| 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. | |||||
Inhyperbolic geometry, auniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which hasregular polygons asfaces and isvertex-transitive (transitive on itsvertices, isogonal, i.e. there is anisometry mapping any vertex onto any other). It follows that all vertices arecongruent, and thetiling has a high degree of rotational and translationalsymmetry.
Uniform tilings can be identified by theirvertex configuration, a sequence of numbers representing the number of sides of the polygons around each vertex. For example, 7.7.7 represents theheptagonal tiling which has 3heptagons around each vertex. It is also regular since all the polygons are the same size, so it can also be given theSchläfli symbol {7,3}.
Uniform tilings may beregular (if also face- and edge-transitive), quasi-regular (if edge-transitive but not face-transitive) orsemi-regular (if neither edge- nor face-transitive). For right triangles (p q 2), there are two regular tilings, represented bySchläfli symbol {p,q} and {q,p}.
There are an infinite number of uniform tilings based on theSchwarz triangles (p q r) where1/p +1/q +1/r < 1, wherep,q,r are each orders of reflection symmetry at three points of thefundamental domain triangle – the symmetry group is a hyperbolictriangle group.
Each symmetry family contains 7 uniform tilings, defined by aWythoff symbol orCoxeter-Dynkin diagram, 7 representing combinations of 3 active mirrors. An 8th represents analternation operation, deleting alternate vertices from the highest form with all mirrors active.
Families withr = 2 containregular hyperbolic tilings, defined by aCoxeter group such as [7,3], [8,3], [9,3], ... [5,4], [6,4], ....
Hyperbolic families withr = 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 ofp,q orr can be replaced by ∞ which defines a paracompact hyperbolic triangle and creates uniform tilings with either infinite faces (calledapeirogons) 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 thePoincaré disk model for the hyperbolic plane). Three of them – (7 3 2), (5 4 2), and (4 3 3) – and no others, areminimal 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 adual uniform tiling, with many of them also given below.
There are infinitely many (p q 2)triangle group families. This article shows the regular tiling up top,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).
The simplest set of hyperbolic tilings are regular tilings {p,q}, which exist in a matrix with the regular polyhedra and Euclidean tilings. The regular tiling {p,q} has a dual tiling {q,p} across the diagonal axis of the table. Self-dual tilings {2,2},{3,3},{4,4},{5,5}, etc. pass down the diagonal of the table.
| Regular hyperbolic tiling table | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Spherical(improper/Platonic)/Euclidean/hyperbolic (Poincaré disk:compact/paracompact/noncompact) tessellations with theirSchläfli symbol | |||||||||||
| p \ q | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ... | ∞ | ... | iπ/λ |
| 2 |  {2,2}  |  {2,3} |  {2,4}  |  {2,5} |  {2,6} |  {2,7} |  {2,8}  |  {2,∞} |  {2,iπ/λ}  | ||
| 3 |  {3,2} |  (tetrahedron) {3,3} |  (octahedron) {3,4} |  (icosahedron) {3,5} |  (deltille) {3,6} |  {3,7} |  {3,8} |  {3,∞} |  {3,iπ/λ} | ||
| 4 |  {4,2} |  (cube) {4,3} |  (quadrille) {4,4} |  {4,5} |  {4,6} |  {4,7} |  {4,8} |  {4,∞} |  {4,iπ/λ} | ||
| 5 |  {5,2} | (dodecahedron) {5,3} |  {5,4} |  {5,5} |  {5,6} |  {5,7} |  {5,8} |  {5,∞} |  {5,iπ/λ} | ||
| 6 |  {6,2} | (hextille) {6,3} |  {6,4} |  {6,5} |  {6,6} |  {6,7} |  {6,8} |  {6,∞} |  {6,iπ/λ} | ||
| 7 | {7,2} | {7,3} |  {7,4} |  {7,5} |  {7,6} |  {7,7} |  {7,8} |  {7,∞} | {7,iπ/λ} | ||
| 8 | {8,2} |  {8,3} |  {8,4} |  {8,5} |  {8,6} |  {8,7} |  {8,8} |  {8,∞} | {8,iπ/λ} | ||
| ... | |||||||||||
| ∞ |  {∞,2} | {∞,3} |  {∞,4} |  {∞,5} |  {∞,6} |  {∞,7} |  {∞,8} |  {∞,∞} |  {∞,iπ/λ} | ||
| ... | |||||||||||
| iπ/λ |  {iπ/λ,2} |  {iπ/λ,3} |  {iπ/λ,4} |  {iπ/λ,5} |  {iπ/λ,6} | {iπ/λ,7} | {iπ/λ,8} |  {iπ/λ,∞} |  {iπ/λ, iπ/λ} | ||
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) | ||||||||||
| | | | | | |  | ||||
| | |  |  | | |  | ||||
| {7,3} | t{7,3} | r{7,3} | t{3,7} | {3,7} | rr{7,3} | tr{7,3} | sr{7,3} | ||||
| Uniform duals | |||||||||||
 | | | | | | |  | ||||
|  |  |  | |  |  |  | ||||
| V73 | V3.14.14 | V3.7.3.7 | V6.6.7 | V37 | V3.4.7.4 | V4.6.14 | V3.3.3.3.7 | ||||
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} | |||
| | | | | | | | | |||||
    | |  | |   or | or |  | |||||||
|  |   |   |   |  |  |  |   |   |   | |||
| 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 | |||
| | | | | | | | | | | |||
|  |  |  | |  |  |  |  |  |  | |||
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) | ||||||||
| | | | | | | | | | ||
|  |  |  |  |  |  |  |  |  | ||
| {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 | |||||||||||
| | | | | | | | | | ||
