Uniform tilings in hyperbolic plane

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Examples of uniform tilings
Spherical Euclidean Hyperbolic
Uniform tiling 532-t0.png
{5,3}
5.5.5
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 63-t0.png
{6,3}
6.6.6
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 73-t0.png
{7,3}
7.7.7
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 23i-1.png
{∞,3}
∞.∞.∞
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
Regular tilings of the sphere {p,q}, Euclidean plane, and hyperbolic plane using regular pentagonal, hexagonal and heptagonal and apeirogonal faces.
Uniform tiling 532-t01.png
t{5,3}
10.10.3
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t01.png
t{6,3}
12.12.3
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 73-t01.png
t{7,3}
14.14.3
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png
H2 tiling 23i-3.png
t{∞,3}
∞.∞.3
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png
Truncated tilings have 2p.2p.q vertex figures from regular {p,q}
Uniform tiling 532-t1.png
r{5,3}
3.5.3.5
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t1.png
r{6,3}
3.6.3.6
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 73-t1.png
r{7,3}
3.7.3.7
CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png
H2 tiling 23i-2.png
r{∞,3}
3.∞.3.∞
CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png
Quasiregular tilings are similar to regular tilings but alternate two types of regular polygon around each vertex.
Uniform tiling 532-t02.png
rr{5,3}
3.4.5.4
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t02.png
rr{6,3}
3.4.6.4
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform tiling 73-t02.png
rr{7,3}
3.4.7.4
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png
H2 tiling 23i-5.png
rr{∞,3}
3.4.∞.4
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png
Semiregular tilings have more than one type of regular polygon.
Uniform tiling 532-t012.png
tr{5,3}
4.6.10
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t012.png
tr{6,3}
4.6.12
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform tiling 73-t012.png
tr{7,3}
4.6.14
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png
H2 tiling 23i-7.png
tr{∞,3}
4.6.∞
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Omnitruncated tilings have three or more even-sided regular polygons.

In hyperbolic geometry, a uniform (regular, quasiregular or semiregular) hyperbolic tiling is an edge-to-edge filling of the 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}.

Wythoff construction[edit]

Example Wythoff construction with right triangles (r=2) and the 7 generator points. Lines to the active mirrors are colored red, yellow, and blue with the 3 nodes opposite them as associated by the Wythoff symbol.

There are an infinite number of uniform tilings based on the Schwarz triangles (p q r) where 1/p + 1/q + 1/r < 1, where p,q,r are each orders of reflection symmetry at three points of the fundamental domain triangle – the symmetry group is a hyperbolic triangle group.

Each symmetry family contains 7 uniform tilings, defined by a Wythoff symbol or Coxeter-Dynkin diagram, 7 representing combinations of 3 active mirrors. An 8th represents an alternation operation, deleting alternate vertices from the highest form with all mirrors active.

Families with r=2 contain regular hyperbolic tilings, defined by a Coxeter group such as [7,3], [8,3], [9,3], ... [5,4], [6,4], ....

Hyperbolic families with r=3 or higher are given by (p q r) and include (4 3 3), (5 3 3), (6 3 3) ... (4 4 3), (5 4 3), ... (4 4 4)....

Hyperbolic triangles (p q r) define compact uniform hyperbolic tilings. In the limit any of p,q or r can be replaced by ∞ which defines a paracompact hyperbolic triangle and creates uniform tilings with either infinite faces (called apeirogons) that converge to a single ideal point, or infinite vertex figure with infinitely many edges diverging from the same ideal point.

More symmetry families can be constructed from fundamental domains that are not triangles.

Selected families of uniform tilings are shown below (using the Poincaré disk model for the hyperbolic plane). Three of them – (7 3 2), (5 4 2), and (4 3 3) – and no others, are minimal in the sense that if any of their defining numbers is replaced by a smaller integer the resulting pattern is either Euclidean or spherical rather than hyperbolic; conversely, any of the numbers can be increased (even to infinity) to generate other hyperbolic patterns.

Each uniform tiling generates a dual uniform tiling, with many of them also given below.

Right triangle domains[edit]

There are infinitely many (p q 2) triangle group families. This article shows the regular tiling up to p,q=8, and uniform tilings in 12 families: (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), (7 7 2), (8 6 2), and (8 8 2).

Regular hyperbolic tilings[edit]

The simplest set of hyperbolic tilings are regular tilings {p,q}, which exist in a matrix with the regular polyhedra and Euclidean tilings. Regular tile {p,q} has a dual tiling {q,p} across the diagonal axis of the table. Self-dual tilings {3,3}, {4,4}, {5,5}, etc. pass down the diagonal of the table.

Spherical (Platonic)/Euclidean/hyperbolic (Poincaré disc) tessellations with their Schläfli symbol
p \ q 3 4 5 6 7 8 ...
3 Uniform tiling 332-t0-1-.png
(tetrahedron)
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 432-t2.png
(octahedron)
{3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 532-t2.png
(icosahedron)
{3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 63-t2.png
(deltille)
{3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 37-t0.png

{3,7}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
Uniform tiling 38-t0.png

{3,8}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 23i-4.png

{3,∞}
CDel node 1.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
4 Uniform tiling 432-t0.png
(cube)
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 44-t0.png
(quadrille)
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 45-t0.png

{4,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 46-t0.png

{4,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 47-t0.png

{4,7}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png
Uniform tiling 48-t0.png

{4,8}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 24i-4.png

{4,∞}
CDel node 1.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node.png
5 Uniform tiling 532-t0.png
(dodecahedron)
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 54-t0.png

{5,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 55-t0.png

{5,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 56-t0.png

{5,6}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 57-t0.png

{5,7}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 7.pngCDel node.png
Uniform tiling 58-t0.png

{5,8}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 25i-4.png

{5,∞}
CDel node 1.pngCDel 5.pngCDel node.pngCDel infin.pngCDel node.png
6 Uniform tiling 63-t0.png
(hextille)
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 64-t0.png

{6,4}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 65-t0.png

{6,5}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 66-t2.png

{6,6}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 67-t0.png

{6,7}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 7.pngCDel node.png
Uniform tiling 68-t0.png

{6,8}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 26i-4.png

{6,∞}
CDel node 1.pngCDel 6.pngCDel node.pngCDel infin.pngCDel node.png
7 Uniform tiling 73-t0.png
{7,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 74-t0.png
{7,4}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 75-t0.png
{7,5}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 76-t0.png
{7,6}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 77-t2.png
{7,7}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node.png
Uniform tiling 78-t0.png
{7,8}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 27i-4.png
{7,∞}
CDel node 1.pngCDel 7.pngCDel node.pngCDel infin.pngCDel node.png
8 Uniform tiling 83-t0.png
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 84-t0.png
{8,4}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 85-t0.png
{8,5}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 86-t0.png
{8,6}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 87-t0.png
{8,7}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 7.pngCDel node.png
Uniform tiling 88-t2.png
{8,8}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 28i-4.png
{8,∞}
CDel node 1.pngCDel 8.pngCDel node.pngCDel infin.pngCDel node.png
...
H2 tiling 23i-1.png
{∞,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 24i-1.png
{∞,4}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png
H2 tiling 25i-1.png
{∞,5}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 26i-1.png
{∞,6}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 6.pngCDel node.png
H2 tiling 27i-1.png
{∞,7}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 7.pngCDel node.png
H2 tiling 28i-1.png
{∞,8}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 2ii-1.png
{∞,∞}
CDel node 1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png

(7 3 2)[edit]

The (7 3 2) triangle group, Coxeter group [7,3], orbifold (*732) contains these uniform tilings:

Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732) [7,3]+, (732)
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 7.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 73-t0.png Uniform tiling 73-t01.png Uniform tiling 73-t1.png Uniform tiling 73-t12.png Uniform tiling 73-t2.png Uniform tiling 73-t02.png Uniform tiling 73-t012.png Uniform tiling 73-snub.png
{7,3} t{7,3} r{7,3} 2t{7,3}=t{3,7} 2r{7,3}={3,7} rr{7,3} tr{7,3} sr{7,3}
Uniform duals
CDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Uniform tiling 73-t2.png Ord7 triakis triang til.png Order73 qreg rhombic til.png Order3 heptakis heptagonal til.png Uniform tiling 73-t0.png Deltoidal triheptagonal til.png Order-3 heptakis heptagonal tiling.png Ord7 3 floret penta til.png
V73 V3.14.14 V3.7.3.7 V6.6.7 V37 V3.4.7.4 V4.6.14 V3.3.3.3.7

(8 3 2)[edit]

The (8 3 2) triangle group, Coxeter group [8,3], orbifold (*832) contains these uniform tilings:

Uniform octagonal/triangular tilings
Symmetry: [8,3], (*832) [8,3]+
(832)
[1+,8,3]
(*443)
[8,3+]
(3*4)
{8,3} t{8,3} r{8,3} t{3,8} {3,8} rr{8,3}
s2{3,8}
tr{8,3} sr{8,3} h{8,3} h2{8,3} s{3,8}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node.png or CDel label4.pngCDel branch 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png or CDel label4.pngCDel branch 01rd.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel label4.pngCDel branch hh.pngCDel split2.pngCDel node h.png
Uniform tiling 83-t0.png Uniform tiling 83-t01.png Uniform tiling 83-t1.png
Uniform tiling 433-t02.png
Uniform tiling 83-t12.png
Uniform tiling 433-t012.png
Uniform tiling 83-t2.png Uniform tiling 83-t02.png Uniform tiling 83-t012.png Uniform tiling 83-snub.png Uniform tiling 433-t0.pngUniform tiling 433-t1.png Uniform tiling 433-t02.pngUniform tiling 433-t12.png Uniform tiling 433-snub1.png
Uniform tiling 433-snub2.png
Uniform duals
V83 V3.16.16 V3.8.3.8 V6.6.8 V38 V3.4.8.4 V4.6.16 V34.8 V(3.4)3 V8.6.6 V35.4
CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Uniform tiling 83-t2.png Ord8 triakis triang til.png Uniform dual tiling 433-t01-yellow.png Uniform dual tiling 433-t012.png Uniform tiling 83-t0.png Deltoidal trioctagonal til.png Order-3 octakis octagonal tiling.png Uniform dual tiling 433-t0.png Uniform dual tiling 433-t01.png Uniform dual tiling 433-snub.png

(5 4 2)[edit]

The (5 4 2) triangle group, Coxeter group [5,4], orbifold (*542) contains these uniform tilings:

Uniform pentagonal/square tilings
Symmetry: [5,4], (*542) [5,4]+, (542) [5+,4], (5*2) [5,4,1+], (*552)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node h.png
Uniform tiling 54-t0.png Uniform tiling 54-t01.png Uniform tiling 54-t1.png Uniform tiling 54-t12.png Uniform tiling 54-t2.png Uniform tiling 54-t02.png Uniform tiling 54-t012.png Uniform tiling 54-snub.png Uniform tiling 542-h01.png Uniform tiling 552-t0.png
{5,4} t{5,4} r{5,4} 2t{5,4}=t{4,5} 2r{5,4}={4,5} rr{5,4} tr{5,4} sr{5,4} s{5,4} h{4,5}
Uniform duals
CDel node f1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node fh.png
Uniform tiling 54-t2.png Order-5 tetrakis square tiling.png Order-5-4 quasiregular rhombic tiling.png Order-4 pentakis pentagonal tiling.png Uniform tiling 54-t0.png Deltoidal tetrapentagonal tiling.png Order-4 bisected pentagonal tiling.png Order-5-4 floret pentagonal tiling.png Uniform tiling 552-t2.png
V54 V4.10.10 V4.5.4.5 V5.8.8 V45 V4.4.5.4 V4.8.10 V3.3.4.3.5 V3.3.5.3.5 V55

(6 4 2)[edit]

The (6 4 2) triangle group, Coxeter group [6,4], orbifold (*642) contains these uniform tilings. Because all the elements are even, each uniform dual tiling one represents the fundamental domain of a reflective symmetry: *3333, *662, *3232, *443, *222222, *3222, and *642 respectively. As well, all 7 uniform tiling can be alternated, and those have duals as well.

Uniform tetrahexagonal tilings
Symmetry: [6,4], (*642)
(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries)
(And [(∞,3,∞,3)] (*3232) index 4 subsymmetry)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-66.pngCDel nodes.png
CDel 2.png
= CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png
= CDel branch 11.pngCDel 3a3b-cross.pngCDel branch 11.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-66.pngCDel nodes 11.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node.pngCDel split1-66.pngCDel nodes 11.png
= CDel branch 11.pngCDel split2-44.pngCDel node.png
CDel 2.png
= CDel nodes 11.pngCDel 3a3b-cross.pngCDel nodes 11.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel branch 11.pngCDel split2-44.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel branch.pngCDel split2-44.pngCDel node 1.png
= CDel branch.pngCDel 2a2b-cross.pngCDel nodes 11.png
= CDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
CDel 2.png
= CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png
H2 tiling 246-1.png H2 tiling 246-3.png H2 tiling 246-2.png H2 tiling 246-6.png H2 tiling 246-4.png H2 tiling 246-5.png H2 tiling 246-7.png
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
CDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.png
H2chess 246b.png H2chess 246f.png H2chess 246a.png H2chess 246e.png H2chess 246c.png H2chess 246d.png H2checkers 246.png
V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12
Alternations
[1+,6,4]
(*443)
[6+,4]
(6*2)
[6,1+,4]
(*3222)
[6,4+]
(4*3)
[6,4,1+]
(*662)
[(6,4,2+)]
(2*32)
[6,4]+
(642)
CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
= CDel branch 10ru.pngCDel split2-44.pngCDel node.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node.png
= CDel node h.pngCDel split1-66.pngCDel nodes hh.png
CDel node.pngCDel 6.pngCDel node h1.pngCDel 4.pngCDel node.png
= CDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 10.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png
= CDel branch hh.pngCDel split2-44.pngCDel node h.png
CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.png
= CDel node.pngCDel split1-66.pngCDel nodes 10lu.png
CDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h.png
= CDel branch hh.pngCDel 2a2b-cross.pngCDel nodes hh.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png
Uniform tiling 443-t0.png Uniform tiling 64-h02.png Uniform tiling 64-h1.png Uniform tiling 443-snub2.png Uniform tiling 66-t0.png Uniform tiling 3.4.4.4.4.png Uniform tiling 64-snub.png
h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}

(7 4 2)[edit]

The (7 4 2) triangle group, Coxeter group [7,4], orbifold (*742) contains these uniform tilings:

