User:Tomruen/uniform tilings
Summary
[edit]1-uniform tilings include 3 regular tilings, and 8 semiregular ones, with 2 or more types of regular polygon faces. There are 20 2-uniform tilings. 3-uniform tilings include 22 2-Archimedean tilings and 39 3-Archimedean ones. The 3 regular tilings have regular dual tilings. In general n-isogonal tilings have n-isohedral dual tilings.
| m | ||||||||
|---|---|---|---|---|---|---|---|---|
| k | 1 | 2 | 3 | 4 | 5 | 6 | Total | |
| 1 | 11 | 0 | 0 | 0 | 0 | 0 | 11 | |
| 2 | 0 | 20 | 0 | 0 | 0 | 0 | 20 | |
| 3 | 0 | 22 | 39 | 0 | 0 | 0 | 61 | |
| 4 | 0 | 33 | 85 | 33 | 0 | 0 | 151 | |
| 5 | 0 | 74 | 149 | 94 | 15 | 0 | 332 | |
| 6 | 0 | 100 | 284 | 187 | 92 | 10 | 673 | |
Vertex figures and dual faces
[edit]| Valance 3 | Valance 4 | |||||
|---|---|---|---|---|---|---|
| 6.6.6 | 4.6.12 | 4.8.8 | 3.12.12 | 4.4.4.4 | 3.6.3.6 | 3.3.6.6 |
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| Valance 4 | Valance 5 | Valance 6 | |||||
|---|---|---|---|---|---|---|---|
| 3.4.4.6 | 3.4.6.4 | 3.4.3.12 | 3.3.4.12 | 3.3.4.3.4 | 3.3.3.4.4 | 3.3.3.3.6 | 3.3.3.3.3.3 |
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Semiregular and duals (1-8)
[edit]There are 8 semiregular tilings, and 8 duals.
| cmm, 2*22 | |
|---|---|
 33.42, Elongated triangular tiling |
 V33.42, d Prismatic pentagonal tiling |
| p4m, *442 | p4, 442 | ||
|---|---|---|---|
 4.82, tQ Truncated square tiling |
 V4.82, dtQ Tetrakis square tiling |
 32.4.3.4, sQ Snub square tiling |
 V32.4.3.4, dsQ Cairo pentagonal tiling |
| p6m, *632 | p6, 632 | |||
|---|---|---|---|---|
 3.122, tH Truncated hexagonal tiling |
 (3.6)2, aH Trihexagonal tiling |
 3.4.6.4, eH Rhombitrihexagonal tiling |
 4.6.12, taH Truncated trihexagonal tiling |
 34.6, sH Snub hexagonal tiling |
 V3.122, dtH Triakis triangular tiling |
 V(3.6)2, daH Rhombille tiling |
 V3.4.6.4, deH Deltoidal trihexagonal tiling |
 V4.6.12, dtaH Kisrhombille tiling |
 V34.6, dsH Floret pentagonal tiling |
2-uniform and duals (1-20)
[edit]There are 20 2-uniform tilings, and 20 duals:
| cmm, 2*22 | cmm, 2*22 | pmm, *2222 | cmm, 2*22 | cmm, 2*22 |
|---|---|---|---|---|
 (44; 33.42)1, (t=2, e=4) |
 (44; 33.42)2, (t=3, e=5) |
 (36; 33.42)1, (t=3, e=4) |
 (36; 33.42)2, (t=4, e=5) |
 (3.42.6; (3.6)2)2, aH:e (t=3, e=4) |
 (V44; V33.42)1 |
 (V44; V33.42)2 |
 (V36; V33.42)1 |
 (V36; V33.42)2 |
 (V3.42.6; V(3.6)2)2 |
| pmm, *2222 | pmm, *2222 | cmm, 2*22 | pgg, 2× | |
 (3.42.6; (3.6)2)1 (t=4, e=4) |
 ((3.6)2; 32.62) (t=2, e=3) |
 (32.62; 34.6) (t=2, e=4) |
 (33.42; 32.4.3.4)2 (t=6, e=6) | |
 (V3.42.6; V(3.6)2)1 |
 (V(3.6)2; V32.62) |
