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.

Construction of Archimedean Solids and Tessellations

Symmetry		Triangular dihedral symmetry		Tetrahedral	Octahedral		Icosahedral		p6m symmetry		[3,7] symmetry		[3,8] symmetry
Starting solid Operation	Symbol {p,q}	Triangular hosohedron {2,3}	Triangular dihedron {3,2}	Tetrahedron {3,3}	Cube {4,3}	Octahedron {3,4}	Dodecahedron {5,3}	Icosahedron {3,5}	Hexagonal tiling {6,3}	Triangular tiling {3,6}	Heptagonal tiling {7,3}	Order-7 triangular tiling {3,7}	Octagonal tiling {8,3}	Order-8 triangular tiling {3,8}
Truncation (t)	t{p,q}	triangular prism	truncated triangular dihedron (Half of the "edges" count as degenerate digon faces. The other half are normal edges.)	truncated tetrahedron	truncated cube	truncated octahedron	truncated dodecahedron	truncated icosahedron	Truncated hexagonal tiling	Truncated triangular tiling	Truncated heptagonal tiling	Truncated order-7 triangular tiling	Truncated octagonal tiling	Truncated order-8 triangular tiling
Rectification (r) Ambo (a)	r{p,q}	tridihedron (All of the "edges" count as degenerate digon faces.)		tetratetrahedron	cuboctahedron		icosidodecahedron		Trihexagonal tiling		Triheptagonal tiling		Trioctagonal tiling
Bitruncation (2t) Dual kis (dk)	2t{p,q}	truncated triangular dihedron (Half of the "edges" count as degenerate digon faces. The other half are normal edges.)	triangular prism	truncated tetrahedron	truncated octahedron	truncated cube	truncated icosahedron	truncated dodecahedron	truncated triangular tiling	truncated hexagonal tiling	Truncated order-7 triangular tiling	Truncated heptagonal tiling	Truncated order-8 triangular tiling	Truncated octagonal tiling
Birectification (2r) Dual (d)	2r{p,q}	triangular dihedron {3,2}	triangular hosohedron {2,3}	tetrahedron	octahedron	cube	icosahedron	dodecahedron	triangular tiling	hexagonal tiling	Order-7 triangular tiling	Heptagonal tiling	Order-8 triangular tiling	Octagonal tiling
Cantellation (rr) Expansion (e)	rr{p,q}	triangular prism (The "edge" between each pair of tetragons counts as a degenerate digon face. The other edges (the ones between a trigon and a tetragon) are normal edges.)		rhombitetratetrahedron	rhombicuboctahedron		rhombicosidodecahedron		rhombitrihexagonal tiling		Rhombitriheptagonal tiling		Rhombitrioctagonal tiling
Snub rectified (sr) Snub (s)	sr{p,q}	triangular antiprism (Three yellow-yellow "edges", no two of which share any vertices, count as degenerate digon faces. The other edges are normal edges.)		snub tetratetrahedron	snub cuboctahedron		snub icosidodecahedron		snub trihexagonal tiling		Snub triheptagonal tiling		Snub trioctagonal tiling
Cantitruncation (tr) Bevel (b)	tr{p,q}	hexagonal prism		truncated tetratetrahedron	truncated cuboctahedron		truncated icosidodecahedron		truncated trihexagonal tiling		Truncated triheptagonal tiling		Truncated trioctagonal tiling

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}.