|  |  |  | |  |  |  |  | |||
| 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 | ||
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) | |||||||||||
=    =  =  | =  | = =  = | = | = = =  | = | | |||||
 |  |  |  |  |  |  | |||||
| {6,4} | t{6,4} | r{6,4} | t{4,6} | {4,6} | rr{6,4} | tr{6,4} | |||||
| Uniform duals | |||||||||||
| | | | | | | |||||
 |  |  |  |  |  |  | |||||
| 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) | |||||
= | =  | =   | = | =  | =  | | |||||
 |  |  |  |  |  |  | |||||
| h{6,4} | s{6,4} | hr{6,4} | s{4,6} | h{4,6} | hrr{6,4} | sr{6,4} | |||||
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) | ||||||||
| | | | | | | | | | ||
|  |  |  |  |  |  |  |  |  | ||
| {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 | |||||||||||
| | | | | | | | | | ||
|  |  |  | |  |  | | ||||
| 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 | ||
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) | |||||||||||
=  = =  | = | = = = | = | = = | = | | |||||
 |  |  |  |  |  |  | |||||
| {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 | |||||||||||
| | | | | | | |||||
 |  |  |  |  |  |  | |||||
| 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) | |||||
= | =  | = | = | = | = | | |||||
 |  |  |  |  | |  | |||||
| h{8,4} | s{8,4} | hr{8,4} | s{4,8} | h{4,8} | hrr{8,4} | sr{8,4} | |||||
| Alternation duals | |||||||||||
| | | | | | | |||||
 |  | | | | |||||||
| 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 | |||||
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) | ||||||||||
= | = | = | = | = | = | = | = | ||||
|  |  |  | |  |  |  | ||||
| Order-5 pentagonal tiling {5,5} | Truncated order-5 pentagonal tiling t{5,5} | Order-4 pentagonal tiling r{5,5} | Truncated order-5 pentagonal tiling 2t{5,5} = t{5,5} | Order-5 pentagonal tiling 2r{5,5} = {5,5} | Tetrapentagonal tiling rr{5,5} | Truncated order-4 pentagonal tiling tr{5,5} | Snub pentapentagonal tiling sr{5,5} | ||||
| Uniform duals | |||||||||||
| | | | | | | | ||||
|  | | | | | | |||||
| Order-5 pentagonal tiling V5.5.5.5.5 | V5.10.10 | Order-5 square tiling V5.5.5.5 | V5.10.10 | Order-5 pentagonal tiling V5.5.5.5.5 | V4.5.4.5 | V4.10.10 | V3.3.5.3.5 | ||||
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) | ||||||||
| | | | | | | | | | ||
 |  |  |  |  |  |  |  |  | |||
| {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 | |||||||||||
| | | | | | | | | | ||
 |  |  |  | |  |  | |||||
| 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 | ||
The(6 6 2)triangle group,Coxeter group [6,6],orbifold (*662) contains these uniform tilings:
| Uniform hexahexagonal tilings | ||||||
|---|---|---|---|---|---|---|
| Symmetry:[6,6], (*662) | ||||||
=  = | = = | = = | = = | = = | = = | = = |
 |  |  |  |  |  |  |
| {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 | ||||||
| | | | | | |
 |  |  |  |  |  |  |
| 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) |
| = | | = | | = | | |
| | | | | | |
 |  | | |  | ||
| h{6,6} | s{6,6} | hr{6,6} | s{6,6} | h{6,6} | hrr{6,6} | sr{6,6} |
The(8 6 2) triangle group, Coxeter group [8,6],orbifold (*862) contains these uniform tilings.
| Uniform octagonal/hexagonal tilings | ||||||
|---|---|---|---|---|---|---|
| Symmetry:[8,6], (*862) | ||||||
| | | | | | |
 |  |  |  |  |  |  |
| {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 | ||||||
| | | | | | |
 |  |  |  |  |  |  |
| 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) |
| | | | | | |
 |  |  | ||||
| h{8,6} | s{8,6} | hr{8,6} | s{6,8} | h{6,8} | hrr{8,6} | sr{8,6} |
| Alternation duals | ||||||
| | | | | | |
 | ||||||
| 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 |
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) | ||||||||||
= = | = = | = = | = = | = = | = = | = = | = = | ||||
|  |  |  | |  |  |  | ||||
| {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 | |||||||||||
| | | | | | | | ||||
|  | | | | | | |||||
| V77 | V7.14.14 | V7.7.7.7 | V7.14.14 | V77 | V4.7.4.7 | V4.14.14 | V3.3.7.3.7 | ||||
The(8 8 2)triangle group,Coxeter group [8,8],orbifold (*882) contains these uniform tilings:
| Uniform octaoctagonal tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry:[8,8], (*882) | |||||||||||
= = | = = | = = | = = | = = | = = | = = | |||||
 |  |  |  |  |  |  | |||||
| {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 | |||||||||||
| | | | | | | |||||
 |  |  |  |  |  |  | |||||
| 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) | |||||
| = | | = | | = | = = | = = | |||||
 | | | |  | |||||||
| h{8,8} | s{8,8} | hr{8,8} | s{8,8} | h{8,8} | hrr{8,8} | sr{8,8} | |||||
| Alternation duals | |||||||||||
| | | | | | | |||||
| | ||||||||||
| 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 | |||||
There are infinitely many generaltriangle 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).