 (V32.62; V34.6) |
 (V33.42; V32.4.3.4)2 |
| p4m, *442 | p4g, 4*2 |
|---|---|
 (3.12.12; 3.4.3.12) (t=3, e=3) |
 (33.42; 32.4.3.4)1 (t=4, e=5) |
 (V3.12.12; V3.4.3.12) |
 (V33.42; V32.4.3.4)1 |
| p6m, *632 | p6m, *632 | p6m, *632 | p6, 632 | |
|---|---|---|---|---|
 (36; 32.62) (t=2, e=3) |
 (36; 32.4.3.4) (t=3, e=3) |
 (36; 34.6)1 (t=3, e=3) |
 (36; 34.6)2 (t=5, e=7) |
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 (V36; V32.62) |
 (V36; V32.4.3.4) |
 (V36; V34.6)1 |
 (V36; V34.6)2 |
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| p6m, *632 | ||||
 (3.4.6.4; 32.4.3.4) (t=4, e=4) |
 (3.4.6.4; 33.42) (t=4, e=4) |
 (3.4.6.4; 3.42.6) (t=5, e=5) |
 (4.6.12; 3.4.6.4) (t=4, e=4) |
 (36; 32.4.12) (t=4, e=4) |
 (V3.4.6.4; V32.4.3.4) |
 (V3.4.6.4; V33.42) |
 (V3.4.6.4; V3.42.6) |
 (V4.6.12; V3.4.6.4) |
 (V36; V32.4.12) |
3-uniform and duals
[edit]There are 61 3-uniform tilings, and 61 duals:
 p6m (t=4, e=5) |
 cmm (t=2, e=4) |
 pmg (t=2, e=5) |
 p6m (t=2, e=3) |
 p6m (t=6, e=7) |
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 p3m1 (t=5, e=6) |
 p6m (t=5, e=6) |
 p6m (t=5, e=6) |
 cmm (t=5, e=7) |
 cmm (t=4, e=7) |
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 cmm (t=5, e=7) |
 pmm (t=6, e=7) |
 pmm (t=5, e=7) |
 pmm (t=4, e=6) |
 cmm (t=5, e=7) |
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 cmm (t=3, e=6) |
 cmm (t=4, e=5) |
 pmm (t=5, e=6) |
 pmm (t=4, e=6) |
 pmm (t=3, e=5) |
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 cmm (t=4, e=6) |
 p4g (t=3, e=4) |
 p4 (t=4, e=6) |
 p6m (t=5, e=6) |
 cmm (t=4, e=6) |
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 p6m (t=6, e=6) |
 cmm (t=5, e=8) |
 cmm (t=6, e=8) |
 cmm (t=4, e=7) |
 pmg (t=4, e=6) |
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3-uniform and duals (31-61)
[edit] pmm (t=5, e=7) |
 cmm (t=5, e=6) |
 pmm (t=5, e=7) |
 p6m (t=5, e=6) |
 p6m (t=6, e=8) |
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 p6m (t=6, e=6) |
 p6m (t=6, e=6) |
 pmg (t=5, e=8) |
 cmm (t=3, e=5) |
 pmm (t=3, e=5) |
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 cmm (t=3, e=6) |
 p6m (t=4, e=4) |
 p6m (t=3, e=3) |
 cmm (t=5, e=6) |
 pmg (t=4, e=7) |
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 p2 (t=4, e=7) |
 p6m (t=6, e=7) |
 p6m (t=5, e=6) |
 pmm (t=5, e=6) |
 cmm (t=6, e=7) |
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 cmm (t=4, e=6) |
 pmg (t=5, e=7) |
 pgg (t=5, e=8) |
 p6m (t=4, e=5) |
 p2 (t=6, e=9) |
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 p6m (t=5, e=6) |
 p6m (t=8, e=5) |
 p6m (t=3, e=4) |
 p6m (t=5, e=5) |
 p6 (t=7, e=9) |
