View source for Uniform tilings in hyperbolic plane

{{Short description|Symmetric subdivision in hyperbolic geometry}}
{| class=wikitable align=right width=300
|+ Examples of uniform tilings
|-
!Spherical
!Euclidean 
!colspan=4|Hyperbolic
|- align=center
|[[File:Uniform tiling 532-t0.png|100px]]<BR>[[Dodecahedron|{5,3}]]<BR>5.5.5<BR>{{CDD|node_1|5|node|3|node}}
|[[File:Uniform tiling 63-t0.png|100px]]<BR>[[Hexagonal tiling|{6,3}]]<BR>6.6.6<BR>{{CDD|node_1|6|node|3|node}}
|[[File:Heptagonal tiling.svg|100px]]<BR>[[Heptagonal tiling|{7,3}]]<BR>7.7.7<BR>{{CDD|node_1|7|node|3|node}}
|[[File:H2-I-3-dual.svg|100px]]<BR>[[Order-3 apeirogonal tiling|{∞,3}]]<BR>∞.∞.∞<BR>{{CDD|node_1|infin|node|3|node}}
|-
|colspan=4|''Regular tilings'' {p,q} of the sphere, Euclidean plane, and hyperbolic plane using regular pentagonal, hexagonal and heptagonal and apeirogonal faces.
|- align=center
|[[File:Uniform tiling 532-t01.png|100px]]<BR>[[Truncated dodecahedron|t{5,3}]]<BR>10.10.3<BR>{{CDD|node_1|5|node_1|3|node}}
|[[File:Uniform tiling 63-t01.png|100px]]<BR>[[Truncated hexagonal tiling|t{6,3}]]<BR>12.12.3<BR>{{CDD|node_1|6|node_1|3|node}}
|[[File:Truncated heptagonal tiling.svg|100px]]<BR>[[Truncated heptagonal tiling|t{7,3}]]<BR>14.14.3<BR>{{CDD|node_1|7|node_1|3|node}}
|[[File:H2 tiling 23i-3.png|100px]]<BR>t{∞,3}<BR>∞.∞.3<BR>{{CDD|node_1|infin|node_1|3|node}}
|-
|colspan=4|[[Truncation (geometry)|Truncated tilings]] have 2p.2p.q vertex figures from regular {p,q}.
|- align=center
|[[File:Uniform tiling 532-t1.png|100px]]<BR>[[Icosidodecahedron|r{5,3}]]<BR>3.5.3.5<BR>{{CDD|node|5|node_1|3|node}}
|[[File:Uniform tiling 63-t1.png|100px]]<BR>[[Trihexagonal tiling|r{6,3}]]<BR>3.6.3.6<BR>{{CDD|node|6|node_1|3|node}}
|[[File:Triheptagonal tiling.svg|100px]]<BR>[[Triheptagonal tiling|r{7,3}]]<BR>3.7.3.7<BR>{{CDD|node|7|node_1|3|node}}
|[[File:H2 tiling 23i-2.png|100px]]<BR>r{∞,3}<BR>3.∞.3.∞<BR>{{CDD|node|infin|node_1|3|node}}
|-
|colspan=4|''Quasiregular tilings'' are similar to regular tilings but alternate two types of regular polygon around each vertex.
|- align=center
|[[File:Uniform tiling 532-t02.png|100px]]<BR>[[Rhombicosidodecahedron|rr{5,3}]]<BR>3.4.5.4<BR>{{CDD|node_1|5|node|3|node_1}}
|[[File:Uniform tiling 63-t02.png|100px]]<BR>[[Rhombitrihexagonal tiling|rr{6,3}]]<BR>3.4.6.4<BR>{{CDD|node_1|6|node|3|node_1}}
|[[File:Rhombitriheptagonal tiling.svg|100px]]<BR>[[Rhombitriheptagonal tiling|rr{7,3}]]<BR>3.4.7.4<BR>{{CDD|node_1|7|node|3|node_1}}
|[[File:H2 tiling 23i-5.png|100px]]<BR>[[Rhombitriapeirogonal tiling|rr{∞,3}]]<BR>3.4.∞.4<BR>{{CDD|node_1|infin|node|3|node_1}}
|-
|colspan=4|''Semiregular tilings'' have more than one type of regular polygon.
|- align=center
|[[File:Uniform tiling 532-t012.png|100px]]<BR>[[Truncated icosidodecahedron|tr{5,3}]]<BR>4.6.10<BR>{{CDD|node_1|5|node_1|3|node_1}}
|[[File:Uniform tiling 63-t012.svg|100px]]<BR>[[Truncated trihexagonal tiling|tr{6,3}]]<BR>4.6.12<BR>{{CDD|node_1|6|node_1|3|node_1}}
|[[File:Truncated triheptagonal tiling.svg|100px]]<BR>[[Truncated triheptagonal tiling|tr{7,3}]]<BR>4.6.14<BR>{{CDD|node_1|7|node_1|3|node_1}}
|[[File:H2 tiling 23i-7.png|100px]]<BR>[[Truncated triapeirogonal tiling|tr{∞,3}]]<BR>4.6.∞<BR>{{CDD|node_1|infin|node_1|3|node_1}}
|-
|colspan=4|''Omnitruncated tilings'' have three or more even-sided regular polygons.
|}<div style="font-size:75%;display:inline-block;float:right;clear:right;">
{| class="wikitable collapsible collapsed" align=right
|+ Construction of Archimedean Solids and Tessellations
!colspan=2|Symmetry
!colspan=2|[[Dihedral symmetry in three dimensions|Triangular dihedral symmetry]]<br />[[File:Sphere symmetry group d3h.png|100px]]
![[Tetrahedral symmetry|Tetrahedral]]<br />[[File:Tetrahedral reflection domains.png|100px]]
