Global InformationUniform tilings in hyperbolic plane information
| 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
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 t{∞,3} ∞.∞.3
| ||
| Truncated tilings have 2p.2p.q vertex figures from regular {p,q}. | |||||
 r{5,3} 3.5.3.5
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 r{6,3} 3.6.3.6
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 r{7,3} 3.7.3.7
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 r{∞,3} 3.∞.3.∞
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| Quasiregular tilings are similar to regular tilings but alternate two types of regular polygon around each vertex. | |||||
 rr{5,3} 3.4.5.4
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 rr{6,3} 3.4.6.4
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 rr{7,3} 3.4.7.4
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 rr{∞,3} 3.4.∞.4
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| Semiregular tilings have more than one type of regular polygon. | |||||
 tr{5,3} 4.6.10
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 tr{6,3} 4.6.12
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 tr{7,3} 4.6.14
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 tr{∞,3} 4.6.∞
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| Omnitruncated tilings have three or more even-sided regular polygons. | |||||
| 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}.