The(4 3 3)triangle group,Coxeter group [(4,3,3)],orbifold (*433) contains these uniform tilings. Without right angles in the fundamental triangle, theWythoff constructions are slightly different. For instance in the (4,3,3)triangle family, thesnub form has six polygons around a vertex and its dual has hexagons rather than pentagons. In general thevertex 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.
| Symmetry: [(4,3,3)],(*433) | [(4,3,3)]+, (433) | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| | | | | | | | |||
| | | | | | | | |||
 |  |  |  |  |  |  | | |||
| h{8,3} t0(4,3,3) | r{3,8}1/2 t0,1(4,3,3) | h{8,3} t1(4,3,3) | h2{8,3} t1,2(4,3,3) | {3,8}1/2 t2(4,3,3) | h2{8,3} t0,2(4,3,3) | t{3,8}1/2 t0,1,2(4,3,3) | s{3,8}1/2 s(4,3,3) | |||
| Uniform duals | ||||||||||
| | |  | | | | | |||
| 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 | |||
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) | |||||||
| | | | | | | | | | |
| | | | | | | | | | |
|  | |  |  |  |  |  | | |  |
| h{6,4} t0(4,4,3) | h2{6,4} t0,1(4,4,3) | {4,6}1/2 t1(4,4,3) | h2{6,4} t1,2(4,4,3) | h{6,4} t2(4,4,3) | r{6,4}1/2 t0,2(4,4,3) | t{4,6}1/2 t0,1,2(4,4,3) | s{4,6}1/2 s(4,4,3) | hr{4,6}1/2 hr(4,3,4) | h{4,6}1/2 h(4,3,4) | q{4,6} h1(4,3,4) |
| Uniform duals | ||||||||||
|  |  | | |||||||
| 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 |
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) | ||||||||
| | | | | | | | | | ||
 |  |  |  |  |  |  | | |  | ||
| t0(4,4,4) h{8,4} | t0,1(4,4,4) h2{8,4} | t1(4,4,4) {4,8}1/2 | t1,2(4,4,4) h2{8,4} | t2(4,4,4) h{8,4} | t0,2(4,4,4) r{4,8}1/2 | t0,1,2(4,4,4) t{4,8}1/2 | s(4,4,4) s{4,8}1/2 | h(4,4,4) h{4,8}1/2 | hr(4,4,4) hr{4,8}1/2 | ||
| Uniform duals | |||||||||||
 |  |  |  |  |  |  | | | | ||
| 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 | ||
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) | ||||||||||
 | | | | | | | | ||||
 | | | | | | | | ||||
 |  |  |  |  |  |  |  | ||||
| h{10,3} t0(5,3,3) | r{3,10}1/2 t0,1(5,3,3) | h{10,3} t1(5,3,3) | h2{10,3} t1,2(5,3,3) | {3,10}1/2 t2(5,3,3) | h2{10,3} t0,2(5,3,3) | t{3,10}1/2 t0,1,2(5,3,3) | s{3,10}1/2 ht0,1,2(5,3,3) | ||||
| Uniform duals | |||||||||||
 |  | ||||||||||
| 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 | ||||
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) | ||||||||||
 | | | | | | | | ||||
 |  |  |  |  |  |  |  | ||||
| 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 | |||||||||||
 | |||||||||||
| 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 | ||||
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) | ||||||||
| | | | | | | | | | ||
| | | | | | | | | | ||
 |  |  |  |  |  |  |  | ||||
| t0(5,4,4) h{10,4} | t0,1(5,4,4) r{4,10}1/2 | t1(5,4,4) h{10,4} | t1,2(5,4,4) h2{10,4} | t2(5,4,4) {4,10}1/2 | t0,2(5,4,4) h2{10,4} | t0,1,2(5,4,4) t{4,10}1/2 | s(4,5,4) s{4,10}1/2 | h(4,5,4) h{4,10}1/2 | hr(4,5,4) hr{4,10}1/2 | ||
| Uniform duals | |||||||||||
 |  |  | |||||||||
| 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 | ||
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) | ||||||||||
 | | | | | | | | ||||
 | | | | | | | | ||||
 |  |  |  |  |  |  |  | ||||
| t0{(6,3,3)} h{12,3} | t0,1{(6,3,3)} r{3,12}1/2 | t1{(6,3,3)} h{12,3} | t1,2{(6,3,3)} h2{12,3} | t2{(6,3,3)} {3,12}1/2 | t0,2{(6,3,3)} h2{12,3} | t0,1,2{(6,3,3)} t{3,12}1/2 | s{(6,3,3)} s{3,12}1/2 | ||||
| Uniform duals | |||||||||||
 |  | ||||||||||
| V(3.6)3 | V3.12.3.12 | V(3.6)3 | V3.6.6.6 | V(3.3)6 {12,3} | V3.6.6.6 | V6.6.12 | V3.3.3.3.3.6 | ||||
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) | ||||||
 | | | | | | | | = | |
 |  |  |  |  |  |  |  | ||
| 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 | |||||||||
 |  |  | |||||||
| 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 |
The(6 4 4)triangle group,Coxeter group [(6,4,4)],orbifold (*644) contains these uniform tilings.
| 6-4-4 uniform tilings | |||||||
|---|---|---|---|---|---|---|---|
| Symmetry: [(6,4,4)], (*644) | (644) | ||||||
| | | | | | | |
 |  |  |  |  |  |  |  |
| (6,4,4) h{12,4} | t0,1(6,4,4) r{4,12}1/2 | t1(6,4,4) h{12,4} | t1,2(6,4,4) h2{12,4} | t2(6,4,4) {4,12}1/2 | t0,2(6,4,4) h2{12,4} | t0,1,2(6,4,4) t{4,12}1/2 | s(6,4,4) s{4,12}1/2 |
| Uniform duals | |||||||
 |  |  |  |  |  |  | |
| V(4.6)4 | V(4.12)2 | V(4.6)4 | V4.8.6.8 | V412 | V4.8.6.8 | V8.8.12 | V4.6.4.6.6.6 |
Quadrilateral fundamental domains also exist in the hyperbolic plane, with the*3222orbifold ([∞,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 andalternated 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 degeneratetetrahedron 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.
| Uniform tilings in symmetry *3222 | ||||
|---|---|---|---|---|
| 64 |  6.6.4.4.png) |  (3.4.4)2 | 4.3.4.3.3.3 | |
6.6.4.4 | 6.4.4.4 | 3.4.4.4.4.png) | ||
| (3.4.4)2 | 3.4.4.4.4 | 46 | ||
| Similar H2 tilings in *3232 symmetry | ||||||||
|---|---|---|---|---|---|---|---|---|
| Coxeter diagrams | | | | | ||||
 | | | | | | | | |
| | | | |||||
| Vertex figure | 66 | (3.4.3.4)2 | 3.4.6.6.4 | 6.4.6.4 | ||||