!colspan=2|[[Octahedral symmetry|Octahedral]]<br />[[File:Octahedral reflection domains.png|100px]]
!colspan=2|[[Icosahedral symmetry|Icosahedral]]<br />[[File:Icosahedral reflection domains.png|100px]]
!colspan=2|p6m symmetry<br />[[File:Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg|100px]]
!colspan=2|[3,7] symmetry<br />[[File:H2checkers 237.png|100px]]
!colspan=2|[3,8] symmetry<br />[[File:H2checkers 238.png|100px]]
|-
! Starting solid<br />Operation||Symbol<br />{p,q}<br />{{CDD|node_1|p|node|q|node}}
! [[Hosohedron|Triangular hosohedron]]<br />{2,3}<br />[[File:Spherical trigonal hosohedron.svg|50px]] || [[Dihedron|Triangular dihedron]]<br />{3,2}<br />50px || [[Tetrahedron|Tetrahedron]]<br />{3,3}<br />50px || [[Cube|Cube]]<br />{4,3}<br />50px || [[Octahedron|Octahedron]]<br />{3,4}<br />50px || [[Dodecahedron|Dodecahedron]]<br />{5,3}<br />50px || [[Regular icosahedron|Icosahedron]]<br />{3,5}<br />50px || [[Hexagonal tiling|Hexagonal tiling]]<br />{6,3}<br />50px || [[Triangular tiling|Triangular tiling]]<br />{3,6}<br />50px || [[Heptagonal tiling|Heptagonal tiling]]<br />{7,3}<br />50px || [[Order-7 triangular tiling|Order-7 triangular tiling]]<br />{3,7}<br />50px || [[Octagonal tiling|Octagonal tiling]]<br />{8,3}<br />50px || [[Order-8 triangular tiling|Order-8 triangular tiling]]<br />{3,8}<br />[[File:H2 tiling 238-4.png|50px]]
|- align=center
! [[Truncation (geometry)|Truncation]] (t)||t{p,q}<br />{{CDD|node_1|p|node_1|q|node}}
| [[Triangular prism|triangular prism]]<br />[[File:Spherical triangular prism.svg|50px]] || truncated triangular dihedron<br />50px<span style="font-size:70%">(Half of the "edges" count as degenerate [[Digon|digon faces]]. The other half are normal edges.)</span> || [[Truncated tetrahedron|truncated tetrahedron]]<br />50px || [[Truncated cube|truncated cube]]<br />50px || [[Truncated octahedron|truncated octahedron]]<br />50px || [[Truncated dodecahedron|truncated dodecahedron]]<br />50px || [[Truncated icosahedron|truncated icosahedron]]<br />50px || [[Truncated hexagonal tiling|Truncated hexagonal tiling]]<br />50px || [[Hexagonal tiling|Truncated triangular tiling]]<br />50px || [[Truncated heptagonal tiling|Truncated heptagonal tiling]]<br />50px || [[Truncated order-7 triangular tiling|Truncated order-7 triangular tiling]]<br />50px  || [[Truncated octagonal tiling|Truncated octagonal tiling]]<br />50px || [[Truncated order-8 triangular tiling|Truncated order-8 triangular tiling]]<br />[[File:H2 tiling 238-6.png|50px]]
|- align=center
! [[Rectification (geometry)|Rectification]] (r)<br />Ambo (a)||r{p,q}<br />{{CDD|node|p|node_1|q|node}}
|colspan=2|  tridihedron<br />[[File:Trigonal_dihedron.svg|50px]]<span style="font-size:70%">(All of the "edges" count as degenerate [[Digon|digon faces]].)</span> ||tetratetrahedron<br />50px ||colspan=2| [[Cuboctahedron|cuboctahedron]]<br />50px ||colspan=2| [[Icosidodecahedron|icosidodecahedron]]<br />50px ||colspan=2| [[Trihexagonal tiling|Trihexagonal tiling]]<br />50px ||colspan=2| [[Triheptagonal tiling|Triheptagonal tiling]]<br />50px ||colspan=2| [[Trioctagonal tiling|Trioctagonal tiling]]<br />[[File:H2 tiling 238-2.png|50px]]
|- align=center
! [[Bitruncation]] (2t)<br />Dual kis (dk)||2t{p,q}<br />{{CDD|node|p|node_1|q|node_1}}
| truncated triangular dihedron<br />[[File:Hexagonal_dihedron.svg|50px]]<span style="font-size:70%">(Half of the "edges" count as degenerate [[Digon|digon faces]]. The other half are normal edges.)</span> || triangular prism<br />50px || truncated tetrahedron<br />50px || truncated octahedron<br />50px || truncated cube<br />50px || truncated icosahedron<br />50px || truncated dodecahedron<br />50px || truncated triangular tiling<br />50px || truncated hexagonal tiling<br />50px || Truncated order-7 triangular tiling<br />50px || Truncated heptagonal tiling<br />50px || Truncated order-8 triangular tiling<br />50px || Truncated octagonal tiling<br />[[File:H2 tiling 238-3.png|50px]]