Uniform heptagonal/square tilings
Symmetry: [7,4], (*742) [7,4]+, (742) [7+,4], (7*2) [7,4,1+], (*772)
CDel node 1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node h.pngCDel 7.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h.pngCDel 7.pngCDel node h.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node h.png
Uniform tiling 74-t0.png Uniform tiling 74-t01.png Uniform tiling 74-t1.png Uniform tiling 74-t12.png Uniform tiling 74-t2.png Uniform tiling 74-t02.png Uniform tiling 74-t012.png Uniform tiling 74-snub.png Uniform tiling 74-h01.png Uniform tiling 77-t0.png
{7,4} t{7,4} r{7,4} 2t{7,4}=t{4,7} 2r{7,4}={4,7} rr{7,4} tr{7,4} sr{7,4} s{7,4} h{4,7}
Uniform duals
CDel node f1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node fh.png
Uniform tiling 74-t2.png Hyperbolic domains 772.png Ord74 qreg rhombic til.png Order4 heptakis heptagonal til.png Uniform tiling 74-t0.png Deltoidal tetraheptagonal til.png Hyperbolic domains 742.png Uniform tiling 77-t2.png
V74 V4.14.14 V4.7.4.7 V7.8.8 V47 V4.4.7.4 V4.8.14 V3.3.4.3.7 V3.3.7.3.7 V77

(8 4 2)[edit]

The (8 4 2) triangle group, Coxeter group [8,4], orbifold (*842) contains these uniform tilings. Because all the elements are even, each uniform dual tiling one represents the fundamental domain of a reflective symmetry: *4444, *882, *4242, *444, *22222222, *4222, and *842 respectively. As well, all 7 uniform tiling can be alternated, and those have duals as well.

Uniform octagonal/square tilings
[8,4], (*842)
(with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries)
(And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-88.pngCDel nodes.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png
= CDel label4.pngCDel branch 11.pngCDel 4a4b-cross.pngCDel branch 11.pngCDel label4.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-88.pngCDel nodes 11.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node.pngCDel split1-88.pngCDel nodes 11.png
= CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel branch 11.pngCDel label4.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node 1.png
= CDel label4.pngCDel branch.pngCDel 2a2b-cross.pngCDel nodes 11.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
H2 tiling 248-1.png H2 tiling 248-3.png H2 tiling 248-2.png H2 tiling 248-6.png H2 tiling 248-4.png H2 tiling 248-5.png H2 tiling 248-7.png
{8,4} t{8,4}
r{8,4} 2t{8,4}=t{4,8} 2r{8,4}={4,8} rr{8,4} tr{8,4}
Uniform duals
CDel node f1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node f1.png
H2chess 248b.png H2chess 248f.png H2chess 248a.png H2chess 248e.png H2chess 248c.png H2chess 248d.png H2checkers 248.png
V84 V4.16.16 V(4.8)2 V8.8.8 V48 V4.4.4.8 V4.8.16
Alternations
[1+,8,4]
(*444)
[8+,4]
(8*2)
[8,1+,4]
(*4222)
[8,4+]
(4*4)
[8,4,1+]
(*882)
[(8,4,2+)]
(2*42)
[8,4]+
(842)
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
= CDel label4.pngCDel branch 10ru.pngCDel split2-44.pngCDel node.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node.png
= CDel node h.pngCDel split1-88.pngCDel nodes hh.png
CDel node.pngCDel 8.pngCDel node h1.pngCDel 4.pngCDel node.png
= CDel label4.pngCDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 10.png
CDel node.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node h.png
= CDel label4.pngCDel branch hh.pngCDel split2-44.pngCDel node h.png
CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node h1.png
= CDel node.pngCDel split1-88.pngCDel nodes 10lu.png
CDel node h.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node h.png
= CDel label4.pngCDel branch hh.pngCDel 2a2b-cross.pngCDel nodes hh.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node h.png
Uniform tiling 444-t0.png Uniform tiling 84-h01.png Uniform tiling 443-t1.png Uniform tiling 444-snub.png Uniform tiling 88-t0.png Uniform tiling 54-t2.png Uniform tiling 84-snub.png
h{8,4} s{8,4} hr{8,4} s{4,8} h{4,8} hrr{8,4} sr{8,4}
Alternation duals
CDel node fh.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node fh.png
Uniform tiling 88-t1.png Uniform tiling 66-t1.png Uniform dual tiling 433-t0.png Uniform tiling 88-t2.png Uniform tiling 54-t0.png
V(4.4)4 V3.(3.8)2 V(4.4.4)2 V(3.4)3 V88 V4.44 V3.3.4.3.8

(5 5 2)[edit]

The (5 5 2) triangle group, Coxeter group [5,5], orbifold (*552) contains these uniform tilings:

Uniform pentapentagonal tilings
Symmetry: [5,5], (*552) [5,5]+, (552)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node 1.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node 1.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node 1.png
CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 5.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 5.pngCDel node 1.png
CDel node h.pngCDel 5.pngCDel node h.pngCDel 5.pngCDel node h.png
= CDel node h0.pngCDel 4.pngCDel node h.pngCDel 5.pngCDel node h.png
Uniform tiling 552-t0.png Uniform tiling 552-t01.png Uniform tiling 552-t1.png Uniform tiling 552-t12.png Uniform tiling 552-t2.png Uniform tiling 552-t02.png Uniform tiling 552-t012.png Uniform tiling 552-snub.png
{5,5} t{5,5}
r{5,5} 2t{5,5}=t{5,5} 2r{5,5}={5,5} rr{5,5} tr{5,5} sr{5,5}
Uniform duals
CDel node f1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 5.pngCDel node f1.png CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 5.pngCDel node f1.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 5.pngCDel node fh.png
Uniform tiling 552-t2.png Order5 pentakis pentagonal til.png Uniform tiling 54-t2.png Order5 pentakis pentagonal til.png Uniform tiling 552-t0.png Order-5-4 quasiregular rhombic tiling.png Order-5 tetrakis square tiling.png
V5.5.5.5.5 V5.10.10 V5.5.5.5 V5.10.10 V5.5.5.5.5 V4.5.4.5 V4.10.10 V3.3.5.3.5

(6 5 2)[edit]

The (6 5 2) triangle group, Coxeter group [6,5], orbifold (*652) contains these uniform tilings:

Uniform hexagonal/pentagonal tilings
Symmetry: [6,5], (*652) [6,5]+, (652) [6,5+], (5*3) [1+,6,5], (*553)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 5.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 5.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 5.pngCDel node 1.png CDel node.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 5.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 5.pngCDel node h.png CDel node.pngCDel 6.pngCDel node h.pngCDel 5.pngCDel node h.png CDel node h.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 256-1.png H2 tiling 256-3.png H2 tiling 256-2.png H2 tiling 256-6.png H2 tiling 256-4.png H2 tiling 256-5.png H2 tiling 256-7.png Uniform tiling 65-snub.png H2 tiling 355-1.png
{6,5} t{6,5} r{6,5} 2t{6,5}=t{5,6} 2r{6,5}={5,6} rr{6,5} tr{6,5} sr{6,5} s{5,6} h{6,5}
Uniform duals
CDel node f1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 5.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 5.pngCDel node f1.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 5.pngCDel node fh.png CDel node.pngCDel 6.pngCDel node fh.pngCDel 5.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png
H2chess 256b.png Order-6 pentakis pentagonal tiling.png Order-6-5 quasiregular rhombic tiling.png H2chess 256e.png H2 tiling 256-1.png Deltoidal pentahexagonal tiling.png H2checkers 256.png
V65 V5.12.12 V5.6.5.6 V6.10.10 V56 V4.5.4.6 V4.10.12 V3.3.5.3.6 V3.3.3.5.3.5 V(3.5)5

(6 6 2)[edit]

The (6 6 2) triangle group, Coxeter group [6,6], orbifold (*662) contains these uniform tilings:

