H2 tiling 27i-7.png|65px]]<BR>[[Truncated heptaapeirogonal tiling|4.14.&infin;]]
|valign=bottom|[[Snub heptaapeirogonal tiling|3.3.7.3.&infin;]]
|- align=center
|(Hyperbolic plane)<BR>(&infin; 8 2)
|[[File:H2checkers_28i.png|72px]]<BR>V4.16.&infin;
|[[File:H2 tiling 28i-1.png|65px]]<BR>[[Order-8 apeirogonal tiling|{&infin;,8}]]
|[[File:H2 tiling 28i-3.png|65px]]<BR>[[Truncated order-8 apeirogonal tiling|8.&infin;.&infin;]]
|[[File:H2 tiling 28i-2.png|65px]]<BR>[[Octaapeirogonal tiling|8.&infin;.8.&infin;]]
|[[File:H2 tiling 28i-6.png|65px]]<BR>[[Truncated infinite-order octagonal tiling|&infin;.16.16]]
|[[File:H2 tiling 28i-4.png|65px]]<BR>[[Infinite-order octagonal tiling|{8,&infin;}]]
|[[File:H2 tiling 28i-5.png|65px]]<BR>[[Rhombioctaapeirogonal tiling|8.4.&infin;.4]]
|[[File:H2 tiling 28i-7.png|65px]]<BR>[[Truncated octaapeirogonal tiling|4.16.&infin;]]
|valign=bottom|[[Snub octaapeirogonal tiling|3.3.8.3.&infin;]]
|- align=center
|(Hyperbolic plane)<BR>(&infin; &infin; 2)
|[[File:H2checkers_2ii.png|72px]]<BR>V4.&infin;.&infin;
|[[File:H2 tiling 2ii-1.png|65px]]<br>[[Infinite-order apeirogonal tiling|{&infin;,&infin;}]]
|[[File:H2 tiling 2ii-3.png|65px]]<br>[[Order-3 apeirogonal tiling|&infin;.&infin;.&infin;]]
|[[File:H2 tiling 2ii-2.png|65px]]<BR>[[Order-4 apeirogonal tiling|&infin;.&infin;.&infin;.&infin;]]
|[[File:H2 tiling 2ii-6.png|65px]]<BR>[[Order-3 apeirogonal tiling|&infin;.&infin;.&infin;]]
|[[File:H2 tiling 2ii-4.png|65px]]<br>[[Infinite-order apeirogonal tiling|{&infin;,&infin;}]]
|[[File:H2 tiling 2ii-5.png|65px]]<BR>[[Tetraapeirogonal tiling|&infin;.4.&infin;.4]]
|[[File:H2 tiling 2ii-7.png|65px]]<br>[[Truncated order-4 apeirogonal tiling|4.&infin;.&infin;]]
|[[File:Uniform tiling ii2-snub.png|65px]]<BR>[[Snub apeiroapeirogonal tiling|3.3.&infin;.3.&infin;]]
|}
 
=== Euclidean and hyperbolic tilings (''r'' > 2) ===
 
The [[Coxeter–Dynkin diagram]] is given in a linear form, although it is actually a triangle, with the trailing segment r connecting to the first node.
 
{| class="wikitable"
![[Wythoff construction|Wythoff symbol]]<BR>(p q r)
!Fund.<BR>triangles
! q &#124; p r
! r q &#124; p
! r &#124; p q
! r p &#124; q
! p &#124; q r
! p q &#124; r
! p q r &#124;
! &#124; p q r
|-
!rowspan=2|[[Schläfli symbol]]
!rowspan=2|
!(p,q,r)
!r(r,q,p)
!(q,r,p)
!r(p,q,r)
!(q,p,r)
!r(p,r,q)
!tr(p,q,r)
!rowspan=2|s(p,q,r)
|-
!t<sub>0</sub>(p,q,r)
!t<sub>0,1</sub>(p,q,r)
!t<sub>1</sub>(p,q,r)
!t<sub>1,2</sub>(p,q,r)
!t<sub>2</sub>(p,q,r)
!t<sub>0,2</sub>(p,q,r)
!t<sub>0,1,2</sub>(p,q,r)
|-
![[Coxeter diagram]]
!