|- align=center
! Birectification (2r)<br />[[Dual polyhedron|Dual]] (d)||2r{p,q}<br />{{CDD|node|p|node|q|node_1}}
| triangular dihedron<br />{3,2}<br />[[File:Trigonal dihedron.svg|50px]] || triangular hosohedron<br />{2,3}<br />50px || tetrahedron<br />50px|| octahedron<br />50px || cube<br />50px || icosahedron<br />50px || dodecahedron<br />50px || triangular tiling<br />50px || hexagonal tiling<br />50px || Order-7 triangular tiling<br />50px || Heptagonal tiling<br />50px || Order-8 triangular tiling<br />50px || Octagonal tiling<br />[[File:H2 tiling 238-1.png|50px]]
|- align=center
! Cantellation (rr)<br />[[Expansion (geometry)|Expansion]] (e)||rr{p,q}<br />{{CDD|node_1|p|node|q|node_1}}
| colspan=2|[[Triangular prism|triangular prism]]<br />[[File:Spherical triangular prism.svg|50px]]<span style="font-size:70%">(The "edge" between each pair of tetragons counts as a degenerate [[Digon|digon face]]. The other edges (the ones between a trigon and a tetragon) are normal edges.)</span> || rhombitetratetrahedron<br />50px ||colspan=2| [[Rhombicuboctahedron|rhombicuboctahedron]]<br />50px || colspan=2|[[Rhombicosidodecahedron|rhombicosidodecahedron]] <br />50px || colspan=2|[[Rhombitrihexagonal tiling|rhombitrihexagonal tiling]]<br />50px || colspan=2|[[Rhombitriheptagonal tiling|Rhombitriheptagonal tiling]] <br />50px || colspan=2|[[Rhombitrioctagonal tiling|Rhombitrioctagonal tiling]] <br />[[File:H2 tiling 238-5.png|50px]]
|- align=center
! Snub rectified (sr)<br />[[Snub (geometry)|Snub]] (s)||sr{p,q}<br />{{CDD|node_h|p|node_h|q|node_h}}
|colspan=2| triangular antiprism<br />[[File:Trigonal antiprism.png|50px]]<span style="font-size:70%">(Three yellow-yellow "edges", no two of which share any vertices, count as degenerate [[Digon|digon faces]]. The other edges are normal edges.)</span> || snub tetratetrahedron<br />50px ||colspan=2| snub cuboctahedron<br />50px ||colspan=2| snub icosidodecahedron<br />50px ||colspan=2| [[Snub trihexagonal tiling|snub trihexagonal tiling]]<br />50px ||colspan=2| [[Snub triheptagonal tiling|Snub triheptagonal tiling]]<br />50px ||colspan=2| [[Snub trioctagonal tiling|Snub trioctagonal tiling]]<br />[[File:Uniform tiling 83-snub.png|50px]]
|- align=center
! Cantitruncation (tr)<br />Bevel (b)||tr{p,q}<br />{{CDD|node_1|p|node_1|q|node_1}}
|colspan=2| [[Hexagonal prism|hexagonal prism]]<br />[[File:Truncated triangle prism.png|50px]] || truncated tetratetrahedron<br />50px ||colspan=2| [[Truncated cuboctahedron|truncated cuboctahedron]]<br />50px ||colspan=2| [[Truncated icosidodecahedron|truncated icosidodecahedron]]<br />50px ||colspan=2| [[Truncated trihexagonal tiling|truncated trihexagonal tiling]]<br />50px ||colspan=2| [[Truncated triheptagonal tiling|Truncated triheptagonal tiling]]<br />50px ||colspan=2| [[Truncated trioctagonal tiling|Truncated trioctagonal tiling]]<br />[[File:H2 tiling 238-7.png|50px]]
|}</div>
In [[Hyperbolic geometry|hyperbolic]] [[Geometry|geometry]], a '''uniform hyperbolic tiling''' (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has [[Regular polygon|regular polygon]]s as [[Face (geometry)|faces]] and is vertex-transitive (transitive on its [[Vertex (geometry)|vertices]], isogonal, i.e. there is an [[Isometry|isometry]] mapping any vertex onto any other). It follows that all vertices are [[Congruence (geometry)|congruent]], and the [[Tessellation|tiling]] has a high degree of rotational and translational [[Symmetry|symmetry]].