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 cmm (t=4, e=6) | ||||
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4-uniform and duals
[edit]There are 151 4-uniform tilings, and 151 duals:
4-uniform and duals (1-25)
[edit] p6m, *632 |
 p6m, *632 |
 cmm, 2*22 |
 cmm, 2*22 |
 cmm, 2*22 |
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 p6m, *632 |
 p6m, *632 |
 p6m, *632 |
 p6, 632 |
 p6, 632 |
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 cmm, 2*22 |
 cmm, 2*22 |
 cmm, 2*22 |
 p6m, *632 |
 pmm, *2222 |
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 pmm, *2222 |
 p6, 632 |
 pmm, *2222 |
 pmm, *2222 |
 pmm, *2222 |
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 pmm, *2222 |
 p6m, *632 |
 p6, 632 |
 p6, 632 |
 p6m, *632 |
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4-uniform and duals (26-50)
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4-uniform and duals (51-75)
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4-uniform and duals (76-100)
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4-uniform and duals (101-125)
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4-uniform and duals (125-151)
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5-uniform tilings and duals (1-332)
[edit] [3342; 3262; 3446; (63)2] |
 [36; (3262)2; (63)2] |
 [36; 346; 3262; (63)2] |
 [3262; 3636; (63)3] |
 [346; 3262; (63)3] |
 [3342; 3262; 3446; (63)2] |
 [36; (3262)2; (63)2] |
 [36; 346; 3262; (63)2] |
 [3262; 3636; (63)3] |
 [346; 3262; (63)3] |
 [36; (3262)3; 63] |
 [36; 346; (3262)2; 63] |
 [346; (3262)2; 3636; 63] |
 [(3262)2; (3636)2; 63] |
 [(346)2; 3262; 3636; 63] |
 [36; (3262)3; 63] |
 [36; 346; (3262)2; 63] |
 [346; (3262)2; 3636; 63] |
 [(3262)2; (3636)2; 63] |
 [(346)2; 3262; 3636; 63] |
 [(36)2; 346; 3262; 63] |
 [36; (346)2; 3262; 63] |
 [(346)2; (3262)2; 63] |
 [(36)2; 346; 3262; 63] |
 [(36)2; 346; 3262; 63] |
 [(36)2; 346; 3262; 63] |
 [36; (346)2; 3262; 63] |
 [(346)2; (3262)2; 63] |
 [(36)2; 346; 3262; 63] |
 [(36)2; 346; 3262; 63] |
 [3342; 3262; 3446; (63)2] |
 [36; (3262)2; (63)2] |
 [36; 346; 3262; (63)2] |
 [(3262)2; 3636; (63)2] |
 [36; 346; 3262; (63)2] |
 [3342; 3262; 3446; (63)2] |
 [36; (3262)2; (63)2] |
 [36; 346; 3262; (63)2] |
 [(3262)2; 3636; (63)2] |
 [36; 346; 3262; (63)2] |
 [36; 346; 3262; (63)2] |
 [36; (3262)3; 63] |
 [36; 346; (3262)2; 63] |
 [346; (3262)2; 3636; 63] |
 [3262; (3636)3; 63] |
 [36; 346; 3262; (63)2] |
 [36; (3262)3; 63] |
 [36; 346; (3262)2; 63] |
 [346; (3262)2; 3636; 63] |
 [3262; (3636)3; 63] |
 [346; (3262)2; 3636; 63] |
 [346; (3262)2; 3636; 63] |