!{{CDD|3|node_1|p|node|q|node|r}}
!{{CDD|3|node_1|p|node_1|q|node|r}}
!{{CDD|3|node|p|node_1|q|node|r}}
!{{CDD|3|node|p|node_1|q|node_1|r}}
!{{CDD|3|node|p|node|q|node_1|r}}
!{{CDD|3|node_1|p|node|q|node_1|r}}
!{{CDD|3|node_1|p|node_1|q|node_1|r}}
!{{CDD|3|node_h|p|node_h|q|node_h|r}}
|-
![[Vertex configuration|Vertex figure]]
!
!(p.r)<sup>q</sup>
!(r.2p.q.2p)
!(p.q)<sup>r</sup>
!(q.2r.p.2r)
!(q.r)<sup>p</sup>
!(p.2r.q.2r)
!(2p.2q.2r)
!(3.r.3.q.3.p)
|- align=center
|Euclidean<BR>(3 3 3)<BR>{{CDD|branch|split2|node}}
|[[Image:Tile 3,6.svg|72px]]<BR>V6.6.6
|[[Image:Uniform tiling 333-t0.png|64px]]<BR>[[Triangular tiling|(3.3)<sup>3</sup>]]
|[[Image:Uniform tiling 333-t01.png|64px]]<BR>[[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 333-t1.png|64px]]<BR>[[Triangular tiling|(3.3)<sup>3</sup>]]
|[[Image:Uniform tiling 333-t12.png|64px]]<BR>[[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 333-t2.png|64px]]<BR>[[Triangular tiling|(3.3)<sup>3</sup>]]
|[[Image:Uniform tiling 333-t02.png|64px]]<BR>[[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 333-t012.png|64px]]<BR>[[Hexagonal tiling|6.6.6]]
|[[Image:Uniform tiling 333-snub.png|64px]]<BR>[[Triangular tiling|3.3.3.3.3.3]]
|- align=center
|Hyperbolic<BR>(4 3 3)<BR>{{CDD|label4|branch|split2|node}}
|[[Image:Hyperbolic_domains_433.png|72px]]<BR>V6.6.8
|[[Image:Uniform tiling 433-t0.png|64px]]<BR>[[Tritetragonal tiling|(3.4)<sup>3</sup>]]
|[[Image:Uniform tiling 433-t01.png|64px]]<BR>[[Trioctagonal tiling|3.8.3.8]]
|[[Image:Uniform tiling 433-t1.png|64px]]<BR>[[Tritetragonal tiling|(3.4)<sup>3</sup>]]
|[[Image:Uniform tiling 433-t12.png|64px]]<BR>[[Tritetratrigonal tiling|3.6.4.6]]
|[[Image:Uniform tiling 433-t2.png|64px]]<BR>[[Order-8 triangular tiling|(3.3)<sup>4</sup>]]
|[[Image:Uniform tiling 433-t02.png|64px]]<BR>[[Tritetratrigonal tiling|3.6.4.6]]
|[[Image:Uniform tiling 433-t012.png|64px]]<BR>[[Truncated order-8 triangular tiling|6.6.8]]
|[[Image:Uniform tiling 433-snub2.png|64px]]<BR>[[Snub tritetratrigonal tiling|3.3.3.3.3.4]]
|- align=center
|Hyperbolic<BR>(4 4 3)<BR>{{CDD|branch|split2-44|node}}
|[[Image:Hyperbolic_domains_443.png|72px]]<BR>V6.8.8
|[[Image:Uniform tiling 443-t0.png|64px]]<BR>[[Ditetragonal tritetragonal tiling|(3.4)<sup>4</sup>]]
|[[Image:Uniform tiling 443-t01.png|64px]]<BR>[[Tritetratetragonal tiling|3.8.4.8]]
|[[Image:Uniform tiling 443-t1.png|64px]]<BR>[[Order-6 square tiling|(4.4)<sup>3</sup>]]
|[[Image:Uniform tiling 443-t12.png|64px]]<BR>[[Tritetratetragonal tiling|3.8.4.8]]
|[[Image:Uniform tiling 443-t2.png|64px]]<BR>[[Ditetragonal tritetragonal tiling|(3.4)<sup>4</sup>]]