| Image |  | | |  | ||||
| Dual |  | | ||||||
There are infinitely manytriangle 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 (∞ ∞ ∞).
The ideal(∞ 3 2)triangle group,Coxeter group [∞,3],orbifold (*∞32) contains these uniform tilings:
| Paracompact uniform tilings in [∞,3] family | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry:[∞,3], (*∞32) | [∞,3]+ (∞32) | [1+,∞,3] (*∞33) | [∞,3+] (3*∞) | |||||||
| | | | | | | | | | |
=  | = | = | | = or | = or | = | ||||
| | |  | | | |  |  |  | |
| {∞,3} | t{∞,3} | r{∞,3} | t{3,∞} | {3,∞} | rr{∞,3} | tr{∞,3} | sr{∞,3} | h{∞,3} | h2{∞,3} | s{3,∞} |
| Uniform duals | ||||||||||
| | | | | | | | | | |
|  |  |  | |  |  |  |  | ||
| 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.∞ | |
The ideal(∞ 4 2)triangle group,Coxeter group [∞,4],orbifold (*∞42) contains these uniform tilings:
| Paracompact uniform tilings in [∞,4] family | |||||||
|---|---|---|---|---|---|---|---|
| | | | | | | |
|  |  |  | |  |  | |
| {∞,4} | t{∞,4} | r{∞,4} | 2t{∞,4}=t{4,∞} | 2r{∞,4}={4,∞} | rr{∞,4} | tr{∞,4} | |
| Dual figures | |||||||
| | | | | | | |
 |  |  |  |  |  |  | |
| 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) | |
= | | | | =  | | | |
| h{∞,4} | s{∞,4} | hr{∞,4} | s{4,∞} | h{4,∞} | hrr{∞,4} | s{∞,4} | |
 |  | |  | ||||
| Alternation duals | |||||||
| | | | | | | |
 |  | ||||||
| V(∞.4)4 | V3.(3.∞)2 | V(4.∞.4)2 | V3.∞.(3.4)2 | V∞∞ | V∞.44 | V3.3.4.3.∞ | |
The ideal(∞ 5 2)triangle group,Coxeter group [∞,5],orbifold (*∞52) contains these uniform tilings:
| Paracompact uniform apeirogonal/pentagonal tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [∞,5], (*∞52) | [∞,5]+ (∞52) | [1+,∞,5] (*∞55) | [∞,5+] (5*∞) | ||||||||
| | | | | | | | | | | |
|  |  |  | |  |  |  |  | |||
| {∞,5} | t{∞,5} | r{∞,5} | 2t{∞,5}=t{5,∞} | 2r{∞,5}={5,∞} | rr{∞,5} | tr{∞,5} | sr{∞,5} | h{∞,5} | h2{∞,5} | s{5,∞} | |
| Uniform duals | |||||||||||
| | | | | | | | | | | |
 |  | |  | ||||||||
| V∞5 | V5.∞.∞ | V5.∞.5.∞ | V∞.10.10 | V5∞ | V4.5.4.∞ | V4.10.∞ | V3.3.5.3.∞ | V(∞.5)5 | V3.5.3.5.3.∞ | ||
The ideal(∞ ∞ 2)triangle group,Coxeter group [∞,∞],orbifold (*∞∞2) contains these uniform tilings:
| Paracompact uniform tilings in [∞,∞] family | ||||||
|---|---|---|---|---|---|---|
= = | = = | = =  | = = | = = | = | = |
|  |  |  | |  |  |
| {∞,∞} | t{∞,∞} | r{∞,∞} | 2t{∞,∞}=t{∞,∞} | 2r{∞,∞}={∞,∞} | rr{∞,∞} | tr{∞,∞} |
| Dual tilings | ||||||
| | | | | | |
 |  |  |  |  |  |  |
| V∞∞ | V∞.∞.∞ | V(∞.∞)2 | V∞.∞.∞ | V∞∞ | V4.∞.4.∞ | V4.4.∞ |
| Alternations | ||||||
| [1+,∞,∞] (*∞∞2) | [∞+,∞] (∞*∞) | [∞,1+,∞] (*∞∞∞∞) | [∞,∞+] (∞*∞) | [∞,∞,1+] (*∞∞2) | [(∞,∞,2+)] (2*∞∞) | [∞,∞]+ (2∞∞) |
| | | | | | |
| | |  | |  | |
| h{∞,∞} | s{∞,∞} | hr{∞,∞} | s{∞,∞} | h2{∞,∞} | hrr{∞,∞} | sr{∞,∞} |
| Alternation duals | ||||||
| | | | | | |
| | |  | |||
| V(∞.∞)∞ | V(3.∞)3 | V(∞.4)4 | V(3.∞)3 | V∞∞ | V(4.∞.4)2 | V3.3.∞.3.∞ |
The ideal(∞ 3 3)triangle group,Coxeter group [(∞,3,3)],orbifold (*∞33) contains these uniform tilings.
| Paracompact hyperbolic uniform tilings in [(∞,3,3)] family | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [(∞,3,3)], (*∞33) | [(∞,3,3)]+, (∞33) | ||||||||||
| | | | | | | | ||||
| | | | | | | | ||||
|  | |  |  |  |  | | ||||
| (∞,∞,3) | t0,1(∞,3,3) | t1(∞,3,3) | t1,2(∞,3,3) | t2(∞,3,3) | t0,2(∞,3,3) | t0,1,2(∞,3,3) | s(∞,3,3) | ||||
| Dual tilings | |||||||||||
| | | | | | | | ||||
| | | | | | | | ||||
| | ||||||||||
| V(3.∞)3 | V3.∞.3.∞ | V(3.∞)3 | V3.6.∞.6 | V(3.3)∞ | V3.6.∞.6 | V6.6.∞ | V3.3.3.3.3.∞ | ||||
The ideal(∞ 4 3)triangle group,Coxeter group [(∞,4,3)],orbifold (*∞43) contains these uniform tilings:
| Paracompact hyperbolic uniform tilings in [(∞,4,3)] family | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [(∞,4,3)] (*∞43) | [(∞,4,3)]+ (∞43) | [(∞,4,3+)] (3*4∞) | [(∞,1+,4,3)] (*∞323) | ||||||||
 | | | | | | | | | = | ||
 |  |  |  |  |  |  |  | ||||
| (∞,4,3) | t0,1(∞,4,3) | t1(∞,4,3) | t1,2(∞,4,3) | t2(∞,4,3) | t0,2(∞,4,3) | t0,1,2(∞,4,3) | s(∞,4,3) | ht0,2(∞,4,3) | ht1(∞,4,3) | ||
| Dual tilings | |||||||||||
 |  |  | |||||||||
| V(3.∞)4 | V3.∞.4.∞ | V(4.∞)3 | V3.8.∞.8 | V(3.4)∞ | 4.6.∞.6 | V6.8.∞ | V3.3.3.4.3.∞ | V(4.3.4)2.∞ | V(6.∞.6)3 | ||
The ideal(∞ 4 4)triangle group,Coxeter group [(∞,4,4)],orbifold (*∞44) contains these uniform tilings.
| Paracompact hyperbolic uniform tilings in [(4,4,∞)] family | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [(4,4,∞)], (*44∞) | (44∞) | ||||||||||
| | | | | | | | ||||
|  |  |  |  |  |  | | ||||
| (4,4,∞) h{∞,4} | t0,1(4,4,∞) r{4,∞}1/2 | t1(4,4,∞) h{4,∞}1/2 | t1,2(4,4,∞) h2{∞,4} | t2(4,4,∞) {4,∞}1/2 | t0,2(4,4,∞) h2{∞,4} | t0,1,2(4,4,∞) t{4,∞}1/2 | s(4,4,∞) s{4,∞}1/2 | ||||
| Dual tilings | |||||||||||
|  |  |  |  |  |  | |||||
| V(4.∞)4 | V4.∞.4.∞ | V(4.∞)4 | V4.8.∞.8; | V4∞ | V4.8.∞.8; | V8.8.∞ | V3.4.3.4.3.∞ | ||||
The ideal(∞ ∞ 3)triangle group,Coxeter group [(∞,∞,3)],orbifold (*∞∞3) contains these uniform tilings.
