쌍곡면에서의 균일한 타일링
Uniform tilings in hyperbolic plane| 구면 | 유클리드 | 쌍곡선 | |||
|---|---|---|---|---|---|
 {5,3} 5.5.5     |  {6,3} 6.6.6  |  {7,3} 7.7.7  |  {∞,3} ∞.∞.∞  | ||
| 정오각형, 육각형 및 칠각형 및 편평면을 사용하는 구체의 정타일링 {p,q}, 유클리드 평면 및 쌍곡면. | |||||
 t{5,3} 10.10.3 |  t{6,3} 12.12.3 |  t{7,3} 14.14.3 |  t{buffic,3} ∞.∞.3 | ||
| 잘린 타일링은 정규 {p,q}의 2p.2p.q 꼭지점 숫자를 가집니다. | |||||
 r{5,3} 3.5.3.5 |  r{6,3} 3.6.3.6 |  r{7,3} 3.7.3.7 |  r{syslog,3} 3.∞.3.∞ | ||
| 준규격 타일링은 정규 타일링과 유사하지만 각 정점을 중심으로 두 가지 유형의 규칙 폴리곤을 번갈아 사용합니다. | |||||
 rr{5,3} 3.4.5.4 |  rr{6,3} 3.4.6.4 |  rr{7,3} 3.4.7.4 |  rr{param,3} 3.4.∞.4 | ||
| 반규칙 타일링에는 둘 이상의 유형의 정규 폴리곤이 있습니다. | |||||
 tr{5,3} 4.6.10 |  tr{6,3} 4.6.12 |  tr{7,3} 4.6.14 |  tr {syslog,3} 4.6.∞ | ||
| 옴니트런티드 타일링에는 세 개 이상의 짝수 면 정규 폴리곤이 있습니다. | |||||
| 대칭 | 삼각 이면체 대칭 | 사면체 | 팔면체 | 이십면체 | p6m 대칭 | [3,7] 대칭 | [3,8] 대칭 | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 솔리드 시작 작동 | 기호. {p,q}   | 삼각 호소면체 {2,3}  | 삼각 이면체 {3,2}  | 사면체 {3,3}  | 큐브 {4,3}  | 팔면체 {3,4}  | 십이면체 {5,3}  | 이십면체 {3,5}  | 육각형 타일링 {6,3}  | 삼각 타일링 {3,6}  | 칠각형 타일링 {7,3} | 7차 삼각 타일링 {3,7}  | 팔각형 타일링 {8,3}  | 8차 삼각 타일링 {3,8}  |
| 잘라내기(t) | t{p,q} | 삼각 프리즘 | 잘린 삼각 이면체 ('에지'의 절반은 퇴화 디곤 면으로 카운트됩니다.나머지 절반은 정상 가장자리입니다.) | 깎은 사면체 | 잘린 입방체 | 잘린 팔면체 | 잘린 12면체 | 깎은 20면체 | 잘린 육각형 타일링 | 잘린 삼각형 타일링 | 잘린 7각형 타일링 | 잘린 순서-7 삼각형 타일링 | 잘린 팔각형 타일링 | 잘린 순서-8 삼각형 타일링 |
| 수정(r) Ambo(a) | r{p,q} | 삼면체(모든 "edge"는 퇴화 디곤 면으로 카운트됩니다). | 사면체 | 육팔면체 | 이십이면체 | 삼육각 타일링 | 삼육각 타일링 | 삼팔각형 타일링 | ||||||
| 비트런케이션 (2t) 듀얼키(dk) | 2t{p,q} | 잘린 삼각 이면체('에지'의 절반은 퇴화 디곤 면으로 카운트됩니다.나머지 절반은 정상 가장자리입니다.) | 삼각 프리즘 | 깎은 사면체 | 잘린 팔면체 | 잘린 입방체 | icosahedron는 잘리 | 잘려진 12면체 | 잘려진 타일 붙이기 삼각형 | 육방 tiling들의 길이를 줄였다 | order-7 삼각 타일 Truncated | 7각형의. tiling Truncated | order-8 삼각 타일 Truncated | Truncatedtiling 팔모의 |