|[[Image:Uniform tiling 443-t02.png|64px]]<BR>[[Tetrahexagonal tiling|4.6.4.6]]
|[[Image:Uniform tiling 443-t012.png|64px]]<BR>[[Truncated order-6 square tiling|6.8.8]]
|[[Image:Uniform tiling 443-snub1.png|64px]]<BR>[[Snub tetratritetragonal tiling|3.3.3.4.3.4]]
|- align=center
|Hyperbolic<BR>(4 4 4)<BR>{{CDD|label4|branch|split2-44|node}}
|[[Image:Hyperbolic_domains_444.png|72px]]<BR>[[Order-8 triangular tiling|V8.8.8]]
|[[Image:Uniform tiling 444-t0.png|64px]]<BR>[[Order-8 square tiling|(4.4)<sup>4</sup>]]
|[[Image:Uniform tiling 444-t01.png|64px]]<BR>[[Tetraoctagonal tiling|4.8.4.8]]
|[[Image:Uniform tiling 444-t1.png|64px]]<BR>[[Order-8 square tiling|(4.4)<sup>4</sup>]]
|[[Image:Uniform tiling 444-t12.png|64px]]<BR>[[Tetraoctagonal tiling|4.8.4.8]]
|[[Image:Uniform tiling 444-t2.png|64px]]<BR>[[Order-8 square tiling|(4.4)<sup>4</sup>]]
|[[Image:Uniform tiling 444-t02.png|64px]]<BR>[[Tetraoctagonal tiling|4.8.4.8]]
|[[Image:Uniform tiling 444-t012.png|64px]]<BR>[[Octagonal tiling|8.8.8]]
|[[Image:Uniform tiling 444-snub.png|64px]]<BR>[[Tritetragonal tiling|3.4.3.4.3.4]]
|- align=center
|Hyperbolic<br>(5 3 3)<BR>{{CDD|label5|branch|split2|node}}
|[[File:Hyperbolic_domains_533.png|72px]]<br>V6.6.10
|[[File:H2 tiling 335-1.png|65px]]<br>[[Tripentagonal tiling|(3.5)<sup>3</sup>]]
|[[File:H2 tiling 335-3.png|65px]]<br>[[Tridecagonal tiling|3.10.3.10]]
|[[File:H2 tiling 335-2.png|65px]]<br>[[Tripentagonal tiling|(3.5)<sup>3</sup>]]
|[[File:H2 tiling 335-6.png|65px]]<br>[[Tripentatrigonal tiling|3.6.5.6]]
|[[File:H2 tiling 335-4.png|65px]]<br>[[Order-10 triangular tiling|(3.3)<sup>5</sup>]]
|[[File:H2 tiling 335-5.png|65px]]<br>[[Tripentatrigonal tiling|3.6.5.6]]
|[[File:H2 tiling 335-7.png|65px]]<br>[[Truncated order-10 triangular tiling|6.6.10]]
|valign=bottom|[[Snub tripentatrigonal tiling|3.3.3.3.3.5]]
|- align=center
|Hyperbolic<br>(5 4 3)<BR>{{CDD|label5|branch|split2-43|node}}
|[[File:Hyperbolic_domains_543.png|72px]]<br>V6.8.10
|[[File:H2 tiling 345-1.png|65px]]<br>[[Ditetragonal tripentagonal tiling|(3.5)<sup>4</sup>]]
|[[File:H2 tiling 345-3.png|65px]]<br>[[Tritetrapentagonal tiling|3.10.4.10]]
|[[File:H2 tiling 345-2.png|65px]]<br>[[Ditrigonal tetrapentagonal tiling|(4.5)<sup>3</sup>]]
|[[File:H2 tiling 345-6.png|65px]]<br>[[Tripentatetragonal tiling|3.8.5.8]]
|[[File:H2 tiling 345-4.png|65px]]<br>[[Dipentagonal tritetragonal tiling|(3.4)<sup>5</sup>]]
|[[File:H2 tiling 345-5.png|65px]]<br>[[Tetrapentatrigonal tiling|4.6.5.6]]
|[[File:H2 tiling 345-7.png|65px]]<br>[[Trigonally truncated tetrapentagonal tiling|6.8.10]]