 [346; 3262; 3636; (63)2] |
 [33434; 3262; 3464; 3446; 63] |
 [36; 346; 3262; 3636; 63] |
 [346; (3262)2; 3636; 63] |
 [346; (3262)2; 3636; 63] |
 [346; 3262; 3636; (63)2] |
 [33434; 3262; 3464; 3446; 63] |
 [36; 346; 3262; 3636; 63] |
 [346; (3262)2; 3636; 63] |
 [36; 334.12; (46.12)3] |
 [36; 33434; (3446)2; 46.12] |
 [(3446)2; (3636)2; 46.12] |
 [36; 346; 3342; 3446; 46.12] |
 [346; (3262)2; 3636; 63] |
 [36; 334.12; (46.12)3] |
 [36; 33434; (3446)2; 46.12] |
 [(3446)2; (3636)2; 46.12] |
 [36; 346; 3342; 3446; 46.12] |
 [(3464)4; 46.12] |
 [36; 33434; 3446; (46.12)2] |
 [36; 33434; 3464; (46.12)2] |
 [(3464)2; (46.12)3] |
 [3464; 3446; (46.12)3] |
 [(3464)4; 46.12] |
 [36; 33434; 3446; (46.12)2] |
 [36; 33434; 3464; (46.12)2] |
 [(3464)2; (46.12)3] |
 [3464; 3446; (46.12)3] |
 [(3464)2; (46.12)3] |
 [36; (334.12)3; 46.12] |
 [33434; 334.12; 3464; 3.12.12; 46.12] |
 [343.12; (3.12.12)4] |
 [334.12; 343.12; (3.12.12)3] |
 [(3464)2; (46.12)3] |
 [36; (334.12)3; 46.12] |
 [33434; 334.12; 3464; 3.12.12; 46.12] |
 [343.12; (3.12.12)4] |
 [334.12; 343.12; (3.12.12)3] |
 [36; 33434; 334.12; (3.12.12)2] |
 [36; 33434; 343.12; (3.12.12)2] |
 [(3342)2; 334.12; 343.12; 3.12.12] |
 [(3342)2; 334.12; 343.12; 3.12.12] |
 [3342; (44)4] |
 [36; 33434; 334.12; (3.12.12)2] |
 [36; 33434; 343.12; (3.12.12)2] |
 [(3342)2; 334.12; 343.12; 3.12.12] |
 [(3342)2; 334.12; 343.12; 3.12.12] |
 [3342; (44)4] |
 [3446; 3636; (44)3] |
 [3446; 3636; (44)3] |
 [(3342)2; (44)3] |
 [36; 3342; (44)3] |
 [36; 3342; (44)3] |
 [3446; 3636; (44)3] |
 [3446; 3636; (44)3] |
 [(3342)2; (44)3] |
 [36; 3342; (44)3] |
 [36; 3342; (44)3] |
 [(3446)2; 3636; (44)2] |
 [(3446)2; 3636; (44)2] |
 [3262; 3446; 3636; (44)2] |
 [3446; (3636)2; (44)2] |
 [(3446)2; 3636; (44)2] |
 [(3446)2; 3636; (44)2] |
 [(3446)2; 3636; (44)2] |
 [3262; 3446; 3636; (44)2] |
 [3446; (3636)2; (44)2] |
 [(3446)2; 3636; (44)2] |
 [(3446)2; 3636; (44)2] |
 [3262; 3446; 3636; (44)2] |
 [3446; (3636)2; (44)2] |
 [(3342)2; (44)3] |
 [(3342)2; (44)3] |
 [(3446)2; 3636; (44)2] |
 [3262; 3446; 3636; (44)2] |
 [3446; (3636)2; (44)2] |
 [(3342)2; (44)3] |
 [(3342)2; (44)3] |
 [(3342)3; (44)2] |
 [36; (3342)2; (44)2] |
 [(36)2; 3342; (44)2] |
 [(36)2; 3342; (44)2] |
 [(3446)2; 3636; (44)2] |
 [(3342)3; (44)2] |
 [36; (3342)2; (44)2] |
 [(36)2; 3342; (44)2] |
 [(36)2; 3342; (44)2] |
 [(3446)2; 3636; (44)2] |
 [(3446)2; 3636; (44)2] |
 [(3446)2; 3636; (44)2] |
 [3262; 3446; (3636)2; 44] |
 [3446; (3636)3; 44] |
 [36; 346; 3446; 3636; 44] |
 [(3446)2; 3636; (44)2] |
 [(3446)2; 3636; (44)2] |
 [3262; 3446; (3636)2; 44] |
 [3446; (3636)3; 44] |