Uniform tilings can be identified by their [[Vertex configuration|vertex configuration]], a sequence of numbers representing the number of sides of the polygons around each vertex. For example, 7.7.7 represents the [[Heptagonal tiling|heptagonal tiling]] which has 3 [[Heptagon|heptagon]]s around each vertex. It is also regular since all the polygons are the same size, so it can also be given the [[Schläfli symbol]] {7,3}.

Uniform tilings may be [[Regular polyhedron|regular]] (if also face- and edge-transitive), quasi-regular (if edge-transitive but not face-transitive) or [[Semiregular polyhedron|semi-regular]] (if neither edge- nor face-transitive). For right triangles (''p''&nbsp;''q''&nbsp;2), there are two regular tilings, represented by [[Schläfli symbol]] {''p'',''q''} and {''q'',''p''}.

== Wythoff construction ==
[[Image:Wythoffian construction diagram.svg|400px|thumb|Example Wythoff construction with right triangles (''r''&nbsp;=&nbsp;2) and the 7 generator points. Lines to the active mirrors are colored red, yellow, and blue with the&nbsp;3&nbsp;nodes opposite them as associated by the Wythoff symbol.]]
There are an infinite number of uniform tilings based on the [[Schwarz triangle#Triangles for the hyperbolic plane|Schwarz triangle]]s (''p''&nbsp;''q''&nbsp;''r'') where {{sfrac|''p''}}&nbsp;+&nbsp;{{sfrac|''q''}}&nbsp;+&nbsp;{{sfrac|''r''}}&nbsp;<&nbsp;1, where ''p'', ''q'', ''r'' are each orders of reflection symmetry at three points of the fundamental domain triangle – the symmetry group is a hyperbolic [[Triangle group|triangle group]].