|[[File:Uniform tiling 543-snub.png|65px]]<br>[[Snub tritetrapentagonal tiling|3.5.3.4.3.3]]
|- align=center
|Hyperbolic<BR>(5 4 4)<BR>{{CDD|label5|branch|split2-44|node}}
|[[Image:Hyperbolic_domains_544.png|72px]]<BR>V8.8.10
|[[Image:H2 tiling 445-1.png|64px]]<BR>[[Ditetragonal tetrapentagonal tiling|(4.5)<sup>4</sup>]]
|[[Image:H2 tiling 445-3.png|64px]]<BR>[[Tetradecagonal tiling|4.10.4.10]]
|[[Image:H2 tiling 445-2.png|64px]]<BR>[[Ditetragonal tetrapentagonal tiling|(4.5)<sup>4</sup>]]
|[[Image:H2 tiling 445-6.png|64px]]<BR>[[Tetrapentatetragonal tiling|4.8.5.8]]
|[[Image:H2 tiling 445-4.png|64px]]<BR>[[Order-10 square tiling|(4.4)<sup>5</sup>]]
|[[Image:H2 tiling 445-5.png|64px]]<BR>[[Tetrapentatetragonal tiling|4.8.5.8]]
|[[Image:H2 tiling 445-7.png|64px]]<BR>[[Truncated order-10 square tiling|8.8.10]]
|valign=bottom|[[Snub tetrapentatetragonal tiling|3.4.3.4.3.5]]
|- align=center
|Hyperbolic<br>(6 3 3)<BR>{{CDD|label6|branch|split2|node}}
|[[File:Hyperbolic_domains_633.png|72px]]<br>V6.6.12
|[[File:H2 tiling 336-1.png|65px]]<br>[[Ditrigonal trihexagonal tiling|(3.6)<sup>3</sup>]]
|[[File:H2 tiling 336-3.png|65px]]<br>[[Tridodecagonal tiling|3.12.3.12]]
|[[File:H2 tiling 336-2.png|65px]]<br>[[Ditrigonal trihexagonal tiling|(3.6)<sup>3</sup>]]
|[[File:H2 tiling 336-6.png|65px]]<br>[[Trihexatrigonal tiling|3.6.6.6]]
|[[File:H2 tiling 336-4.png|65px]]<br>[[Order-12 triangular tiling|(3.3)<sup>6</sup>]]
|[[File:H2 tiling 336-5.png|65px]]<br>[[Trihexatrigonal tiling|3.6.6.6]]
|[[File:H2 tiling 336-7.png|65px]]<br>[[Truncated order-12 triangular tiling|6.6.12]]
|valign=bottom|[[Snub trihexatrigonal tiling|3.3.3.3.3.6]]
|- align=center
|Hyperbolic<br>(6 4 3)<BR>{{CDD|label6|branch|split2-43|node}}
|[[File:Hyperbolic_domains_643.png|72px]]<br>V6.8.12
|[[File:H2 tiling 346-1.png|65px]]<br>[[Ditetragonal trihexagonal tiling|(3.6)<sup>4</sup>]]
|[[File:H2 tiling 346-3.png|65px]]<br>[[Tritetrahexagonal tiling|3.12.4.12]]
|[[File:H2 tiling 346-2.png|65px]]<br>[[Ditrigonal tetrahexagonal tiling|(4.6)<sup>3</sup>]]
|[[File:H2 tiling 346-6.png|65px]]<br>[[Trihexatetragonal tiling|3.8.6.8]]
|[[File:H2 tiling 346-4.png|65px]]<br>[[Dihexagonal tritetragonal tiling|(3.4)<sup>6</sup>]]
|[[File:H2 tiling 346-5.png|65px]]<br>[[Tetrahexatrigonal tiling|4.6.6.6]]
|[[File:H2 tiling 346-7.png|65px]]<br>[[Trigonally truncated tetrahexagonal tiling|6.8.12]]
|valign=bottom|[[Snub tritetrahexagonal tiling|3.6.3.4.3.3]]
|- align=center
|Hyperbolic<br>(6 4 4)<BR>{{CDD|label6|branch|split2-44|node}}
|[[File:Hyperbolic_domains_644.png|72px]]<br>V8.8.12