Uniform hexahexagonal tilings
Symmetry: [6,6], (*662)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 6.pngCDel node 1.png
H2 tiling 266-1.png H2 tiling 266-3.png H2 tiling 266-2.png H2 tiling 266-6.png H2 tiling 266-4.png H2 tiling 266-5.png H2 tiling 266-7.png
{6,6}
= h{4,6}
t{6,6}
= h2{4,6}
r{6,6}
{6,4}
t{6,6}
= h2{4,6}
{6,6}
= h{4,6}
rr{6,6}
r{6,4}
tr{6,6}
t{6,4}
Uniform duals
CDel node f1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node f1.png
H2chess 266b.png H2chess 266f.png H2chess 266a.png H2chess 266e.png H2chess 266c.png H2chess 266d.png H2checkers 266.png
V66 V6.12.12 V6.6.6.6 V6.12.12 V66 V4.6.4.6 V4.12.12
Alternations
[1+,6,6]
(*663)
[6+,6]
(6*3)
[6,1+,6]
(*3232)
[6,6+]
(6*3)
[6,6,1+]
(*663)
[(6,6,2+)]
(2*33)
[6,6]+
(662)
CDel node h1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node h1.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node h.png CDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node h1.png CDel node h.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node h.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node h.png
Uniform tiling 66-h0.png Uniform tiling 443-t0.png Uniform tiling 66-h0.png Uniform tiling 64-h1.png Uniform tiling 66-snub.png
h{6,6} s{6,6} hr{6,6} s{6,6} h{6,6} hrr{6,6} sr{6,6}

(8 6 2)[edit]

The (8 6 2) triangle group, Coxeter group [8,6], orbifold (*862) contains these uniform tilings.

Uniform octagonal/hexagonal tilings
Symmetry: [8,6], (*862)
CDel node 1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.png CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 6.pngCDel node 1.png CDel node.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 6.pngCDel node 1.png
H2 tiling 268-1.png H2 tiling 268-3.png H2 tiling 268-2.png H2 tiling 268-6.png H2 tiling 268-4.png H2 tiling 268-5.png H2 tiling 268-7.png
{8,6} t{8,6}
r{8,6} 2t{8,6}=t{6,8} 2r{8,6}={6,8} rr{8,6} tr{8,6}
Uniform duals
CDel node f1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 6.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 6.pngCDel node f1.png
H2chess 268b.png H2chess 268f.png H2chess 268a.png H2chess 268e.png H2chess 268c.png H2chess 268d.png H2checkers 268.png
V86 V6.16.16 V(6.8)2 V8.12.12 V68 V4.6.4.8 V4.12.16
Alternations
[1+,8,6]
(*466)
[8+,6]
(8*3)
[8,1+,6]
(*4232)
[8,6+]
(6*4)
[8,6,1+]
(*883)
[(8,6,2+)]
(2*43)
[8,6]+
(862)
CDel node h1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.png CDel node h.pngCDel 8.pngCDel node h.pngCDel 6.pngCDel node.png CDel node.pngCDel 8.pngCDel node h1.pngCDel 6.pngCDel node.png CDel node.pngCDel 8.pngCDel node h.pngCDel 6.pngCDel node h.png CDel node.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node h1.png CDel node h.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node h.png CDel node h.pngCDel 8.pngCDel node h.pngCDel 6.pngCDel node h.png
H2 tiling 466-1.png H2 tiling 388-1.png Uniform tiling 86-snub.png
h{8,6} s{8,6} hr{8,6} s{6,8} h{6,8} hrr{8,6} sr{8,6}
Alternation duals
CDel node fh.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 6.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 6.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 6.pngCDel node fh.png CDel node.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 6.pngCDel node fh.png
H2chess 466b.png
V(4.6)6 V3.3.8.3.8.3 V(3.4.4.4)2 V3.4.3.4.3.6 V(3.8)8 V3.45 V3.3.6.3.8

(7 7 2)[edit]

The (7 7 2) triangle group, Coxeter group [7,7], orbifold (*772) contains these uniform tilings:

Uniform heptaheptagonal tilings
Symmetry: [7,7], (*772) [7,7]+, (772)
CDel node 1.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node 1.png
CDel node.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node 1.png
CDel node.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node 1.png
CDel node.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 7.pngCDel node.png
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 7.pngCDel node 1.png
CDel node h.pngCDel 7.pngCDel node h.pngCDel 7.pngCDel node h.png
= CDel node h0.pngCDel 4.pngCDel node h.pngCDel 7.pngCDel node h.png
Uniform tiling 77-t0.png Uniform tiling 77-t01.png Uniform tiling 77-t1.png Uniform tiling 77-t12.png Uniform tiling 77-t2.png Uniform tiling 77-t02.png Uniform tiling 77-t012.png Uniform tiling 77-snub.png
{7,7} t{7,7}
r{7,7} 2t{7,7}=t{7,7} 2r{7,7}={7,7} rr{7,7} tr{7,7} sr{7,7}
Uniform duals
CDel node f1.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 7.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 7.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 7.pngCDel node f1.png CDel node.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 7.pngCDel node f1.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 7.pngCDel node fh.png
Uniform tiling 77-t2.png Order7 heptakis heptagonal til.png Uniform tiling 74-t2.png Order7 heptakis heptagonal til.png Uniform tiling 77-t0.png Ord74 qreg rhombic til.png Hyperbolic domains 772.png
V77 V7.14.14 V7.7.7.7 V7.14.14 V77 V4.7.4.7 V4.14.14 V3.3.7.3.7

(8 8 2)[edit]

The (8 8 2) triangle group, Coxeter group [8,8], orbifold (*882) contains these uniform tilings:

Uniform octaoctagonal tilings
Symmetry: [8,8], (*882)
CDel node 1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node 1.png
H2 tiling 288-1.png H2 tiling 288-3.png H2 tiling 288-2.png H2 tiling 288-6.png H2 tiling 288-4.png H2 tiling 288-5.png H2 tiling 288-7.png
{8,8} t{8,8}
r{8,8} 2t{8,8}=t{8,8} 2r{8,8}={8,8} rr{8,8} tr{8,8}
Uniform duals
CDel node f1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node f1.png
H2chess 288b.png H2chess 288f.png H2chess 288a.png H2chess 288e.png H2chess 288c.png H2chess 288d.png H2checkers 288.png
V88 V8.16.16 V8.8.8.8 V8.16.16 V88 V4.8.4.8 V4.16.16
Alternations
[1+,8,8]
(*884)
[8+,8]
(8*4)
[8,1+,8]
(*4242)
[8,8+]
(8*4)
[8,8,1+]
(*884)
[(8,8,2+)]
(2*44)
[8,8]+
(882)
CDel node h1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png CDel node h.pngCDel 8.pngCDel node h.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node h1.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node h.pngCDel 8.pngCDel node h.png CDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node h1.png CDel node h.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node h.png CDel node h.pngCDel 8.pngCDel node h.pngCDel 8.pngCDel node h.png
Uniform tiling 88-h0.png Uniform tiling 444-t0.png Uniform tiling 88-h0.png Uniform tiling 443-t1.png Uniform tiling 88-snub.png
h{8,8} s{8,8} hr{8,8} s{8,8} h{8,8} hrr{8,8} sr{8,8}
Alternation duals
CDel node fh.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node fh.png CDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node fh.png
Uniform tiling 88-t1.png Uniform tiling 66-t1.png
V(4.8)8 V3.4.3.8.3.8 V(4.4)4 V3.4.3.8.3.8 V(4.8)8 V46 V3.3.8.3.8

General triangle domains[edit]

There are infinitely many general triangle group families (p q r). This article shows uniform tilings in 9 families: (4 3 3), (4 4 3), (4 4 4), (5 3 3), (5 4 3), (5 4 4), (6 3 3), (6 4 3), and (6 4 4).