| Paracompact hyperbolic uniform tilings in [(∞,∞,3)] family | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [(∞,∞,3)], (*∞∞3) | [(∞,∞,3)]+ (∞∞3) | [(∞,∞,3+)] (3*∞∞) | [(∞,1+,∞,3)] (*∞3∞3) | ||||||
| | | | | | | | | = |
| | | | | | | | | |
 |  |  |  |  |  |  |  |  | |
| (∞,∞,3) h{6,∞} | t0,1(∞,∞,3) h2{6,∞} | t1(∞,∞,3) {∞,6}1/2 | t1,2(∞,∞,3) h2{6,∞} | t2(∞,∞,3) h{6,∞} | t0,2(∞,∞,3) r{∞,6}1/2 | t0,1,2(∞,∞,3) t{∞,6}1/2 | s(∞,∞,3) s{∞,6}1/2 | hr0,2(∞,∞,3) hr{∞,6}1/2 | hr1(∞,∞,3) h{∞,6}1/2 |
| Dual tilings | |||||||||
 |  |  |  | ||||||
| V(3.∞)∞ | V3.∞.∞.∞ | V(∞.∞)3 | V3.∞.∞.∞ | V(3.∞)∞ | V(6.∞)2 | V6.∞.∞ | V3.∞.3.∞.3.3 | V(3.4.∞.4)2 | V(∞.6)6 |
The ideal(∞ ∞ 4)triangle group,Coxeter group [(∞,∞,4)],orbifold (*∞∞4) contains these uniform tilings.
| Paracompact hyperbolic uniform tilings in [(∞,∞,4)] family | ||||||
|---|---|---|---|---|---|---|
| Symmetry: [(∞,∞,4)], (*∞∞4) | ||||||
| | | | | | |
| | | | | | |
 |  |  |  |  |  |  |
| (∞,∞,4) h{8,∞} | t0,1(∞,∞,4) h2{8,∞} | t1(∞,∞,4) {∞,8} | t1,2(∞,∞,4) h2{∞,8} | t2(∞,∞,4) h{8,∞} | t0,2(∞,∞,4) r{∞,8} | t0,1,2(∞,∞,4) t{∞,8} |
| Dual tilings | ||||||
 |  |  |  |  |  |  |
| V(4.∞)∞ | V∞.∞.∞.4 | V∞4 | V∞.∞.∞.4 | V(4.∞)∞ | V∞.∞.∞.4 | V∞.∞.8 |
| Alternations | ||||||
| [(1+,∞,∞,4)] (*2∞∞∞) | [(∞+,∞,4)] (∞*2∞) | [(∞,1+,∞,4)] (*2∞∞∞) | [(∞,∞+,4)] (∞*2∞) | [(∞,∞,1+,4)] (*2∞∞∞) | [(∞,∞,4+)] (2*∞∞) | [(∞,∞,4)]+ (4∞∞) |
| | | | | | |
 | | |  | | | |
| | | | |||
| Alternation duals | ||||||
| | | ||||
| V∞∞ | V∞.44 | V(∞.4)4 | V∞.44 | V∞∞ | V∞.44 | V3.∞.3.∞.3.4 |
The ideal(∞ ∞ ∞)triangle group,Coxeter group [(∞,∞,∞)],orbifold (*∞∞∞) contains these uniform tilings.
| Paracompact uniform tilings in [(∞,∞,∞)] family | ||||||
|---|---|---|---|---|---|---|
| | | | | | |
| | | | | | |
 |  |  |  |  |  |  |
| (∞,∞,∞) h{∞,∞} | r(∞,∞,∞) h2{∞,∞} | (∞,∞,∞) h{∞,∞} | r(∞,∞,∞) h2{∞,∞} | (∞,∞,∞) h{∞,∞} | r(∞,∞,∞) r{∞,∞} | t(∞,∞,∞) t{∞,∞} |
| Dual tilings | ||||||
 |  |  |  |  |  |  |
| V∞∞ | V∞.∞.∞.∞ | V∞∞ | V∞.∞.∞.∞ | V∞∞ | V∞.∞.∞.∞ | V∞.∞.∞ |
| Alternations | ||||||
| [(1+,∞,∞,∞)] (*∞∞∞∞) | [∞+,∞,∞)] (∞*∞) | [∞,1+,∞,∞)] (*∞∞∞∞) | [∞,∞+,∞)] (∞*∞) | [(∞,∞,∞,1+)] (*∞∞∞∞) | [(∞,∞,∞+)] (∞*∞) | [∞,∞,∞)]+ (∞∞∞) |
| | | | | | |
| | | | | |  |
| Alternation duals | ||||||
| | | | | | |
| V(∞.∞)∞ | V(∞.4)4 | V(∞.∞)∞ | V(∞.4)4 | V(∞.∞)∞ | V(∞.4)4 | V3.∞.3.∞.3.∞ |
For a table of all uniform hyperbolic tilings with fundamental domains (p q r), where 2 ≤p,q,r ≤ 8, and one or more as ∞.
| Infinite triangular hyperbolic tilings | |||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| (p q r) | t0 | h0 | t01 | h01 | t1 | h1 | t12 | h12 | t2 | h2 | t02 | h02 | t012 | s | |||||
(∞ 3 2) | t0{∞,3} ∞3 | h0{∞,3} (3.∞)3 | t01{∞,3} ∞.3.∞ | t1{∞,3} (3.∞)2 | t12{∞,3} 6.∞.6 | h12{∞,3} 3.3.3.∞.3.3 | t2{∞,3} 3∞ | t02{∞,3} 3.4.∞.4 | t012{∞,3} 4.6.∞ | s{∞,3} 3.3.3.3.∞ | |||||||||
