Uniform tilings in hyperbolic plane

Examples of uniform tilings
Spherical Euclidean Hyperbolic

{5,3}
5.5.5
{6,3}
6.6.6
{7,3}
7.7.7
{∞,3}
∞.∞.∞
Regular tilings {p,q} of the sphere, Euclidean plane, and hyperbolic plane using regular pentagonal, hexagonal and heptagonal and apeirogonal faces.

t{5,3}
10.10.3
t{6,3}
12.12.3
t{7,3}
14.14.3
t{∞,3}
∞.∞.3
Truncated tilings have 2p.2p.q vertex figures from regular {p,q}.

r{5,3}
3.5.3.5
r{6,3}
3.6.3.6
r{7,3}
3.7.3.7
r{∞,3}
3.∞.3.∞
Quasiregular tilings are similar to regular tilings but alternate two types of regular polygon around each vertex.

rr{5,3}
3.4.5.4
rr{6,3}
3.4.6.4
rr{7,3}
3.4.7.4
rr{∞,3}
3.4.∞.4
Semiregular tilings have more than one type of regular polygon.

tr{5,3}
4.6.10
tr{6,3}
4.6.12
tr{7,3}
4.6.14
tr{∞,3}
4.6.∞
Omnitruncated tilings have three or more even-sided regular polygons.

In hyperbolic 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 polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry.

Uniform tilings can be identified by their 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 which has 3 heptagons 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 (if also face- and edge-transitive), quasi-regular (if edge-transitive but not face-transitive) or semi-regular (if neither edge- nor face-transitive). For right triangles (p q 2), there are two regular tilings, represented by Schläfli symbol {p,q} and {q,p}.