(4 3 3)[edit]

The (4 3 3) triangle group, Coxeter group [(4,3,3)], orbifold (*433) contains these uniform tilings. Without right angles in the fundamental triangle, the Wythoff constructions are slightly different. For instance in the (4,3,3) triangle family, the snub form has six polygons around a vertex and its dual has hexagons rather than pentagons. In general the vertex figure of a snub tiling in a triangle (p,q,r) is p. 3.q.3.r.3, being 4.3.3.3.3.3 in this case below.

Uniform (4,3,3) tilings
Symmetry: [(4,3,3)], (*433) [(4,3,3)]+, (433)
CDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.png
CDel label4.pngCDel branch 01rd.pngCDel split2.pngCDel node.png CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node.png CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node.png CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png CDel label4.pngCDel branch.pngCDel split2.pngCDel node 1.png CDel label4.pngCDel branch 01rd.pngCDel split2.pngCDel node 1.png CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node 1.png CDel label4.pngCDel branch hh.pngCDel split2.pngCDel node h.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 433-t0.png Uniform tiling 433-t01.png Uniform tiling 433-t1.png Uniform tiling 433-t12.png Uniform tiling 433-t2.png Uniform tiling 433-t02.png Uniform tiling 433-t012.png Uniform tiling 433-snub2.png
h{8,3}
t0{(4,3,3)}
{(4,3,3)}
r{8,3}
t0,1{(4,3,3)}
r{(3,4,3)}
h{8,3}
t1{(4,3,3)}]]
{(3,3,4)}
h2{8,3}
t1,2{(4,3,3)}
r{(4,3,3)}
{3,8}
t2{(4,3,3)}]
{(3,4,3)}
h2{8,3}
t0,2{(4,3,3)}
r{(3,3,4)}
t{3,8}
t0,1,2{(4,3,3)}
t{(3,4,3)}
s{3,8}
 
s{(3,4,3)}
Uniform duals
CDel 3.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.png
Uniform dual tiling 433-t0.png Uniform dual tiling 433-t01.png Uniform dual tiling 433-t0.png Uniform dual tiling 433-t12.png Uniform dual tiling 433-t2.png Uniform dual tiling 433-t12.png Uniform dual tiling 433-t012.png Uniform dual tiling 433-snub.png
V(3.4)3 V3.8.3.8 V(3.4)3 V3.6.4.6 V(3.3)4 V3.6.4.6 V6.6.8 V3.3.3.3.3.4

(4 4 3)[edit]

The (4 4 3) triangle group, Coxeter group [(4,4,3)], orbifold (*443) contains these uniform tilings.

Uniform (4,4,3) tilings
Symmetry: [(4,4,3)] (*443) [(4,4,3)]+
(443)
[(4,4,3+)]
(3*22)
[(4,1+,4,3)]
(*3232)
CDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.png CDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.png
CDel branch 01rd.pngCDel split2-44.pngCDel node.png CDel branch 01rd.pngCDel split2-44.pngCDel node 1.png CDel branch.pngCDel split2-44.pngCDel node 1.png CDel branch 10ru.pngCDel split2-44.pngCDel node 1.png CDel branch 10ru.pngCDel split2-44.pngCDel node.png CDel branch 11.pngCDel split2-44.pngCDel node.png CDel branch 11.pngCDel split2-44.pngCDel node 1.png CDel branch hh.pngCDel split2-44.pngCDel node h.png CDel branch hh.pngCDel split2-44.pngCDel node.png CDel branch.pngCDel split2-44.pngCDel node h1.png CDel branch 10ru.pngCDel split2-44.pngCDel node h1.png
CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node h0.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node h0.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h0.pngCDel 6.pngCDel node h1.pngCDel 4.pngCDel node.png CDel node h0.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.png
Uniform tiling 443-t0.png Uniform tiling 443-t01.png Uniform tiling 443-t1.png Uniform tiling 443-t12.png Uniform tiling 443-t2.png Uniform tiling 443-t02.png Uniform tiling 443-t012.png Uniform tiling 443-snub1.png Uniform tiling 64-h1.png Uniform tiling 66-t2.png Uniform tiling verf 34664.png
h{6,4}
t0{(4,4,3)}
{(4,4,3)}
h2{6,4}
t0,1{(4,4,3)}
r{(3,4,4)}
{4,6}
t1{(4,4,3)}
{(4,3,4)}
h2{6,4}
t1,2{(4,4,3)}
r{(4,4,3)}
h{6,4}
t2{(4,4,3)}
{(3,4,4)}
r{6,4}
t0,2{(4,4,3)}
r{(4,3,4)}
t{4,6}
t0,1,2{(4,4,3)}
t{(4,3,4)}
s{4,6}
 
s{(4,4,3)}
hr{6,4}
 
hr{(4,3,4)}
h{4,6}
 
h{(4,3,4)}
q{4,6}
 
h2{(4,3,4)}
Uniform duals
CDel 3.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.png CDel 3.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node fh.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 3.png
Uniform tiling 66-t1.png Ord64 qreg rhombic til.png Order4 hexakis hexagonal til.png Uniform tiling 66-t0.png
V(3.4)4 V3.8.4.8 V(4.4)3 V3.8.4.8 V(3.4)4 V4.6.4.6 V6.8.8 V3.3.3.4.3.4 V(4.4.3)2 V66 V4.3.4.6.6

(4 4 4)[edit]

The (4 4 4) triangle group, Coxeter group [(4,4,4)], orbifold (*444) contains these uniform tilings.

Uniform (4,4,4) tilings
Symmetry: [(4,4,4)], (*444) [(4,4,4)]+
(444)
[(1+,4,4,4)]
(*4242)
[(4+,4,4)]
(4*22)
CDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 4.png
CDel label4.pngCDel branch 01rd.pngCDel split2-44.pngCDel node.png CDel label4.pngCDel branch 01rd.pngCDel split2-44.pngCDel node 1.png CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node 1.png CDel label4.pngCDel branch 10ru.pngCDel split2-44.pngCDel node 1.png CDel label4.pngCDel branch 10ru.pngCDel split2-44.pngCDel node.png CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node.png CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.png CDel label4.pngCDel branch hh.pngCDel split2-44.pngCDel node h.png CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node h1.png CDel label4.pngCDel branch hh.pngCDel split2-44.pngCDel node.png
H2 tiling 444-1.png H2 tiling 444-3.png H2 tiling 444-2.png H2 tiling 444-6.png H2 tiling 444-4.png H2 tiling 444-5.png H2 tiling 444-7.png Uniform tiling 444-snub.png H2 tiling 288-4.png H2 tiling 344-2.png
t0{(4,4,4)} t0,1{(4,4,4)} t1{(4,4,4)} t1,2{(4,4,4)} t2{(4,4,4)} t0,2{(4,4,4)} t0,1,2{(4,4,4)} s{(4,4,4)} h{(4,4,4)} hr{(4,4,4)}
Uniform duals
CDel 3.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node fh.pngCDel 4.png
H2chess 444b.png H2chess 444f.png H2chess 444a.png H2chess 444e.png H2chess 444c.png H2chess 444d.png H2checkers 444.png Uniform dual tiling 433-t0.png H2 tiling 288-1.png H2 tiling 266-2.png
V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V8.8.8 V3.4.3.4.3.4 V88 V(4,4)3

(5 3 3)[edit]

The (5 3 3) triangle group, Coxeter group [(5,3,3)], orbifold (*533) contains these uniform tilings.