Each symmetry family contains 7 uniform tilings, defined by a [[Wythoff symbol]] or Coxeter-Dynkin diagram, 7 representing combinations of&nbsp;3&nbsp;active mirrors. An 8th represents an [[Alternation (geometry)|alternation]] operation, deleting alternate vertices from the highest form with all mirrors active.

Families with ''r''&nbsp;=&nbsp;2 contain regular hyperbolic tilings, defined by a Coxeter group such as [7,3], [8,3], [9,3], ... [5,4], [6,4], ....

Hyperbolic families with ''r''&nbsp;=&nbsp;3 or higher are given by (''p''&nbsp;''q''&nbsp;''r'') and include (4&nbsp;3&nbsp;3), (5&nbsp;3&nbsp;3), (6&nbsp;3&nbsp;3) ... (4&nbsp;4&nbsp;3), (5&nbsp;4&nbsp;3), ... (4&nbsp;4&nbsp;4)....

Hyperbolic triangles (''p''&nbsp;''q''&nbsp;''r'') define compact uniform hyperbolic tilings. In the limit any of ''p'', ''q'' or ''r'' can be replaced by ∞ which defines a paracompact hyperbolic triangle and creates uniform tilings with either infinite faces (called [[Apeirogon|apeirogon]]s) that converge to a single ideal point, or infinite vertex figure with infinitely many edges diverging from the same ideal point.

More symmetry families can be constructed from fundamental domains that are not triangles.

Selected families of uniform tilings are shown below (using the [[Poincaré disk model]] for the hyperbolic plane).  Three of them – (7&nbsp;3&nbsp;2), (5&nbsp;4&nbsp;2), and (4&nbsp;3&nbsp;3) – and no others, are ''minimal'' in the sense that if any of their defining numbers is replaced by a smaller integer the resulting pattern is either Euclidean or spherical rather than hyperbolic; conversely, any of the numbers can be increased (even to infinity) to generate other hyperbolic patterns.

Each uniform tiling generates a dual uniform tiling, with many of them also given below.

== Right triangle domains ==
There are infinitely many (''p''&nbsp;''q''&nbsp;2) [[Triangle group|triangle group]] families. This article shows the regular tiling up to ''p'', ''q''&nbsp;=&nbsp;8, and uniform tilings in 12 families: (7&nbsp;3&nbsp;2), (8&nbsp;3&nbsp;2), (5&nbsp;4&nbsp;2), (6&nbsp;4&nbsp;2), (7&nbsp;4&nbsp;2), (8&nbsp;4&nbsp;2), (5&nbsp;5&nbsp;2), (6&nbsp;5&nbsp;2) (6&nbsp;6&nbsp;2), (7&nbsp;7&nbsp;2), (8&nbsp;6&nbsp;2), and (8&nbsp;8&nbsp;2).

=== Regular hyperbolic tilings ===
The simplest set of hyperbolic tilings are regular tilings {''p'',''q''}, which exist in a matrix with the regular polyhedra and Euclidean tilings. The regular tiling {''p'',''q''} has a dual tiling {''q'',''p''} across the diagonal axis of the table. Self-dual tilings {2,2}, [[Tetrahedron|{3,3}]], [[Square tiling|{4,4}]], [[Order-5 pentagonal tiling|{5,5}]], etc. pass down the diagonal of the table.