|[[File:H2 tiling 446-1.png|65px]]<br>[[Ditetragonal tetrahexagonal tiling|(4.6)<sup>4</sup>]]
|[[File:H2 tiling 446-3.png|65px]]<br>[[Tetradodecagonal tiling|4.12.4.12]]
|[[File:H2 tiling 446-2.png|65px]]<br>[[Ditetragonal tetrahexagonal tiling|(4.6)<sup>4</sup>]]
|[[File:H2 tiling 446-6.png|65px]]<br>[[Tetrahexatetragonal tiling|4.8.6.8]]
|[[File:H2 tiling 446-4.png|65px]]<br>[[Order-12 square tiling|(4.4)<sup>6</sup>]]
|[[File:H2 tiling 446-5.png|65px]]<br>[[Tetrahexatetragonal tiling|4.8.6.8]]
|[[File:H2 tiling 446-7.png|65px]]<br>[[Truncated order-12 square tiling|8.8.12]]
|valign=bottom|[[Snub tetrahexatetragonal tiling|3.6.3.4.3.4]]
|- align=center
|Hyperbolic<BR>(&infin; 3 3)<BR>{{CDD|labelinfin|branch|split2|node}}
|[[File:H2checkers_33i.png|72px]]<br>V6.6.&infin;
|[[File:H2 tiling 33i-1.png|65px]]<br>[[Ditrigonal triapeirogonal tiling|(3.&infin;)<sup>3</sup>]]
|[[File:H2 tiling 33i-3.png|65px]]<br>[[Triapeirogonal tiling|3.&infin;.3.&infin;]]
|[[File:H2 tiling 33i-2.png|65px]]<br>[[Ditrigonal triapeirogonal tiling|(3.&infin;)<sup>3</sup>]]
|[[File:H2 tiling 33i-6.png|65px]]<br>[[Triapeirotrigonal tiling|3.6.&infin;.6]]
|[[File:H2 tiling 33i-4.png|65px]]<br>[[Infinite-order triangular tiling|(3.3)<sup>&infin;</sup>]]
|[[File:H2 tiling 33i-5.png|65px]]<br>[[Triapeirotrigonal tiling|3.6.&infin;.6]]
|[[File:H2 tiling 33i-7.png|65px]]<br>[[Truncated infinite-order triangular tiling|6.6.&infin;]]
|valign=bottom|[[Snub triapeirotrigonal tiling|3.3.3.3.3.&infin;]]
|- align=center
|Hyperbolic<BR>(&infin; 4 3)<BR>{{CDD|labelinfin|branch|split2-43|node}}
|[[File:H2checkers_34i.png|72px]]<br>V6.8.&infin;
|[[File:H2 tiling 34i-1.png|65px]]<br>[[Ditetragonal triapeirogonal tiling|(3.&infin;)<sup>4</sup>]]
|[[File:H2 tiling 34i-3.png|65px]]<br>[[Tritetraapeirogonal tiling|3.&infin;.4.&infin;]]
|[[File:H2 tiling 34i-2.png|65px]]<br>[[Ditrigonal tetraapeirogonal tiling|(4.&infin;)<sup>3</sup>]]
|[[File:H2 tiling 34i-6.png|65px]]<br>[[Triapeirotetragonal tiling|3.8.&infin;.8]]
|[[File:H2 tiling 34i-4.png|65px]]<br>[[Diapeirogonal tritetragonal tiling|(3.4)<sup>&infin;</sup>]]
|[[File:H2 tiling 34i-5.png|65px]]<br>[[Tetraapeirotrigonal tiling|4.6.&infin;.6]]
|[[File:H2 tiling 34i-7.png|65px]]<br>[[Trigonally truncated tetraapeirogonal tiling|6.8.&infin;]]
|valign=bottom|[[Snub tritetraapeirogonal tiling|3.&infin;.3.4.3.3]]
|- align=center
|Hyperbolic<BR>(&infin; 4 4)<BR>{{CDD|labelinfin|branch|split2-44|node}}
|[[File:H2checkers_44i.png|72px]]<br>V8.8.&infin;
|[[File:H2 tiling 44i-1.png|65px]]<br>[[Ditetragonal tetraapeirogonal tiling|(4.&infin;)<sup>4</sup>]]