(∞ 4 2) | t0{∞,4} ∞4 | h0{∞,4} (4.∞)4 | t01{∞,4} ∞.4.∞ | h01{∞,4} 3.∞.3.3.∞ | t1{∞,4} (4.∞)2 | h1{∞,4} (4.4.∞)2 | t12{∞,4} 8.∞.8 | h12{∞,4} 3.4.3.∞.3.4 | t2{∞,4} 4∞ | h2{∞,4} ∞∞ | t02{∞,4} 4.4.∞.4 | h02{∞,4} 4.4.4.∞.4 | t012{∞,4} 4.8.∞ | s{∞,4} 3.3.4.3.∞ | |||||
(∞ 5 2) | t0{∞,5} ∞5 | h0{∞,5} (5.∞)5 | t01{∞,5} ∞.5.∞ | t1{∞,5} (5.∞)2 | t12{∞,5} 10.∞.10 | h12{∞,5} 3.5.3.∞.3.5 | t2{∞,5} 5∞ | t02{∞,5} 5.4.∞.4 | t012{∞,5} 4.10.∞ | s{∞,5} 3.3.5.3.∞ | |||||||||
 (∞ 6 2) | t0{∞,6} ∞6 | h0{∞,6} (6.∞)6 | t01{∞,6} ∞.6.∞ | h01{∞,6} 3.∞.3.3.3.∞ | t1{∞,6} (6.∞)2 | h1{∞,6} (4.3.4.∞)2 | t12{∞,6} 12.∞.12 | h12{∞,6} 3.6.3.∞.3.6 | t2{∞,6} 6∞ | h2{∞,6} (∞.3)∞ | t02{∞,6} 6.4.∞.4 | h02{∞,6} 4.3.4.4.∞.4 | t012{∞,6} 4.12.∞ | s{∞,6} 3.3.6.3.∞ | |||||
 (∞ 7 2) | t0{∞,7} ∞7 | h0{∞,7} (7.∞)7 | t01{∞,7} ∞.7.∞ | t1{∞,7} (7.∞)2 | t12{∞,7} 14.∞.14 | h12{∞,7} 3.7.3.∞.3.7 | t2{∞,7} 7∞ | t02{∞,7} 7.4.∞.4 | t012{∞,7} 4.14.∞ | s{∞,7} 3.3.7.3.∞ | |||||||||
 (∞ 8 2) | t0{∞,8} ∞8 | h0{∞,8} (8.∞)8 | t01{∞,8} ∞.8.∞ | h01{∞,8} 3.∞.3.4.3.∞ | t1{∞,8} (8.∞)2 | h1{∞,8} (4.4.4.∞)2 | t12{∞,8} 16.∞.16 | h12{∞,8} 3.8.3.∞.3.8 | t2{∞,8} 8∞ | h2{∞,8} (∞.4)∞ | t02{∞,8} 8.4.∞.4 | h02{∞,8} 4.4.4.4.∞.4 | t012{∞,8} 4.16.∞ | s{∞,8} 3.3.8.3.∞ | |||||
(∞ ∞ 2) | t0{∞,∞} ∞∞ | h0{∞,∞} (∞.∞)∞ | t01{∞,∞} ∞.∞.∞ | h01{∞,∞} 3.∞.3.∞.3.∞ | t1{∞,∞} ∞4 | h1{∞,∞} (4.∞)4 | t12{∞,∞} ∞.∞.∞ | h12{∞,∞} 3.∞.3.∞.3.∞ | t2{∞,∞} ∞∞ | h2{∞,∞} (∞.∞)∞ | t02{∞,∞} (∞.4)2 | h02{∞,∞} (4.∞.4)2 | t012{∞,∞} 4.∞.∞ | s{∞,∞} 3.3.∞.3.∞ | |||||
(∞ 3 3) | t0(∞,3,3) (∞.3)3 | t01(∞,3,3) (3.∞)2 | t1(∞,3,3) (3.∞)3 | t12(∞,3,3) 3.6.∞.6 | t2(∞,3,3) 3∞ | t02(∞,3,3) 3.6.∞.6 | t012(∞,3,3) 6.6.∞ | s(∞,3,3) 3.3.3.3.3.∞ | |||||||||||
(∞ 4 3) | t0(∞,4,3) (∞.3)4 | t01(∞,4,3) 3.∞.4.∞ | t1(∞,4,3) (4.∞)3 | h1(∞,4,3) (6.6.∞)3 | t12(∞,4,3) 3.8.∞.8 | t2(∞,4,3) (4.3)∞ | t02(∞,4,3) 4.6.∞.6 | h02(∞,4,3) 4.4.3.4.∞.4.3 | t012(∞,4,3) 6.8.∞ | s(∞,4,3) 3.3.3.4.3.∞ | |||||||||
 (∞ 5 3) | t0(∞,5,3) (∞.3)5 | t01(∞,5,3) 3.∞.5.∞ | t1(∞,5,3) (5.∞)3 | t12(∞,5,3) 3.10.∞.10 | t2(∞,5,3) (5.3)∞ | t02(∞,5,3) 5.6.∞.6 | t012(∞,5,3) 6.10.∞ | s(∞,5,3) 3.3.3.5.3.∞ | |||||||||||
 (∞ 6 3) | t0(∞,6,3) (∞.3)6 | t01(∞,6,3) 3.∞.6.∞ | t1(∞,6,3) (6.∞)3 | h1(∞,6,3) (6.3.6.∞)3 | t12(∞,6,3) 3.12.∞.12 | t2(∞,6,3) (6.3)∞ | t02(∞,6,3) 6.6.∞.6 | h02(∞,6,3) 4.3.4.3.4.∞.4.3 | t012(∞,6,3) 6.12.∞ | s(∞,6,3) 3.3.3.6.3.∞ | |||||||||
 (∞ 7 3) | t0(∞,7,3) (∞.3)7 | t01(∞,7,3) 3.∞.7.∞ | t1(∞,7,3) (7.∞)3 | t12(∞,7,3) 3.14.∞.14 | t2(∞,7,3) (7.3)∞ | t02(∞,7,3) 7.6.∞.6 | t012(∞,7,3) 6.14.∞ | s(∞,7,3) 3.3.3.7.3.∞ | |||||||||||
 (∞ 8 3) | t0(∞,8,3) (∞.3)8 | t01(∞,8,3) 3.∞.8.∞ | t1(∞,8,3) (8.∞)3 | h1(∞,8,3) (6.4.6.∞)3 | t12(∞,8,3) 3.16.∞.16 | t2(∞,8,3) (8.3)∞ | t02(∞,8,3) 8.6.∞.6 | h02(∞,8,3) 4.4.4.3.4.∞.4.3 | t012(∞,8,3) 6.16.∞ | s(∞,8,3) 3.3.3.8.3.∞ | |||||||||
(∞ ∞ 3) | t0(∞,∞,3) (∞.3)∞ | t01(∞,∞,3) 3.∞.∞.∞ | t1(∞,∞,3) ∞6 | h1(∞,∞,3) (6.∞)6 | t12(∞,∞,3) 3.∞.∞.∞ | t2(∞,∞,3) (∞.3)∞ | t02(∞,∞,3) (∞.6)2 | h02(∞,∞,3) (4.∞.4.3)2 | t012(∞,∞,3) 6.∞.∞ | s(∞,∞,3) 3.3.3.∞.3.∞ | |||||||||