 [36; 346; 3446; 3636; 44] |
 [(3446)2; 3636; (44)2] |
 [(3446)2; 3636; (44)2] |
 [3262; 3446; (3636)2; 44] |
 [3446; (3636)3; 44] |
 [36; 346; 3446; 3636; 44] |
 [(3446)2; 3636; (44)2] |
 [(3446)2; 3636; (44)2] |
 [3262; 3446; (3636)2; 44] |
 [3446; (3636)3; 44] |
 [36; 346; 3446; 3636; 44] |
 [(3342)2; (44)3] |
 [(3342)3; (44)2] |
 [(3342)4; 44] |
 [36; (3342)3; 44] |
 [36; (3342)3; 44] |
 [(3342)2; (44)3] |
 [(3342)3; (44)2] |
 [(3342)4; 44] |
 [36; (3342)3; 44] |
 [36; (3342)3; 44] |
 [36; (3342)2; (44)2] |
 [36; (3342)3; 44] |
 [(36)2; (3342)2; 44] |
 [(36)3; 3342; 44] |
 [(36)3; 3342; 44] |
 [36; (3342)2; (44)2] |
 [36; (3342)3; 44] |
 [(36)2; (3342)2; 44] |
 [(36)3; 3342; 44] |
 [(36)3; 3342; 44] |
 [3342; 3262; 3446; (44)2] |
 [346; 3342; 3446; (44)2] |
 [33434; 3262; (3446)2; 44] |
 [(3342)2; 334.12; 343.12; 44] |
 [3342; (44)4] |
 [3342; 3262; 3446; (44)2] |
 [346; 3342; 3446; (44)2] |
 [33434; 3262; (3446)2; 44] |
 [(3342)2; 334.12; 343.12; 44] |
 [3342; (44)4] |
 [3446; 3636; (44)3] |
 [3446; 3636; (44)3] |
 [(3342)2; (44)3] |
 [36; 3342; (44)3] |
 [36; 3342; (44)3] |
 [3446; 3636; (44)3] |
 [3446; 3636; (44)3] |
 [(3342)2; (44)3] |
 [36; 3342; (44)3] |
 [36; 3342; (44)3] |
 [(3446)2; 3636; (44)2] |
 [(3446)2; 3636; (44)2] |
 [3262; 3446; 3636; (44)2] |
 [3446; (3636)2; (44)2] |
 [(3446)2; 3636; (44)2] |
 [(3446)2; 3636; (44)2] |
 [(3446)2; 3636; (44)2] |
 [3262; 3446; 3636; (44)2] |
 [3446; (3636)2; (44)2] |
 [(3446)2; 3636; (44)2] |
 [(3446)2; 3636; (44)2] |
 [3262; 3446; 3636; (44)2] |
 [3446; (3636)2; (44)2] |
 [(3342)2; (44)3] |
 [(3342)3; (44)2] |
 [(3446)2; 3636; (44)2] |
 [3262; 3446; 3636; (44)2] |
 [3446; (3636)2; (44)2] |
 [(3342)2; (44)3] |
 [(3342)3; (44)2] |
 [36; (3342)2; (44)2] |
 [(36)2; 3342; (44)2] |
 [(36)2; 3342; (44)2] |
 [(3342)3; (44)2] |
 [36; (3342)2; (44)2] |
 [36; (3342)2; (44)2] |
 [(36)2; 3342; (44)2] |
 [(36)2; 3342; (44)2] |
 [(3342)3; (44)2] |
 [36; (3342)2; (44)2] |
 [36; (3342)2; (44)2] |
 [(3446)2; 3636; (44)2] |
 [3262; 3446; (3636)2; 44] |
 [3446; (3636)3; 44] |
 [36; 346; 3446; 3636; 44] |
 [36; (3342)2; (44)2] |
 [(3446)2; 3636; (44)2] |
 [3262; 3446; (3636)2; 44] |
 [3446; (3636)3; 44] |
 [36; 346; 3446; 3636; 44] |
 [3262; 3446; (3636)2; 44] |
 [3446; (3636)3; 44] |
 [36; 346; 3446; 3636; 44] |
 [36; 3342; 3446; 3636; 44] |
 [36; 346; 3342; 3446; 44] |
 [3262; 3446; (3636)2; 44] |
 [3446; (3636)3; 44] |
 [36; 346; 3446; 3636; 44] |
 [36; 3342; 3446; 3636; 44] |
 [36; 346; 3342; 3446; 44] |
 [3342; 3262; 3446; (44)2] |
 [346; 3342; 3446; (44)2] |
 [346; 3342; 33434; 3446; 44] |
 [(3342)4; 44] |
 [36; (3342)3; 44] |