=== (7&nbsp;3&nbsp;2) ===
The (7&nbsp;3&nbsp;2) [[Triangle group|triangle group]], Coxeter group [7,3], [[Orbifold notation#Hyperbolic plane|orbifold]] (*732) contains these uniform tilings:
{{Heptagonal tiling table}}

=== (8&nbsp;3&nbsp;2) ===
The (8&nbsp;3&nbsp;2) [[Triangle group|triangle group]], Coxeter group [8,3], [[Orbifold notation#Hyperbolic plane|orbifold]] (*832) contains these uniform tilings:
{{Octagonal tiling table}}

=== (5&nbsp;4&nbsp;2) ===
The (5&nbsp;4&nbsp;2) [[Triangle group|triangle group]], Coxeter group [5,4], [[Orbifold notation#Hyperbolic plane|orbifold]] (*542) contains these uniform tilings:

=== (6&nbsp;4&nbsp;2) ===
The (6&nbsp;4&nbsp;2) [[Triangle group|triangle group]], Coxeter group [6,4], [[Orbifold notation#Hyperbolic plane|orbifold]] (*642) contains these uniform tilings. Because all the elements are even, each uniform dual tiling one represents the fundamental domain of a reflective symmetry: *3333, *662, *3232, *443, *222222, *3222, and *642 respectively. As well, all 7 uniform tiling can be alternated, and those have duals as well.

=== (7&nbsp;4&nbsp;2) ===
The (7&nbsp;4&nbsp;2) [[Triangle group|triangle group]], Coxeter group [7,4], [[Orbifold notation#Hyperbolic plane|orbifold]] (*742) contains these uniform tilings:

=== (8&nbsp;4&nbsp;2) ===
The (8&nbsp;4&nbsp;2) [[Triangle group|triangle group]], Coxeter group [8,4], [[Orbifold notation#Hyperbolic plane|orbifold]] (*842) contains these uniform tilings. Because all the elements are even, each uniform dual tiling one represents the fundamental domain of a reflective symmetry: *4444, *882, *4242, *444, *22222222, *4222, and *842 respectively. As well, all 7 uniform tiling can be alternated, and those have duals as well.

=== (5&nbsp;5&nbsp;2) ===
The (5&nbsp;5&nbsp;2) [[Triangle group|triangle group]], Coxeter group [5,5], [[Orbifold notation#Hyperbolic plane|orbifold]] (*552) contains these uniform tilings:

=== (6&nbsp;5&nbsp;2) ===
The (6&nbsp;5&nbsp;2) [[Triangle group|triangle group]], Coxeter group [6,5], [[Orbifold notation#Hyperbolic plane|orbifold]] (*652) contains these uniform tilings:

=== (6 6 2) ===
The (6&nbsp;6&nbsp;2) [[Triangle group|triangle group]], Coxeter group [6,6], [[Orbifold notation#Hyperbolic plane|orbifold]] (*662) contains these uniform tilings:

=== (8 6 2) ===
The (8&nbsp;6&nbsp;2) [[Triangle group|triangle group]], Coxeter group [8,6], [[Orbifold notation#Hyperbolic plane|orbifold]] (*862) contains these uniform tilings.

=== (7 7 2) ===
The (7&nbsp;7&nbsp;2) [[Triangle group|triangle group]], Coxeter group [7,7], [[Orbifold notation#Hyperbolic plane|orbifold]] (*772) contains these uniform tilings:

=== (8 8 2) ===
The (8&nbsp;8&nbsp;2) [[Triangle group|triangle group]], Coxeter group [8,8], [[Orbifold notation#Hyperbolic plane|orbifold]] (*882) contains these uniform tilings:

== General triangle domains ==
There are infinitely many general [[Triangle group|triangle group]] families (''p''&nbsp;''q''&nbsp;''r''). This article shows uniform tilings in 9 families: (4&nbsp;3&nbsp;3), (4&nbsp;4&nbsp;3), (4&nbsp;4&nbsp;4), (5&nbsp;3&nbsp;3), (5&nbsp;4&nbsp;3), (5&nbsp;4&nbsp;4), (6&nbsp;3&nbsp;3), (6&nbsp;4&nbsp;3), and (6&nbsp;4&nbsp;4).

=== (4&nbsp;3&nbsp;3) ===
The (4&nbsp;3&nbsp;3) [[Triangle group|triangle group]], Coxeter group [(4,3,3)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*433) contains these uniform tilings. Without right angles in the fundamental triangle, the [[Wythoff construction]]s are slightly different. For instance in the (4,3,3) [[Triangle group|triangle family]], the ''snub'' form has six polygons around a vertex and its dual has hexagons rather than pentagons. In general the [[Vertex figure|vertex figure]] of a snub tiling in a triangle (''p'',''q'',''r'') is p.&nbsp;3.q.3.r.3, being 4.3.3.3.3.3 in this case below.