|[[File:H2 tiling 44i-3.png|65px]]<br>[[Tetraapeirogonal tiling|4.&infin;.4.&infin;]]
|[[File:H2 tiling 44i-2.png|65px]]<br>[[Ditetragonal tetraapeirogonal tiling|(4.&infin;)<sup>4</sup>]]
|[[File:H2 tiling 44i-6.png|65px]]<br>[[Tetraapeirotetragonal tiling|4.8.&infin;.8]]
|[[File:H2 tiling 44i-4.png|65px]]<br>[[Infinite-order square tiling|(4.4)<sup>&infin;</sup>]]
|[[File:H2 tiling 44i-5.png|65px]]<br>[[Tetraapeirotetragonal tiling|4.8.&infin;.8]]
|[[File:H2 tiling 44i-7.png|65px]]<br>[[Truncated infinite-order square tiling|8.8.&infin;]]
|valign=bottom|[[Snub tetraapeirotetragonal tiling|3.&infin;.3.4.3.4]]
|- align=center
|Hyperbolic<BR>(&infin; &infin; 3)<BR>{{CDD|branch|split2-ii|node}}
|[[File:H2checkers_3ii.png|72px]]<br>V6.&infin;.&infin;
|[[File:H2 tiling 3ii-1.png|65px]]<br>[[Diapeirogonal triapeirogonal tiling|(3.&infin;)<sup>&infin;</sup>]]
|[[File:H2 tiling 3ii-3.png|65px]]<br>[[Triapeiroapeirogonal tiling|3.&infin;.&infin;.&infin;]]
|[[File:H2 tiling 3ii-2.png|65px]]<br>[[Order-6 apeirogonal tiling|(&infin;.&infin;)<sup>3</sup>]]
|[[File:H2 tiling 3ii-6.png|65px]]<br>[[Triapeiroapeirogonal tiling|3.&infin;.&infin;.&infin;]]
|[[File:H2 tiling 3ii-4.png|65px]]<br>[[Diapeirogonal triapeirogonal tiling|(3.&infin;)<sup>&infin;</sup>]]
|[[File:H2 tiling 3ii-5.png|65px]]<br>[[Hexaapeirogonal tiling|&infin;.6.&infin;.6]]
|[[File:H2 tiling 3ii-7.png|65px]]<br>[[Truncated order-6 apeirogonal tiling|6.&infin;.&infin;]]
|valign=bottom|[[Snub apeirotriapeirogonal tiling|3.&infin;.3.&infin;.3.3]]
|- align=center
|Hyperbolic<BR>(&infin; &infin; 4)<BR>{{CDD|label4|branch|split2-ii|node}}
|[[File:H2checkers_4ii.png|72px]]<br>V8.&infin;.&infin;
|[[File:H2 tiling 4ii-1.png|65px]]<br>[[Diapeirogonal tetraapeirogonal tiling|(4.&infin;)<sup>&infin;</sup>]]
|[[File:H2 tiling 4ii-3.png|65px]]<br>[[Tetraapeiroapeirogonal tiling|4.&infin;.&infin;.&infin;]]
|[[File:H2 tiling 4ii-2.png|65px]]<br>[[Order-8 apeirogonal tiling|(&infin;.&infin;)<sup>4</sup>]]
|[[File:H2 tiling 4ii-6.png|65px]]<br>[[Tetraapeiroapeirogonal tiling|4.&infin;.&infin;.&infin;]]
|[[File:H2 tiling 4ii-4.png|65px]]<br>[[Diapeirogonal tetraapeirogonal tiling|(4.&infin;)<sup>&infin;</sup>]]
|[[File:H2 tiling 4ii-5.png|65px]]<br>[[Octaapeirogonal tiling|&infin;.8.&infin;.8]]
|[[File:H2 tiling 4ii-7.png|65px]]<br>[[Truncated order-8 apeirogonal tiling|8.&infin;.&infin;]]
|valign=bottom|[[Snub apeirotetraapeirogonal tiling|3.&infin;.3.&infin;.3.4]]
|- align=center
|Hyperbolic<BR>(&infin; &infin; &infin;)<BR>{{CDD|labelinfin|branch|split2-ii|node}}