Uniform (5,3,3) tilings
Symmetry: [(5,3,3)], (*533) [(5,3,3)]+, (533)
CDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.png
CDel label5.pngCDel branch 01rd.pngCDel split2.pngCDel node.png CDel label5.pngCDel branch 11.pngCDel split2.pngCDel node.png CDel label5.pngCDel branch 10ru.pngCDel split2.pngCDel node.png CDel label5.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png CDel label5.pngCDel branch.pngCDel split2.pngCDel node 1.png CDel label5.pngCDel branch 01rd.pngCDel split2.pngCDel node 1.png CDel label5.pngCDel branch 11.pngCDel split2.pngCDel node 1.png CDel label5.pngCDel branch hh.pngCDel split2.pngCDel node h.png
CDel node h1.pngCDel 10.pngCDel node.pngCDel 3.pngCDel node.png CDel node h0.pngCDel 10.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node h1.pngCDel 10.pngCDel node.pngCDel 3.pngCDel node.png CDel node h1.pngCDel 10.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 10.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h1.pngCDel 10.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 10.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 10.pngCDel node h.pngCDel 3.pngCDel node h.png
H2 tiling 335-1.png H2 tiling 335-3.png H2 tiling 335-2.png H2 tiling 335-6.png H2 tiling 335-4.png H2 tiling 335-5.png H2 tiling 335-7.png
h{10,3}
t0{(5,3,3)}
{(5,3,3)}
r{10,3}
t0,1{(5,3,3)}
r{(3,5,3)}
h{10,3}
t1{(5,3,3)}]]
{(3,3,5)}
h2{10,3}
t1,2{(5,3,3)}
r{(5,3,3)}
{3,10}
{(4,3,3)}]
{(3,5,3)}
h2{10,3}
t0,2{(4,3,3)}
r{(3,3,5)}
t{3,10}
t0,1,2{(4,3,3)}
t{(3,5,3)}
s{3,10}
 
s{(3,5,3)}
Uniform duals
CDel 3.pngCDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node fh.pngCDel 5.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.png
H2 tiling 555-7.png Hyperbolic domains 533.png
V(3.5)3 V3.10.3.10 V(3.5)3 V3.6.5.6 V(3.3)5 V3.6.5.6 V6.6.10 V3.3.3.3.3.5

(5 4 3)[edit]

The (5 4 3) triangle group, Coxeter group [(5,4,3)], orbifold (*543) contains these uniform tilings.

(5,4,3) uniform tilings
Symmetry: [(5,4,3)], (*543) [(5,4,3)]+, (543)
CDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.png
CDel branch 01rd.pngCDel split2-45.pngCDel node.png CDel branch 01rd.pngCDel split2-45.pngCDel node 1.png CDel branch.pngCDel split2-45.pngCDel node 1.png CDel branch 10ru.pngCDel split2-45.pngCDel node 1.png CDel branch 10ru.pngCDel split2-45.pngCDel node.png CDel branch 11.pngCDel split2-45.pngCDel node.png CDel branch 11.pngCDel split2-45.pngCDel node 1.png CDel branch hh.pngCDel split2-45.pngCDel node h.png
H2 tiling 345-1.png H2 tiling 345-3.png H2 tiling 345-2.png H2 tiling 345-6.png H2 tiling 345-4.png H2 tiling 345-5.png H2 tiling 345-7.png Uniform tiling 543-snub.png
t0{(5,4,3)}
{(5,4,3)}
t0,1{(5,4,3)}
r{(3,5,4)}
t1{(5,4,3)}
{(4,3,5)}
t1,2{(5,4,3)}
r{(5,4,3)}
t2{(5,4,3)}
{(3,5,4)}
t0,2{(5,4,3)}
r{(4,3,5)}
t0,1,2{(5,4,3)}
t{(5,4,3)}
s{(5,4,3)}
Uniform duals
CDel 3.pngCDel node f1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node fh.pngCDel 5.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.png
Hyperbolic domains 543.png
V(3.5)4 V3.10.4.10 V(4.5)3 V3.8.5.8 V(3.4)5 V4.6.5.6 V6.8.10 V3.5.3.4.3.3

(5 4 4)[edit]

The (5 4 4) triangle group, Coxeter group [(5,4,4)], orbifold (*544) contains these uniform tilings.

Uniform (5,4,4) tilings
Symmetry: [(5,4,4)]
(*544)
[(5,4,4)]+
(544)
[(5+,4,4)]
(5*22)
[(5,4,1+,4)]
(*5222)
CDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 4.png CDel 3.pngCDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node h.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 4.png
CDel label5.pngCDel branch 01rd.pngCDel split2-44.pngCDel node.png CDel label5.pngCDel branch 11.pngCDel split2-44.pngCDel node.png CDel label5.pngCDel branch 10ru.pngCDel split2-44.pngCDel node.png CDel label5.pngCDel branch 10ru.pngCDel split2-44.pngCDel node 1.png CDel label5.pngCDel branch.pngCDel split2-44.pngCDel node 1.png CDel label5.pngCDel branch 01rd.pngCDel split2-44.pngCDel node 1.png CDel label5.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.png CDel label5.pngCDel branch hh.pngCDel split2-44.pngCDel node h.png CDel label5.pngCDel branch.pngCDel split2-44.pngCDel node h.png CDel label5.pngCDel branch hh.pngCDel split2-44.pngCDel node.png
CDel node h1.pngCDel 10.pngCDel node.pngCDel 4.pngCDel node.png CDel node h0.pngCDel 10.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node h1.pngCDel 10.pngCDel node.pngCDel 4.pngCDel node.png CDel node h1.pngCDel 10.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node h0.pngCDel 10.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node h1.pngCDel 10.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node h0.pngCDel 10.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node h0.pngCDel 10.pngCDel node h.pngCDel 4.pngCDel node h.png
H2 tiling 445-1.png H2 tiling 445-3.png H2 tiling 445-2.png H2 tiling 445-6.png H2 tiling 445-4.png H2 tiling 445-5.png H2 tiling 445-7.png
t0{(5,4,4)} t0,1{(5,4,4)} t1{(5,4,4)} t1,2{(5,4,4)} t2{(5,4,4)} t0,2{(5,4,4)} t0,1,2{(5,4,4)} s{(4,5,4)} h{(4,5,4)} hr{(4,5,4)}
Uniform duals
CDel 3.pngCDel node f1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node f1.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node f1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node f1.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node fh.pngCDel 5.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.png CDel 3.pngCDel node.pngCDel 5.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node fh.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node fh.pngCDel 4.png
H2chess 445f.png H2chess 445c.png Hyperbolic domains 544.png
V(4.5)4 V4.10.4.10 V(4.5)4 V4.8.5.8 V(4.4)5 V4.8.5.8 V8.8.10 V3.4.3.4.3.5 V1010 V(4.4.5)2

(6 3 3)[edit]

The (6 3 3) triangle group, Coxeter group [(6,3,3)], orbifold (*633) contains these uniform tilings.