| Birectification(2r) 듀얼(d) | 2r{p,q} | dihedron 삼각 {3,2} | 삼각 hosohedron {2,3} | 4면체 | 8면체 | 큐브 | 20면체 | 12면체 | 삼각 타일링 | 육각형 타일링 | 7차 삼각 타일링 | 칠각형 타일링 | 8차 삼각 타일링 | 팔각형 타일링 |
| 칸테레이션(rr) 확장(e) | rr{p,q} | 삼각 프리즘(각 사각형 쌍 사이의 "엣지"는 퇴화된 사각형 면으로 간주됩니다.다른 에지(트리거와 사각형 사이의 에지)는 정규 에지입니다. | 마름비테트라면체 | 마름모꼴 팔면체 | 마름모꼴 십이면체  | 로밋리헥사각형 타일링 | 롬비트리헵타일링  | 롬빗리옥타일링  | ||||||
| 스너브 수정 완료(sr) 스누브 | sr{p,q} | 삼각 반체제 (3개의 황색 "에지"는 정점을 공유하지 않으며 퇴화 디곤 면으로 간주됩니다.다른 엣지는 일반 엣지입니다). | 스너브 사면체 | 스너브 육팔면체 | 스누브 이십이면체 | 스눕 삼육각 타일 | 스눕 삼칠각 타일링 | 스눕 삼팔각형 타일링 | ||||||
| 캔티트런케이션(tr) 베벨(b) | tr{p,q} | 육각 프리즘 | 절삭 사면체 | 깎은 정육면체 | 깎은 이십이면체 | 잘린 삼육각 타일링 | 잘린 삼육각 타일링 | 잘린 삼팔각형 타일링 | ||||||
쌍곡선 기하학에서 균일한 쌍곡선 타일링(또는 규칙, 준규칙 또는 반규칙 쌍곡선 타일링)은 정다각형들을 면으로 가지며 정점-추이적이다(정점에서의 추이, 즉 정점을 다른 정점에 매핑하는 등각도가 있다).따라서 모든 정점은 일치하며 타일링은 높은 회전 및 변환 대칭을 가집니다.
균일한 타일링은 각 정점 주변의 폴리곤의 변 수를 나타내는 일련의 숫자인 정점 구성으로 식별할 수 있습니다.예를 들어, 7.7.7은 각 정점 주위에 3개의 헵타곤이 있는 7각형 타일을 나타냅니다.또한 모든 폴리곤의 크기가 같기 때문에 규칙적이기 때문에 슐래플리 기호 {7,3}도 지정할 수 있습니다.
균일한 타일링은 정규(면 전이 및 가장자리 전이인 경우), 준정규(면 전이이지만 면 전이 아닌 경우) 또는 반정규(면 전이인 경우)일 수 있습니다.직각 삼각형(p q 2)의 경우 슐레플리 기호 {p,q} 및 {q,p}으로 표시되는 두 개의 규칙 타일링이 있습니다.
위토프 공사
슈바르츠 삼각형(pqr)에 기초한 균일한 타일링의 수는 무한히 많습니다.1/p + 1/q + 1/r < 1 (여기서 p, q, r은 기본 영역 삼각형의 세 점에서의 반사 대칭의 각 순서)대칭군은 쌍곡선 삼각군입니다.
각 대칭 패밀리는 와이토프 기호 또는 콕서터-딘킨 다이어그램에 의해 정의된 7개의 균일한 타일링을 포함하며, 7개는 3개의 활성 미러 조합을 나타냅니다.8번째는 모든 미러가 활성화된 상태에서 가장 높은 형태에서 대체 정점을 삭제하는 교대 연산을 나타냅니다.
r = 2인 패밀리는 [7,3], [8,3], [9,3], ...와 같은 Coxeter 그룹에 의해 정의된 규칙적인 쌍곡 타일링을 포함합니다.[5,4], [6,4], ....
r = 3 이상인 쌍곡선군은 (p q r)로 나타내며 (4 3), (5 3), (6 3), (5 4 3), ... (4 4 4 )...... (4 4 4 )......