|[[File:H2checkers_iii.png|72px]]<br>V&infin;.&infin;.&infin;
|[[File:H2 tiling iii-1.png|65px]]<br>[[Infinite-order apeirogonal tiling|(&infin;.&infin;)<sup>&infin;</sup>]]
|[[File:H2 tiling iii-3.png|65px]]<br>[[Order-4 apeirogonal tiling|&infin;.&infin;.&infin;.&infin;]]
|[[File:H2 tiling iii-2.png|65px]]<br>[[Infinite-order apeirogonal tiling|(&infin;.&infin;)<sup>&infin;</sup>]]
|[[File:H2 tiling iii-6.png|65px]]<br>[[Order-4 apeirogonal tiling|&infin;.&infin;.&infin;.&infin;]]
|[[File:H2 tiling iii-4.png|65px]]<br>[[Infinite-order apeirogonal tiling|(&infin;.&infin;)<sup>&infin;</sup>]]
|[[File:H2 tiling iii-5.png|65px]]<br>[[Order-4 apeirogonal tiling|&infin;.&infin;.&infin;.&infin;]]
|[[File:H2 tiling iii-7.png|65px]]<br>[[Order-3 apeirogonal tiling|&infin;.&infin;.&infin;]]
|[[File:Uniform_tiling_iii-snub.png|65px]]<br>[[Ditrigonal triapeirogonal tiling|3.&infin;.3.&infin;.3.&infin;]]
|}
 
== See also ==
*[[Regular polytope]]
*[[Regular polyhedron]]
*[[List of uniform tilings]]
*[[Uniform tilings in hyperbolic plane]]
*[[List of uniform polyhedra]]
*[[List of uniform polyhedra by Schwarz triangle]]
 
==References ==
* [[Harold Scott MacDonald Coxeter|Coxeter]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
* [[Harold Scott MacDonald Coxeter|Coxeter]] ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
* [[Coxeter]], Longuet-Higgins, Miller, ''Uniform polyhedra'', '''Phil. Trans.''' 1954, 246 A, 401–50.
* {{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger  | title=Polyhedron Models | publisher=Cambridge University Press | year=1974 | isbn=0-521-09859-9 }} pp.&nbsp;9–10.
 
==External links==
* {{MathWorld|title=Wythoff symbol|urlname=WythoffSymbol}}
*[http://www.mathconsult.ch/showroom/unipoly/wythoff.html The Wythoff symbol]
*[http://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=2788&langcode=en Wythoff symbol]
*[http://gregegan.customer.netspace.net.au/APPLETS/26/26.html Displays Uniform Polyhedra using Wythoff's construction method]
*[http://gregegan.customer.netspace.net.au/APPLETS/26/WythoffNotes.html Description of Wythoff Constructions]
*[http://geometrygames.org/KaleidoTile/index.html KaleidoTile 3] Free educational software for Windows by [[Jeffrey Weeks (mathematician)|Jeffrey Weeks]] that generated many of the images on the page.
* {{cite web | author = Hatch, Don | title = Hyperbolic Planar Tessellations | url = http://www.plunk.org/~hatch/HyperbolicTesselations }}
 
[[Category:Polyhedra]]
[[Category:Polytopes]]
[[Category:Mathematical notation]]

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