Uniform (6,3,3) tilings
Symmetry: [(6,3,3)], (*633) [(6,3,3)]+, (633)
CDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.png
CDel label6.pngCDel branch 01rd.pngCDel split2.pngCDel node.png CDel label6.pngCDel branch 11.pngCDel split2.pngCDel node.png CDel label6.pngCDel branch 10ru.pngCDel split2.pngCDel node.png CDel label6.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png CDel label6.pngCDel branch.pngCDel split2.pngCDel node 1.png CDel label6.pngCDel branch 01rd.pngCDel split2.pngCDel node 1.png CDel label6.pngCDel branch 11.pngCDel split2.pngCDel node 1.png CDel label6.pngCDel branch hh.pngCDel split2.pngCDel node h.png
CDel node h1.pngCDel 12.pngCDel node.pngCDel 3.pngCDel node.png CDel node h0.pngCDel 12.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node h1.pngCDel 12.pngCDel node.pngCDel 3.pngCDel node.png CDel node h1.pngCDel 12.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 12.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h1.pngCDel 12.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 12.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 12.pngCDel node h.pngCDel 3.pngCDel node h.png
H2 tiling 336-1.png H2 tiling 336-3.png H2 tiling 336-2.png H2 tiling 336-6.png H2 tiling 336-4.png H2 tiling 336-5.png H2 tiling 336-7.png
t0{(6,3,3)} t0,1{(6,3,3)} t1{(6,3,3)} t1,2{(6,3,3)} t2{(6,3,3)} t0,2{(6,3,3)} t0,1,2{(6,3,3)} s{(6,3,3)}
Uniform duals
CDel 3.pngCDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node fh.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.png
H2 tiling 666-7.png Hyperbolic domains 633.png
V(3.6)3 V3.12.3.12 V(3.6)3 V3.6.6.6 V(3.3)6 V3.6.6.6 V6.6.12 V3.3.3.3.3.6

(6 4 3)[edit]

The (6 4 3) triangle group, Coxeter group [(6,4,3)], orbifold (*643) contains these uniform tilings.

(6,4,3) uniform tilings
Symmetry: [(6,4,3)]
(*643)
[(6,4,3)]+
(643)
[(6,1+,4,3)]
(*3332)
[(6,4,3+)]
(3*32)
CDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.png CDel 3.pngCDel node.pngCDel 6.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 3.png
CDel branch 01rd.pngCDel split2-46.pngCDel node.png CDel branch 01rd.pngCDel split2-46.pngCDel node 1.png CDel branch.pngCDel split2-46.pngCDel node 1.png CDel branch 10ru.pngCDel split2-46.pngCDel node 1.png CDel branch 10ru.pngCDel split2-46.pngCDel node.png CDel branch 11.pngCDel split2-46.pngCDel node.png CDel branch 11.pngCDel split2-46.pngCDel node 1.png CDel branch hh.pngCDel split2-46.pngCDel node h.png CDel branch.pngCDel split2-46.pngCDel node h1.png CDel branch hh.pngCDel split2-46.pngCDel node.png
H2 tiling 346-1.png H2 tiling 346-3.png H2 tiling 346-2.png H2 tiling 346-6.png H2 tiling 346-4.png H2 tiling 346-5.png H2 tiling 346-7.png
t0{(6,4,3)} t0,1{(6,4,3)} t1{(6,4,3)} t1,2{(6,4,3)} t2{(6,4,3)} t0,2{(6,4,3)} t0,1,2{(6,4,3)} s{(6,4,3)} h{(6,4,3)} hr{(6,4,3)}
Uniform duals
CDel 3.pngCDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node fh.pngCDel 6.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.png CDel 3.pngCDel node.pngCDel 6.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 3.png CDel 3.pngCDel node fh.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node fh.pngCDel 3.png
H2chess 346a.png H2chess 346d.png Hyperbolic domains 643.png
V(3.6)4 V3.12.4.12 V(4.6)3 V3.8.6.8 V(3.4)6 V4.6.6.6 V6.8.12 V3.6.3.4.3.3 V(3.6.6)3 V4.(3.4)3

(6 4 4)[edit]

The (6 4 4) triangle group, Coxeter group [(6,4,4)], orbifold (*644) contains these uniform tilings.

Symmetry: [(6,4,4)], (*644)
CDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.png
CDel label6.pngCDel branch 01rd.pngCDel split2-44.pngCDel node.png CDel label6.pngCDel branch 11.pngCDel split2-44.pngCDel node.png CDel label6.pngCDel branch 10ru.pngCDel split2-44.pngCDel node.png CDel label6.pngCDel branch 10ru.pngCDel split2-44.pngCDel node 1.png CDel label6.pngCDel branch.pngCDel split2-44.pngCDel node 1.png CDel label6.pngCDel branch 01rd.pngCDel split2-44.pngCDel node 1.png CDel label6.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.png
H2 tiling 446-1.png H2 tiling 446-3.png H2 tiling 446-2.png H2 tiling 446-6.png H2 tiling 446-4.png H2 tiling 446-5.png H2 tiling 446-7.png
{(6,4,4)} t0,1{(6,4,4)} t1{(6,4,4)} t1,2{(6,4,4)} t2{(6,4,4)} t0,2{(6,4,4)} t0,1,2{(6,4,4)}
Uniform duals
CDel 3.pngCDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.png
H2chess 446b.png H2chess 446f.png H2chess 446a.png H2chess 446e.png H2chess 446c.png H2chess 446d.png H2checkers 446.png
V(4.6)4 V(4.12)2 V(4.6)4 V4.8.6.8 V412 V4.8.6.8 V8.8.12
Alternations
[(1+,6,4,4)]
(*4342)
[(6+,4,4)]
(6*22)
[(6,1+,4,4)]
(*4232)
[(6,4+,4)]
(4*32)
[(6,4,1+,4)]
(*6262)
[(6,4,4+)]
(4*32)
[(6,4,4)]+
(644)
CDel 3.pngCDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 6.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 4.png CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.pngCDel 4.png CDel 3.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 4.png CDel 3.pngCDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 4.png
CDel label6.pngCDel branch 0hr.pngCDel split2-44.pngCDel node.png CDel label6.pngCDel branch hh.pngCDel split2-44.pngCDel node.png CDel label6.pngCDel branch h0r.pngCDel split2-44.pngCDel node.png CDel label6.pngCDel branch h0r.pngCDel split2-44.pngCDel node h.png CDel label6.pngCDel branch.pngCDel split2-44.pngCDel node h.png CDel label6.pngCDel branch 0hr.pngCDel split2-44.pngCDel node h.png CDel label6.pngCDel branch hh.pngCDel split2-44.pngCDel node h.png
ht0,1{(6,4,4)} ht0,1{(6,4,4)} ht1{(6,4,4)} ht1,2{(6,4,4)} ht2{(6,4,4)} ht0,2{(6,4,4)} s{(6,4,4)}
Alternation duals
CDel 3.pngCDel node fh.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node fh.pngCDel 6.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 6.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 6.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.png CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node fh.pngCDel 4.png CDel 3.pngCDel node fh.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node fh.pngCDel 4.png CDel 3.pngCDel node fh.pngCDel 6.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.png
V(8.3.8)4 V(4.6.4)2 V(8.3.8)4 V3.46 V1212 V3.46 V4.3.4.3.6.3

Summary of tilings with finite triangular fundamental domains[edit]

For a table of all uniform hyperbolic tilings with fundamental domains (p q r), where 2 ≤ p,q,r ≤ 8.

See Template:Finite triangular hyperbolic tilings table

Quadrilateral domains[edit]

A quadrilateral domain has 9 generator point positions that define uniform tilings. Vertex figures are listed for general orbifold symmetry *pqrs, with 2-gonal faces degenerating into edges.