 [3342; 3262; 3446; (44)2] |
 [346; 3342; 3446; (44)2] |
 [346; 3342; 33434; 3446; 44] |
 [(3342)4; 44] |
 [36; (3342)3; 44] |
 [36; (3342)3; 44] |
 [36; (3342)3; 44] |
 [(36)2; (3342)2; 44] |
 [(36)3; 3342; 44] |
 [(36)3; 3342; 44] |
 [36; (3342)3; 44] |
 [36; (3342)3; 44] |
 [(36)2; (3342)2; 44] |
 [(36)3; 3342; 44] |
 [(36)3; 3342; 44] |
 [(3342)4; 44] |
 [36; (3342)3; 44] |
 [(36)2; (3342)2; 44] |
 [(36)2; (3342)2; 44] |
 [36; (3342)3; 44] |
 [(3342)4; 44] |
 [36; (3342)3; 44] |
 [(36)2; (3342)2; 44] |
 [(36)2; (3342)2; 44] |
 [36; (3342)3; 44] |
 [36; (3342)3; 44] |
 [(36)2; (3342)2; 44] |
 [346; (3342)2; 3636; 44] |
 [36; (3342)2; 33434; 44] |
 [346; (3342)2; 33434; 44] |
 [36; (3342)3; 44] |
 [(36)2; (3342)2; 44] |
 [346; (3342)2; 3636; 44] |
 [36; (3342)2; 33434; 44] |
 [346; (3342)2; 33434; 44] |
 [36; 3342; (3464)2; 3446] |
 [3342; 3262; 3464; (3446)2] |
 [33434; 3262; 3464; (3446)2] |
 [36; 33434; (3446)2; 3636] |
 [(33434)2; 3262; (3446)2] |
 [36; 3342; (3464)2; 3446] |
 [3342; 3262; 3464; (3446)2] |
 [33434; 3262; 3464; (3446)2] |
 [36; 33434; (3446)2; 3636] |
 [(33434)2; 3262; (3446)2] |
 [(3464)3; (3446)2] |
 [3342; 33434; 3464; (3446)2] |
 [36; 33434; (3262)2; 3446] |
 [(3342)3; 3262; 3446] |
 [3342; (3262)2; (3446)2] |
 [(3464)3; (3446)2] |
 [3342; 33434; 3464; (3446)2] |
 [36; 33434; (3262)2; 3446] |
 [(3342)3; 3262; 3446] |
 [3342; (3262)2; (3446)2] |
 [3342; (3262)2; (3446)2] |
 [3342; 3262; (3464)2; 3446] |
 [33434; 3262; (3464)2; 3446] |
 [(3446)3; (3636)2] |
 [36; 3342; (3446)2; 3636] |
 [3342; (3262)2; (3446)2] |
 [3342; 3262; (3464)2; 3446] |
 [33434; 3262; (3464)2; 3446] |
 [(3446)3; (3636)2] |
 [36; 3342; (3446)2; 3636] |
 [3262; (3446)2; (3636)2] |
 [(3262)2; 3446; (3636)2] |
 [3262; 3446; (3636)3] |
 [(3446)2; (3636)3] |
 [3262; 3446; (3636)3] |
 [3262; (3446)2; (3636)2] |
 [(3262)2; 3446; (3636)2] |
 [3262; 3446; (3636)3] |
 [(3446)2; (3636)3] |
 [3262; 3446; (3636)3] |
 [3446; (3636)4] |
 [346; (3342)2; 3446; 3636] |
 [346; (3342)2; 3446; 3636] |
 [(36)2; 346; 3446; 3636] |
 [(3446)3; (3636)2] |
 [3446; (3636)4] |
 [346; (3342)2; 3446; 3636] |
 [346; (3342)2; 3446; 3636] |
 [(36)2; 346; 3446; 3636] |
 [(3446)3; (3636)2] |
 [36; 3342; (3446)2; 3636] |
 [(3262)2; 3446; (3636)2] |
 [3262; 3446; (3636)3] |
 [3262; 3446; (3636)3] |
 [3446; (3636)4] |
 [36; 3342; (3446)2; 3636] |
 [(3262)2; 3446; (3636)2] |
 [3262; 3446; (3636)3] |
 [3262; 3446; (3636)3] |
 [3446; (3636)4] |
 [346; (3342)2; 3446; 3636] |
 [346; (3342)2; 3446; 3636] |
 [(36)2; 346; 3446; 3636] |
 [(3464)2; (3446)2; 3636] |
 [3262; (3446)2; (3636)2] |
 [346; (3342)2; 3446; 3636] |