(∞ 4 4) | t0(∞,4,4) (∞.4)4 | h0(∞,4,4) (8.∞.8)4 | t01(∞,4,4) (4.∞)2 | h01(∞,4,4) (4.4.∞)2 | t1(∞,4,4) (4.∞)4 | h1(∞,4,4) (8.8.∞)4 | t12(∞,4,4) 4.8.∞.8 | h12(∞,4,4) 4.4.4.4.∞.4.4 | t2(∞,4,4) 4∞ | h2(∞,4,4) ∞∞ | t02(∞,4,4) 4.8.∞.8 | h02(∞,4,4) 4.4.4.4.∞.4.4 | t012(∞,4,4) 8.8.∞ | s(∞,4,4) 3.4.3.4.3.∞ | |||||
 (∞ 5 4) | t0(∞,5,4) (∞.4)5 | h0(∞,5,4) (10.∞.10)5 | t01(∞,5,4) 4.∞.5.∞ | t1(∞,5,4) (5.∞)4 | t12(∞,5,4) 4.10.∞.10 | h12(∞,5,4) 4.4.5.4.∞.4.5 | t2(∞,5,4) (5.4)∞ | t02(∞,5,4) 5.8.∞.8 | t012(∞,5,4) 8.10.∞ | s(∞,5,4) 3.4.3.5.3.∞ | |||||||||
 (∞ 6 4) | t0(∞,6,4) (∞.4)6 | h0(∞,6,4) (12.∞.12)6 | t01(∞,6,4) 4.∞.6.∞ | h01(∞,6,4) 4.4.∞.4.3.4.∞ | t1(∞,6,4) (6.∞)4 | h1(∞,6,4) (8.3.8.∞)4 | t12(∞,6,4) 4.12.∞.12 | h12(∞,6,4) 4.4.6.4.∞.4.6 | t2(∞,6,4) (6.4)∞ | h2(∞,6,4) (∞.3.∞)∞ | t02(∞,6,4) 6.8.∞.8 | h02(∞,6,4) 4.3.4.4.4.∞.4.4 | t012(∞,6,4) 8.12.∞ | s(∞,6,4) 3.4.3.6.3.∞ | |||||
 (∞ 7 4) | t0(∞,7,4) (∞.4)7 | h0(∞,7,4) (14.∞.14)7 | t01(∞,7,4) 4.∞.7.∞ | t1(∞,7,4) (7.∞)4 | t12(∞,7,4) 4.14.∞.14 | h12(∞,7,4) 4.4.7.4.∞.4.7 | t2(∞,7,4) (7.4)∞ | t02(∞,7,4) 7.8.∞.8 | t012(∞,7,4) 8.14.∞ | s(∞,7,4) 3.4.3.7.3.∞ | |||||||||
 (∞ 8 4) | t0(∞,8,4) (∞.4)8 | h0(∞,8,4) (16.∞.16)8 | t01(∞,8,4) 4.∞.8.∞ | h01(∞,8,4) 4.4.∞.4.4.4.∞ | t1(∞,8,4) (8.∞)4 | h1(∞,8,4) (8.4.8.∞)4 | t12(∞,8,4) 4.16.∞.16 | h12(∞,8,4) 4.4.8.4.∞.4.8 | t2(∞,8,4) (8.4)∞ | h2(∞,8,4) (∞.4.∞)∞ | t02(∞,8,4) 8.8.∞.8 | h02(∞,8,4) 4.4.4.4.4.∞.4.4 | t012(∞,8,4) 8.16.∞ | s(∞,8,4) 3.4.3.8.3.∞ | |||||
(∞ ∞ 4) | t0(∞,∞,4) (∞.4)∞ | h0(∞,∞,4) (∞.∞.∞)∞ | t01(∞,∞,4) 4.∞.∞.∞ | h01(∞,∞,4) 4.4.∞.4.∞.4.∞ | t1(∞,∞,4) ∞8 | h1(∞,∞,4) (8.∞)8 | t12(∞,∞,4) 4.∞.∞.∞ | h12(∞,∞,4) 4.4.∞.4.∞.4.∞ | t2(∞,∞,4) (∞.4)∞ | h2(∞,∞,4) (∞.∞.∞)∞ | t02(∞,∞,4) (∞.8)2 | h02(∞,∞,4) (4.∞.4.4)2 | t012(∞,∞,4) 8.∞.∞ | s(∞,∞,4) 3.4.3.∞.3.∞ | |||||
 (∞ 5 5) | t0(∞,5,5) (∞.5)5 | t01(∞,5,5) (5.∞)2 | t1(∞,5,5) (5.∞)5 | t12(∞,5,5) 5.10.∞.10 | t2(∞,5,5) 5∞ | t02(∞,5,5) 5.10.∞.10 | t012(∞,5,5) 10.10.∞ | s(∞,5,5) 3.5.3.5.3.∞ | |||||||||||
 (∞ 6 5) | t0(∞,6,5) (∞.5)6 | t01(∞,6,5) 5.∞.6.∞ | t1(∞,6,5) (6.∞)5 | h1(∞,6,5) (10.3.10.∞)5 | t12(∞,6,5) 5.12.∞.12 | t2(∞,6,5) (6.5)∞ | t02(∞,6,5) 6.10.∞.10 | h02(∞,6,5) 4.3.4.5.4.∞.4.5 | t012(∞,6,5) 10.12.∞ | s(∞,6,5) 3.5.3.6.3.∞ | |||||||||
 (∞ 7 5) | t0(∞,7,5) (∞.5)7 | t01(∞,7,5) 5.∞.7.∞ | t1(∞,7,5) (7.∞)5 | t12(∞,7,5) 5.14.∞.14 | t2(∞,7,5) (7.5)∞ | t02(∞,7,5) 7.10.∞.10 | t012(∞,7,5) 10.14.∞ | s(∞,7,5) 3.5.3.7.3.∞ | |||||||||||
 (∞ 8 5) | t0(∞,8,5) (∞.5)8 | t01(∞,8,5) 5.∞.8.∞ | t1(∞,8,5) (8.∞)5 | h1(∞,8,5) (10.4.10.∞)5 | t12(∞,8,5) 5.16.∞.16 | t2(∞,8,5) (8.5)∞ | t02(∞,8,5) 8.10.∞.10 | h02(∞,8,5) 4.4.4.5.4.∞.4.5 | t012(∞,8,5) 10.16.∞ | s(∞,8,5) 3.5.3.8.3.∞ | |||||||||
 (∞ ∞ 5) | t0(∞,∞,5) (∞.5)∞ | t01(∞,∞,5) 5.∞.∞.∞ | t1(∞,∞,5) ∞10 | h1(∞,∞,5) (10.∞)10 | t12(∞,∞,5) 5.∞.∞.∞ | t2(∞,∞,5) (∞.5)∞ | t02(∞,∞,5) (∞.10)2 | h02(∞,∞,5) (4.∞.4.5)2 | t012(∞,∞,5) 10.∞.∞ | s(∞,∞,5) 3.5.3.∞.3.∞ | |||||||||
