| 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. | |||||
| 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.
Contents
- Wythoff construction
- Right triangle domains
- Regular hyperbolic tilings
- (7 3 2)
- (8 3 2)
- (5 4 2)
- (6 4 2)
- (7 4 2)
- (8 4 2)
- (5 5 2)
- (6 5 2)
- (6 6 2)
- (8 6 2)
- (7 7 2)
- (8 8 2)
- General triangle domains
- (4 3 3)
- (4 4 3)
- (4 4 4)
- (5 3 3)
- (5 4 3)
- (5 4 4)
- (6 3 3)
- (6 4 3)
- (6 4 4)
- Summary of tilings with finite triangular fundamental domains
- Quadrilateral domains
- (3 2 2 2)
- (3 2 3 2)
- Ideal triangle domains
- (∞ 3 2)
- (∞ 4 2)
- (∞ 5 2)
- (∞ ∞ 2)
- (∞ 3 3)
- (∞ 4 3)
- (∞ 4 4)
- (∞ ∞ 3)
- (∞ ∞ 4)
- (∞ ∞ ∞)
- Summary of tilings with infinite triangular fundamental domains
- References
- External links
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}.