(3 2 2 2)[edit]

Example uniform tilings of *3222 symmetry

Quadrilateral fundamental domains also exist in the hyperbolic plane, with the *3222 orbifold ([∞,3,∞] Coxeter notation) as the smallest family. There are 9 generation locations for uniform tiling within quadrilateral domains. The vertex figure can be extracted from a fundamental domain as 3 cases (1) Corner (2) Mid-edge, and (3) Center. When generating points are corners adjacent to order-2 corners, degenerate {2} digon faces at those corners exist but can be ignored. Snub and alternated uniform tilings can also be generated (not shown) if a vertex figure contains only even-sided faces.

Coxeter diagrams of quadrilateral domains are treated as a degenerate tetrahedron graph with 2 of 6 edges labeled as infinity, or as dotted lines. A logical requirement of at least one of two parallel mirrors being active limits the uniform cases to 9, and other ringed patterns are not valid.

Symmetry *3222
CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png 64
Uniform tiling 64-t0.png
CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 01.png 6.6.4.4
Uniform tiling 6.6.4.4 (green).png
CDel branch 01.pngCDel 2a2b-cross.pngCDel nodes 01.png (3.4.4)2
Uniform tiling 3.4.4.3.4.4.png
CDel branch hh.pngCDel 2a2b-cross.pngCDel nodes 01.png 4.3.4.3.3.3
Uniform tiling 4.3.4.3.3.3.png
CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 10.png 6.6.4.4
Uniform tiling 6.6.4.4.png
CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png 6.4.4.4
Uniform tiling 4.4.4.6.png
CDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 11.png 3.4.4.4.4
Uniform tiling 3.4.4.4.4 (green).png
CDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 10.png (3.4.4)2
Uniform tiling 64-h1.png
CDel branch 01.pngCDel 2a2b-cross.pngCDel nodes 11.png 3.4.4.4.4
Uniform tiling 3.4.4.4.4.png
CDel branch.pngCDel 2a2b-cross.pngCDel nodes 11.png 46
Uniform tiling 64-t2.png

(3 2 3 2)[edit]

Similar H2 tilings in *3232 symmetry
Coxeter
diagrams
CDel node h0.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h0.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.png CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node h0.png
CDel labelh.pngCDel node.pngCDel split1-66.pngCDel nodes 10lu.png CDel branch.pngCDel split2-44.pngCDel node h1.png CDel node h1.pngCDel split1-66.pngCDel nodes.png CDel branch 10ru.pngCDel split2-44.pngCDel node.pngCDel labelh.png CDel node h1.pngCDel split1-66.pngCDel nodes 10lu.png CDel branch 10ru.pngCDel split2-44.pngCDel node h1.png CDel labelh.pngCDel node.pngCDel split1-66.pngCDel nodes 11.png CDel branch 11.pngCDel split2-44.pngCDel node.pngCDel labelh.png
CDel branch 11.pngCDel 2a2b-cross.pngCDel branch.png CDel branch 10.pngCDel 2a2b-cross.pngCDel branch 10.png CDel branch 10.pngCDel 2a2b-cross.pngCDel branch 11.png CDel branch 11.pngCDel 2a2b-cross.pngCDel branch 11.png
Vertex
figure
66 (3.4.3.4)2 3.4.6.6.4 6.4.6.4
Image Uniform tiling verf 666666.png Uniform tiling verf 34343434.png Uniform tiling verf 34664.png 3222-uniform tiling-verf4646.png
Dual Uniform tiling verf 666666b.png H2chess 246a.png

Ideal triangle domains[edit]

There are infinitely many triangle group families including infinite orders. This article shows uniform tilings in 9 families: (∞ 3 2), (∞ 4 2), (∞ ∞ 2), (∞ 3 3), (∞ 4 3), (∞ 4 4), (∞ ∞ 3), (∞ ∞ 4), and (∞ ∞ ∞).

(∞ 3 2)[edit]

The ideal (∞ 3 2) triangle group, Coxeter group [∞,3], orbifold (*∞32) contains these uniform tilings:

Noncompact hyperbolic uniform tilings in [∞,3] family
Symmetry: [∞,3], (*∞32) [∞,3]+
(∞32)
[1+,∞,3]
(*∞33)
[∞,3+]
(3*∞)
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node h0.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel labelinfin.pngCDel branch 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel labelinfin.pngCDel branch 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel labelinfin.pngCDel branch.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png =
CDel labelinfin.pngCDel branch 10ru.pngCDel split2.pngCDel node.png or CDel labelinfin.pngCDel branch 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel labelinfin.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png or CDel labelinfin.pngCDel branch 01rd.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png
= CDel labelinfin.pngCDel branch hh.pngCDel split2.pngCDel node h.png
H2 tiling 23i-1.png H2 tiling 23i-3.png H2 tiling 23i-2.png H2 tiling 23i-6.png H2 tiling 23i-4.png H2 tiling 23i-5.png H2 tiling 23i-7.png Uniform tiling i32-snub.png H2 tiling 33i-1.png
{∞,3} t{∞,3} r{∞,3} t{3,∞} {3,∞} rr{∞,3} tr{∞,3} sr{∞,3} h{∞,3} h2{∞,3} s{3,∞}
Uniform duals
CDel node f1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node fh.pngCDel 3.pngCDel node fh.png
H2 tiling 23i-4.png Ord-infin triakis triang til.png Ord3infin qreg rhombic til.png H2checkers 33i.png H2 tiling 23i-1.png Deltoidal triapeirogonal til.png H2checkers 23i.png Order-3-infinite floret pentagonal tiling.png Alternate order-3 apeirogonal tiling.png
V∞3 V3.∞.∞ V(3.∞)2 V6.6.∞ V3 V4.3.4.∞ V4.6.∞ V3.3.3.3.∞ V(3.∞)3 V3.3.3.3.3.∞

(∞ 4 2)[edit]

The ideal (∞ 4 2) triangle group, Coxeter group [∞,4], orbifold (*∞42) contains these uniform tilings:

Noncompact hyperbolic uniform tilings in [∞,4] family
Symmetry: [∞,4], (*∞42)
CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node 1.png
H2 tiling 24i-1.png H2 tiling 24i-3.png H2 tiling 24i-2.png H2 tiling 24i-6.png H2 tiling 24i-4.png H2 tiling 24i-5.png H2 tiling 24i-7.png
{∞,4} t{∞,4} r{∞,4} 2t{∞,4}=t{4,∞} 2r{∞,4}={4,∞} rr{∞,4} tr{∞,4}
Dual figures
CDel node f1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node f1.png
H2chess 24ib.png H2chess 24if.png H2chess 24ia.png H2chess 24ie.png H2chess 24ic.png H2chess 24id.png H2checkers 24i.png
V∞4 V4.∞.∞ V(4.∞)2 V8.8.∞ V4 V43.∞ V4.8.∞
Alternations
[1+,∞,4]
(*44∞)
[∞+,4]
(∞*2)
[∞,1+,4]
(*2∞2∞)
[∞,4+]
(4*∞)
[∞,4,1+]
(*∞∞2)
[(∞,4,2+)]
(2*2∞)
[∞,4]+
(∞42)
CDel node h1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png
= CDel branch 10ru.pngCDel split2-44.pngCDel node.png
CDel node h.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node h1.png
= CDel node.pngCDel split1-ii.pngCDel nodes 10lu.png
CDel node h.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node h.png CDel node h.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node h.png
h{∞,4} s{∞,4} hr{∞,4} s{4,∞} h{4,∞} hrr{∞,4} s{∞,4}
H2 tiling 44i-1.png Uniform tiling i42-h01.png H2 tiling 2ii-1.png Uniform tiling i42-snub.png
Alternation duals
CDel node fh.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 4.png