 [346; (3342)2; 3446; 3636] |
 [(36)2; 346; 3446; 3636] |
 [(3464)2; (3446)2; 3636] |
 [3262; (3446)2; (3636)2] |
 [(3446)2; (3636)3] |
 [3262; (3446)2; (3636)2] |
 [(3446)2; (3636)3] |
 [(3342)3; 3446; 3636] |
 [(3342)3; 3446; 3636] |
 [(3446)2; (3636)3] |
 [3262; (3446)2; (3636)2] |
 [(3446)2; (3636)3] |
 [(3342)3; 3446; 3636] |
 [(3342)3; 3446; 3636] |
 [(36)2; 3342; 3446; 3636] |
 [36; 3342; 3262; 3446; 3636] |
 [36; 3342; 3446; (3636)2] |
 [(346)2; (3446)2; 3636] |
 [346; 3342; (3446)2; 3636] |
 [(36)2; 3342; 3446; 3636] |
 [36; 3342; 3262; 3446; 3636] |
 [36; 3342; 3446; (3636)2] |
 [(346)2; (3446)2; 3636] |
 [346; 3342; (3446)2; 3636] |
 [(346)2; (3446)2; 3636] |
 [(346)2; (3446)2; 3636] |
 [(346)2; (3446)2; 3636] |
 [(3342)2; (3446)2; 3636] |
 [(3342)2; (3446)2; 3636] |
 [(346)2; (3446)2; 3636] |
 [(346)2; (3446)2; 3636] |
 [(346)2; (3446)2; 3636] |
 [(3342)2; (3446)2; 3636] |
 [(3342)2; (3446)2; 3636] |
 [36; 33434; 3464; 3446; 3636] |
 [36; 346; 3464; 3446; 3636] |
 [36; 346; (3342)2; 3446] |
 [36; 346; 3342; 3262; 3446] |
 [36; (346)2; 3342; 3446] |
 [36; 33434; 3464; 3446; 3636] |
 [36; 346; 3464; 3446; 3636] |
 [36; 346; (3342)2; 3446] |
 [36; 346; 3342; 3262; 3446] |
 [36; (346)2; 3342; 3446] |
 [(346)2; (3342)2; 3446] |
 [36; (346)2; 3342; 3446] |
 [36; (346)2; 3342; 3446] |
 [346; (3342)3; 3446] |
 [(346)2; 3342; (3446)2] |
 [(346)2; (3342)2; 3446] |
 [36; (346)2; 3342; 3446] |
 [36; (346)2; 3342; 3446] |
 [346; (3342)3; 3446] |
 [(346)2; 3342; (3446)2] |
 [346; 3342; (3464)2; 3446] |
 [36; 3342; (334.12)2; 3464] |
 [(36)2; (3342)2; 3464] |
 [36; (3342)2; 33434; 3464] |
 [36; (3342)2; 33434; 3464] |
 [346; 3342; (3464)2; 3446] |
 [36; 3342; (334.12)2; 3464] |
 [(36)2; (3342)2; 3464] |
 [36; (3342)2; 33434; 3464] |
 [36; (3342)2; 33434; 3464] |
 [36; (3342)2; 334.12; 3464] |
 [36; 3342; (33434)2; 3464] |
 [36; 33434; (334.12)2; 3464] |
 [(36)2; 3342; 33434; 3464] |
 [36; 3342; (33434)2; 3464] |
 [36; (3342)2; 334.12; 3464] |
 [36; 3342; (33434)2; 3464] |
 [36; 33434; (334.12)2; 3464] |
 [(36)2; 3342; 33434; 3464] |
 [36; 3342; (33434)2; 3464] |
 [(33434)2; (3464)3] |
 [(3342)2; 334.12; (3464)2] |
 [36; (33434)2; (3464)2] |
 [(33434)3; (3464)2] |
 [3342; (33434)2; (3464)2] |
 [(33434)2; (3464)3] |
 [(3342)2; 334.12; (3464)2] |
 [36; (33434)2; (3464)2] |
 [(33434)3; (3464)2] |
 [3342; (33434)2; (3464)2] |
 [3342; (33434)2; (3464)2] |
 [3342; (33434)2; (3464)2] |
 [(33434)2; 343.12; (3464)2] |
 [36; 346; (3342)2; 3262] |
 [(36)3; 346; 3262] |
 [3342; (33434)2; (3464)2] |
 [3342; (33434)2; (3464)2] |
 [(33434)2; 343.12; (3464)2] |
 [36; 346; (3342)2; 3262] |
