Uniform tilings in hyperbolic plane
| 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.
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}.