=== (4&nbsp;4&nbsp;3) ===
The (4&nbsp;4&nbsp;3) [[Triangle group|triangle group]], Coxeter group [(4,4,3)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*443) contains these uniform tilings. 

=== (4&nbsp;4&nbsp;4) ===
The (4&nbsp;4&nbsp;4) [[Triangle group|triangle group]], Coxeter group [(4,4,4)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*444) contains these uniform tilings. 

=== (5&nbsp;3&nbsp;3) ===
The (5&nbsp;3&nbsp;3) [[Triangle group|triangle group]], Coxeter group [(5,3,3)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*533) contains these uniform tilings.


=== (5&nbsp;4&nbsp;3) ===
The (5&nbsp;4&nbsp;3) [[Triangle group|triangle group]], Coxeter group [(5,4,3)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*543) contains these uniform tilings.


=== (5&nbsp;4&nbsp;4) ===
The (5&nbsp;4&nbsp;4) [[Triangle group|triangle group]], Coxeter group [(5,4,4)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*544) contains these uniform tilings.


=== (6&nbsp;3&nbsp;3) ===
The (6&nbsp;3&nbsp;3) [[Triangle group|triangle group]], Coxeter group [(6,3,3)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*633) contains these uniform tilings.


=== (6&nbsp;4&nbsp;3) ===
The (6&nbsp;4&nbsp;3) [[Triangle group|triangle group]], Coxeter group [(6,4,3)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*643) contains these uniform tilings.


=== (6&nbsp;4&nbsp;4) ===
The (6&nbsp;4&nbsp;4) [[Triangle group|triangle group]], Coxeter group [(6,4,4)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*644) contains these uniform tilings.


== Summary of tilings with finite triangular fundamental domains ==
For a table of all uniform hyperbolic tilings with fundamental domains (''p''&nbsp;''q''&nbsp;''r''), where 2 ≤ ''p'',''q'',''r'' ≤ 8.
:See Template:Finite triangular hyperbolic tilings table

== Quadrilateral domains==
[[File:Square_kaleidoscope_generators.png|320px|thumb|A quadrilateral domain has 9 generator point positions that define uniform tilings. Vertex figures are listed for general orbifold symmetry *''pqrs'', with 2-gonal faces degenerating into edges.]]

=== (3&nbsp;2&nbsp;2&nbsp;2) ===
[[File:Example 3222 hyperbolic uniform tilings kaleidoscopes.png|thumb|Example uniform tilings of *3222 symmetry]]
Quadrilateral fundamental domains also exist in the hyperbolic plane, with the *3222 [[Orbifold notation#Hyperbolic plane|orbifold]] ([∞,3,∞] Coxeter notation) as the smallest family. There are 9 generation locations for uniform tiling within quadrilateral domains. The vertex figure can be extracted from a fundamental domain as 3 cases (1) Corner (2) Mid-edge, and (3) Center. When generating points are corners adjacent to order-2 corners, degenerate {2} [[Digon|digon]] faces at those corners exist but can be ignored. [[Snub (geometry)|Snub]] and [[Alternation (geometry)|alternated]] uniform tilings can also be generated (not shown) if a vertex figure contains only even-sided faces.

Coxeter diagrams of quadrilateral domains are treated as a degenerate [[Tetrahedron|tetrahedron]] graph with 2 of 6 edges labeled as infinity, or as dotted lines. A logical requirement of at least one of two parallel mirrors being active limits the uniform cases to 9, and other ringed patterns are not valid.


=== (3&nbsp;2&nbsp;3&nbsp;2) ===

== Ideal triangle domains ==
There are infinitely many [[Triangle group|triangle group]] families including infinite orders. This article shows uniform tilings in 9 families:  (∞&nbsp;3&nbsp;2), (∞&nbsp;4&nbsp;2), (∞ ∞ 2), (∞&nbsp;3&nbsp;3), (∞&nbsp;4&nbsp;3), (∞&nbsp;4&nbsp;4), (∞ ∞ 3), (∞ ∞ 4), and (∞ ∞ ∞).