쌍곡선 삼각형(p q r)은 작고 균일한 쌍곡선 타일링을 정의합니다.한계에서는 p, q 또는 r 중 하나를 파라콤팩트 쌍곡선 삼각형을 정의하고 단일 이상점으로 수렴하는 무한 면(아페이로곤이라고 함) 또는 동일한 이상점에서 무한히 많은 모서리가 분기하는 무한 정점 도형으로 균일한 타일링을 생성하는 θ로 대체할 수 있습니다.
삼각형이 아닌 기본 도메인에서 더 많은 대칭 패밀리를 구성할 수 있습니다.
아래에는 균일한 타일링의 선택된 패밀리가 나와 있습니다(쌍곡 평면에 대한 Poincaré 디스크 모델 사용).이들 중 (7 3 2), (5 4 2), (4 3 ) 등 3개의 정의수가 더 작은 정수로 대체되면 결과 패턴이 쌍곡선이 아닌 유클리드 또는 구면이라는 점에서 최소이며, 반대로 어떤 숫자라도 증가하여 다른 쌍곡선 패턴을 생성할 수 있습니다.
각 균일한 타일링은 이중 균일한 타일링을 생성하며, 그 대부분은 다음과 같습니다.
직각 삼각형 도메인
(p q 2) 삼각형 그룹 패밀리가 무한히 많습니다.이 문서에서는 (7 3 2), (8 3 2), (5 4 2), (6 4 2), (7 4 2), (6 5 2), (6 5 2), (6 5 2), (7 7 2), (7 7 2), (8 2), (8 2), (8 2)의 12 패밀리의 정규 타일을 보여 줍니다.
규칙 쌍곡선 타일링
쌍곡선 타일링의 가장 단순한 세트는 규칙 다면체와 유클리드 타일링을 가진 행렬에 존재하는 규칙 타일링 {p,q}입니다.일반 타일링 {p,q}에는 테이블의 대각선 축에 이중 타일링 {q,p}이(가) 있습니다.자체 이중 타일링 {2,2}, {3,3}, {4,4}, {5,5} 등은 테이블의 대각선으로 전달됩니다.
| 정규 쌍곡선 타일링 테이블 | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Schléfli 기호가 있는 구형(오퍼/플라토닉)/유클리드/하이퍼볼릭(포인카레 디스크: 콤팩트/파라콤팩트/비콤팩트) 테셀레이션 | |||||||||||
| p \ q | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ... | ∞ | ... | i440/i250 |
| 2 |  {2,2}  | {2,3} |  {2,4}  |  {2,5} |  {2,6} |  {2,7} |  {2,8}  |  {2,∞} |  {2,i140/module}  | ||
| 3 |  {3,2} |  (4면체) {3,3} |  (8면체) {3,4} |  (이십면체) {3,5} |  (델틸) {3,6} |  {3,7} |  {3,8} |  {3,∞} |  {3,i140/module} | ||
| 4 |  {4,2} |  (입방체) {4,3} |  (실제) {4,4} |  {4,5} |  {4,6} |  {4,7} |  {4,8} |  {4,∞} |  {4,i440/contract} | ||
| 5 |  {5,2} | (이십이면체) {5,3} |  {5,4} |  {5,5} |  {5,6} |  {5,7} |  {5,8} |  {5,∞} |  {5,i140/160} | ||
| 6 |  {6,2} | (헥스틸) {6,3} |  {6,4} |  {6,5} |  {6,6} |  {6,7} |  {6,8} |  {6,∞} |  {6,i140/contract} | ||
| 7 | {7,2} | {7,3} |  {7,4} |  {7,5} |  {7,6} |  {7,7} |  {7,8} |  {7,∞} | {7,i140/module} | ||
| 8 | {8,2} |  {8,3} |  {8,4} |  {8,5} |  {8,6} |  {8,7} |  {8,8} |  {8,∞} | {8,i140/160} | ||
| ... | |||||||||||
| ∞ |  {∞,2} | {∞,3} |  {∞,4} |  {∞,5} |  {∞,6} |  {∞,7} |  {∞,8} |  {∞,∞} |  {viol,ivi/viol} | ||
| ... | |||||||||||
| i440/i250 |  {i140/module,2} |  {i140/module,3} |  {i140/module,4} |  {i140/1200,5} |  {i140/1600,6} | {i140/1200,7} | {i140/180,8} |  {ii/module,module} |  {ifl/flash, iflash/flash} | ||
(7 3 2)
(7 3 2) 삼각형 그룹, 콕서터 그룹 [7,3], 오르비폴드(*732)에는 다음과 같은 균일한 타일링이 포함되어 있습니다.