 (∞ 6 6) | t0(∞,6,6) (∞.6)6 | h0(∞,6,6) (12.∞.12.3)6 | t01(∞,6,6) (6.∞)2 | h01(∞,6,6) (4.3.4.∞)2 | t1(∞,6,6) (6.∞)6 | h1(∞,6,6) (12.3.12.∞)6 | t12(∞,6,6) 6.12.∞.12 | h12(∞,6,6) 4.3.4.6.4.∞.4.6 | t2(∞,6,6) 6∞ | h2(∞,6,6) (∞.3)∞ | t02(∞,6,6) 6.12.∞.12 | h02(∞,6,6) 4.3.4.6.4.∞.4.6 | t012(∞,6,6) 12.12.∞ | s(∞,6,6) 3.6.3.6.3.∞ | |||||
 (∞ 7 6) | t0(∞,7,6) (∞.6)7 | h0(∞,7,6) (14.∞.14.3)7 | t01(∞,7,6) 6.∞.7.∞ | t1(∞,7,6) (7.∞)6 | t12(∞,7,6) 6.14.∞.14 | h12(∞,7,6) 4.3.4.7.4.∞.4.7 | t2(∞,7,6) (7.6)∞ | t02(∞,7,6) 7.12.∞.12 | t012(∞,7,6) 12.14.∞ | s(∞,7,6) 3.6.3.7.3.∞ | |||||||||
 (∞ 8 6) | t0(∞,8,6) (∞.6)8 | h0(∞,8,6) (16.∞.16.3)8 | t01(∞,8,6) 6.∞.8.∞ | h01(∞,8,6) 4.3.4.∞.4.4.4.∞ | t1(∞,8,6) (8.∞)6 | h1(∞,8,6) (12.4.12.∞)6 | t12(∞,8,6) 6.16.∞.16 | h12(∞,8,6) 4.3.4.8.4.∞.4.8 | t2(∞,8,6) (8.6)∞ | h2(∞,8,6) (∞.4.∞.3)∞ | t02(∞,8,6) 8.12.∞.12 | h02(∞,8,6) 4.4.4.6.4.∞.4.6 | t012(∞,8,6) 12.16.∞ | s(∞,8,6) 3.6.3.8.3.∞ | |||||
 (∞ ∞ 6) | t0(∞,∞,6) (∞.6)∞ | h0(∞,∞,6) (∞.∞.∞.3)∞ | t01(∞,∞,6) 6.∞.∞.∞ | h01(∞,∞,6) 4.3.4.∞.4.∞.4.∞ | t1(∞,∞,6) ∞12 | h1(∞,∞,6) (12.∞)12 | t12(∞,∞,6) 6.∞.∞.∞ | h12(∞,∞,6) 4.3.4.∞.4.∞.4.∞ | t2(∞,∞,6) (∞.6)∞ | h2(∞,∞,6) (∞.∞.∞.3)∞ | t02(∞,∞,6) (∞.12)2 | h02(∞,∞,6) (4.∞.4.6)2 | t012(∞,∞,6) 12.∞.∞ | s(∞,∞,6) 3.6.3.∞.3.∞ | |||||
 (∞ 7 7) | t0(∞,7,7) (∞.7)7 | t01(∞,7,7) (7.∞)2 | t1(∞,7,7) (7.∞)7 | t12(∞,7,7) 7.14.∞.14 | t2(∞,7,7) 7∞ | t02(∞,7,7) 7.14.∞.14 | t012(∞,7,7) 14.14.∞ | s(∞,7,7) 3.7.3.7.3.∞ | |||||||||||
 (∞ 8 7) | t0(∞,8,7) (∞.7)8 | t01(∞,8,7) 7.∞.8.∞ | t1(∞,8,7) (8.∞)7 | h1(∞,8,7) (14.4.14.∞)7 | t12(∞,8,7) 7.16.∞.16 | t2(∞,8,7) (8.7)∞ | t02(∞,8,7) 8.14.∞.14 | h02(∞,8,7) 4.4.4.7.4.∞.4.7 | t012(∞,8,7) 14.16.∞ | s(∞,8,7) 3.7.3.8.3.∞ | |||||||||
 (∞ ∞ 7) | t0(∞,∞,7) (∞.7)∞ | t01(∞,∞,7) 7.∞.∞.∞ | t1(∞,∞,7) ∞14 | h1(∞,∞,7) (14.∞)14 | t12(∞,∞,7) 7.∞.∞.∞ | t2(∞,∞,7) (∞.7)∞ | t02(∞,∞,7) (∞.14)2 | h02(∞,∞,7) (4.∞.4.7)2 | t012(∞,∞,7) 14.∞.∞ | s(∞,∞,7) 3.7.3.∞.3.∞ | |||||||||
 (∞ 8 8) | t0(∞,8,8) (∞.8)8 | h0(∞,8,8) (16.∞.16.4)8 | t01(∞,8,8) (8.∞)2 | h01(∞,8,8) (4.4.4.∞)2 | t1(∞,8,8) (8.∞)8 | h1(∞,8,8) (16.4.16.∞)8 | t12(∞,8,8) 8.16.∞.16 | h12(∞,8,8) 4.4.4.8.4.∞.4.8 | t2(∞,8,8) 8∞ | h2(∞,8,8) (∞.4)∞ | t02(∞,8,8) 8.16.∞.16 | h02(∞,8,8) 4.4.4.8.4.∞.4.8 | t012(∞,8,8) 16.16.∞ | s(∞,8,8) 3.8.3.8.3.∞ | |||||
 (∞ ∞ 8) | t0(∞,∞,8) (∞.8)∞ | h0(∞,∞,8) (∞.∞.∞.4)∞ | t01(∞,∞,8) 8.∞.∞.∞ | h01(∞,∞,8) 4.4.4.∞.4.∞.4.∞ | t1(∞,∞,8) ∞16 | h1(∞,∞,8) (16.∞)16 | t12(∞,∞,8) 8.∞.∞.∞ | h12(∞,∞,8) 4.4.4.∞.4.∞.4.∞ | t2(∞,∞,8) (∞.8)∞ | h2(∞,∞,8) (∞.∞.∞.4)∞ | t02(∞,∞,8) (∞.16)2 | h02(∞,∞,8) (4.∞.4.8)2 | t012(∞,∞,8) 16.∞.∞ | s(∞,∞,8) 3.8.3.∞.3.∞ | |||||
(∞ ∞ ∞) | t0(∞,∞,∞) ∞∞ | h0(∞,∞,∞) (∞.∞)∞ | t01(∞,∞,∞) (∞.∞)2 | h01(∞,∞,∞) (4.∞.4.∞)2 | t1(∞,∞,∞) ∞∞ | h1(∞,∞,∞) (∞.∞)∞ | t12(∞,∞,∞) (∞.∞)2 | h12(∞,∞,∞) (4.∞.4.∞)2 | t2(∞,∞,∞) ∞∞ | h2(∞,∞,∞) (∞.∞)∞ | t02(∞,∞,∞) (∞.∞)2 | h02(∞,∞,∞) (4.∞.4.∞)2 | t012(∞,∞,∞) ∞3 | s(∞,∞,∞) (3.∞)3 | |||||