=== (∞&nbsp;3&nbsp;2) ===
The ideal (∞&nbsp;3&nbsp;2) [[Triangle group|triangle group]], Coxeter group [∞,3], [[Orbifold notation#Hyperbolic plane|orbifold]] (*∞32) contains these uniform tilings:


=== (∞&nbsp;4&nbsp;2) ===
The ideal (∞&nbsp;4&nbsp;2) [[Triangle group|triangle group]], Coxeter group [∞,4], [[Orbifold notation#Hyperbolic plane|orbifold]] (*∞42) contains these uniform tilings:

=== (∞&nbsp;5&nbsp;2) ===
The ideal (∞&nbsp;5&nbsp;2) [[Triangle group|triangle group]], Coxeter group [∞,5], [[Orbifold notation#Hyperbolic plane|orbifold]] (*∞52) contains these uniform tilings:

=== (∞ ∞ 2) ===
The ideal (∞&nbsp;∞&nbsp;2) [[Triangle group|triangle group]], Coxeter group [∞,∞], [[Orbifold notation#Hyperbolic plane|orbifold]] (*∞∞2) contains these uniform tilings:

=== (∞&nbsp;3&nbsp;3) ===
The ideal (∞&nbsp;3&nbsp;3) [[Triangle group|triangle group]], Coxeter group [(∞,3,3)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*∞33) contains these uniform tilings. 

=== (∞&nbsp;4&nbsp;3) ===
The ideal (∞&nbsp;4&nbsp;3) [[Triangle group|triangle group]], Coxeter group [(∞,4,3)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*∞43) contains these uniform tilings:

=== (∞&nbsp;4&nbsp;4) ===
The ideal (∞&nbsp;4&nbsp;4) [[Triangle group|triangle group]], Coxeter group [(∞,4,4)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*∞44) contains these uniform tilings. 

=== (∞ ∞ 3) ===
The ideal (∞&nbsp;∞&nbsp;3) [[Triangle group|triangle group]], Coxeter group [(∞,∞,3)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*∞∞3) contains these uniform tilings. 

=== (∞ ∞ 4) ===
The ideal (∞&nbsp;∞&nbsp;4) [[Triangle group|triangle group]], Coxeter group [(∞,∞,4)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*∞∞4) contains these uniform tilings. 

=== (∞ ∞ ∞) ===
The ideal (∞&nbsp;∞&nbsp;∞) [[Triangle group|triangle group]], Coxeter group [(∞,∞,∞)], [[Orbifold notation#Hyperbolic plane|orbifold]] (*∞∞∞) contains these uniform tilings. 

== Summary of tilings with infinite triangular fundamental domains ==
For a table of all uniform hyperbolic tilings with fundamental domains (''p''&nbsp;''q''&nbsp;''r''), where&nbsp;2&nbsp;≤ ''p'',''q'',''r'' ≤ 8, and one or more as ∞.

== References==
* [[Biography:John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} (Chapter 19, The Hyperbolic Archimedean Tessellations)

== External links ==
* {{cite web 
  | author = Hatch, Don
  | title = Hyperbolic Planar Tessellations
  | url = http://www.plunk.org/~hatch/HyperbolicTesselations/
  | access-date = 2010-08-19}}
* {{cite web 
  | author = Eppstein, David
  | title = The Geometry Junkyard: Hyperbolic Tiling
  | url = http://www.ics.uci.edu/~eppstein/junkyard/hypertile.html
  | access-date = 2010-08-19}}
* {{cite web 
  | author = Joyce, David
  | title = Hyperbolic Tessellations
  | url = http://aleph0.clarku.edu/~djoyce/poincare/poincare.html
  | access-date = 2010-08-19}}
* {{KlitzingPolytopes|hyperbolic.htm|2D Tesselations|Hyperbolic Tesselations}}
* [https://epinet.anu.edu.au/page/home The EPINET project explores 2D hyperbolic (H²) tilings ]
{{Tessellation}}

[[Category:Hyperbolic tilings| ]]
[[Category:Uniform tilings]]
[[Category:Mathematics-related lists|Uniform planar tilings]]

{{Sourceattribution|Uniform tilings in hyperbolic plane}}