| 균일한 7각/삼각 타일링 | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 대칭: [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} | ||||
| 균일한 이중화 | |||||||||||
 | | | | | | |  | ||||
| |  |  |  | |  |  |  | ||||
| V73 | V3.14.14 | V3.7.3.7 | V6.6.7 | V37 | V3.4.7.4 | V4.6.14 | V3.3.3.7 | ||||
(8 3 2)
(8 3 2) 삼각형 그룹, Coxeter 그룹 [8,3], Orbifold(*832)에는 다음과 같은 균일한 타일링이 포함되어 있습니다.
| 균일한 팔각/삼각 타일링 | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 대칭: [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} | |||
| | | | | | | | | | |||||
    | |  | |   또는 | 또는 |  | |||||||
| |  |   |   |   |  |  |  |   |   |   | |||
| 균일한 이중화 | |||||||||||||
| 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 | |||
| | | | | | | | | | | | |||
| |  |  |  | |  |  |  |  |  |  | |||
(5 4 2)
(5 4 2) 삼각형 그룹, 콕서터 그룹 [5,4], 오르비폴드(*542)는 다음과 같은 균일한 타일링을 포함한다.
| 균일한 오각형/사각형 타일링 | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 대칭: [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} | ||
| 균일한 이중화 | |||||||||||
| | | | | | | | | | | ||
| |  |  |  | |  |  |  |  | |||
| 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)
(6 4 2) 삼각형 그룹, 콕서터 그룹 [6,4], 오비폴드(*642)는 이러한 균일한 타일링을 포함한다.모든 요소가 짝수이기 때문에 각 균일한 이중 타일링은 반사 대칭의 기본 영역인 *333, *662, *32, *443, *2222, *3222 및 *642를 나타냅니다.또한 7개의 균일한 타일을 모두 교대로 사용할 수 있으며 이중 타일도 있습니다.
| 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} | |||||
(7 4 2)
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 | ||
(8 4 2)
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 | |||||
(5 5 2)
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 | ||||
(6 5 2)
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 | ||
(6 6 2)
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} |
(8 6 2)
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 |
(7 7 2)
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 | ||||
(8 8 2)
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 | |||||
General triangle domains
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)
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) | ||||||||||
 | | | | | | | | ||||
| | | | | | | | | ||||
 |  |  |  |  |  |  | | ||||
| 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 | ||||
(4 4 3)
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 |
(4 4 4)
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 | ||
(5 3 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 | ||||
(5 4 3)
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 | ||||
(5 4 4)
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 | ||
(6 3 3)
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 | ||||
(6 4 3)
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 |
(6 4 4)
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 |
Summary of tilings with finite triangular fundamental domains
For a table of all uniform hyperbolic tilings with fundamental domains (p q r), where 2 ≤ p,q,r ≤ 8.
Quadrilateral domains
(3 2 2 2)
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.
| 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 | ||
(3 2 3 2)
| 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 |  | | ||||||
Ideal triangle domains
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)
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.∞ | |
(∞ 4 2)
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.∞ | |
(∞ 5 2)
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.∞ | ||
(∞ ∞ 2)
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.∞ |
(∞ 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.∞ | ||||
(∞ 4 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 | ||
(∞ 4 4)
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.∞.4.∞ | V4∞ | V4.∞.4.∞ | V8.8.∞ | V3.4.3.4.3.∞ | ||||
(∞ ∞ 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 |
(∞ ∞ 4)
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.∞ |
Summary of tilings with infinite triangular fundamental domains
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 | |||||
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
- Hatch, Don. "Hyperbolic Planar Tessellations". Retrieved 2010-08-19.
- Eppstein, David. "The Geometry Junkyard: Hyperbolic Tiling". Retrieved 2010-08-19.
- Joyce, David. "Hyperbolic Tessellations". Retrieved 2010-08-19.
- Klitzing, Richard. "2D Tesselations Hyperbolic Tesselations".
- The EPINET project explores 2D hyperbolic (H²) tilings