 [(36)3; 346; 3262] |
 [(36)3; 346; 3262] |
 [(36)3; 346; 3262] |
 [(36)2; 346; 3262; 3636] |
 [(36)2; (346)2; 3262] |
 [36; 346; (3262)2; 3636] |
 [(36)3; 346; 3262] |
 [(36)3; 346; 3262] |
 [(36)2; 346; 3262; 3636] |
 [(36)2; (346)2; 3262] |
 [36; 346; (3262)2; 3636] |
 [36; (346)2; (3262)2] |
 [(36)2; 346; (3262)2] |
 [346; (3262)3; 3636] |
 [36; (346)2; (3262)2] |
 [346; (3262)2; (3636)2] |
 [36; (346)2; (3262)2] |
 [(36)2; 346; (3262)2] |
 [346; (3262)3; 3636] |
 [36; (346)2; (3262)2] |
 [346; (3262)2; (3636)2] |
 [(346)2; (3262)2; 3636] |
 [36; (346)2; (3262)2] |
 [(3262)2; (3636)3] |
 [3262; (3636)4] |
 [346; (3342)2; 3262; 3636] |
 [(346)2; (3262)2; 3636] |
 [36; (346)2; (3262)2] |
 [(3262)2; (3636)3] |
 [3262; (3636)4] |
 [346; (3342)2; 3262; 3636] |
 [346; (3342)2; 3262; 3636] |
 [(36)2; 346; 3262; 3636] |
 [346; (3262)3; 3636] |
 [3262; (3636)4] |
 [36; 346; 3262; (3636)2] |
 [346; (3342)2; 3262; 3636] |
 [(36)2; 346; 3262; 3636] |
 [346; (3262)3; 3636] |
 [3262; (3636)4] |
 [36; 346; 3262; (3636)2] |
 [36; (346)2; 3262; 3636] |
 [36; (346)2; 3262; 3636] |
 [36; (346)2; 3262; 3636] |
 [(346)3; 3262; 3636] |
 [(346)2; 3262; (3636)2] |
 [36; (346)2; 3262; 3636] |
 [36; (346)2; 3262; 3636] |
 [36; (346)2; 3262; 3636] |
 [(346)3; 3262; 3636] |
 [(346)2; 3262; (3636)2] |
 [(346)3; 3262; 3636] |
 [36; (346)2; 3262; 3636] |
 [36; 346; (3262)2; 3636] |
 [36; 346; (3262)2; 3636] |
 [(346)2; (3262)2; 3636] |
 [(346)3; 3262; 3636] |
 [36; (346)2; 3262; 3636] |
 [36; 346; (3262)2; 3636] |
 [36; 346; (3262)2; 3636] |
 [(346)2; (3262)2; 3636] |
 [(36)3; 346; 3262] |
 [(36)3; 346; 3262] |
 [(36)2; (346)2; 3262] |
 [(36)2; (346)2; 3262] |
 [(36)2; (346)2; 3262] |
 [(36)3; 346; 3262] |
 [(36)3; 346; 3262] |
 [(36)2; (346)2; 3262] |
 [(36)2; (346)2; 3262] |
 [(36)2; (346)2; 3262] |
 [(346)3; 3262; 3636] |
 [(346)4; 3262] |
 [(346)4; 3262] |
 [36; (346)2; (3262)2] |
 [36; (346)2; (3262)2] |
 [(346)3; 3262; 3636] |
 [(346)4; 3262] |
 [(346)4; 3262] |
 [36; (346)2; (3262)2] |
 [36; (346)2; (3262)2] |
 [36; 346; (3636)3] |
 [(346)3; (3636)2] |
 [36; 346; (3636)3] |
 [346; (3342)2; (3636)2] |
 [346; (3342)2; (3636)2] |
 [36; 346; (3636)3] |
 [(346)3; (3636)2] |
 [36; 346; (3636)3] |
 [346; (3342)2; (3636)2] |
 [346; (3342)2; (3636)2] |
 [(36)2; 346; (3636)2] |
 [36; 346; (3636)3] |
 [36; 346; (3636)3] |
 [(36)2; (346)2; 3636] |
 [(346)3; (3636)2] |
 [(36)2; 346; (3636)2] |
 [36; 346; (3636)3] |
 [36; 346; (3636)3] |
 [(36)2; (346)2; 3636] |
 [(346)3; (3636)2] |
 [(346)2; (3636)3] |
 [(36)2; 346; 3342; 3636] |
 [(36)2; 346; 3342; 3636] |
 [(36)3; 346; 3636] |
 [(36)3; 346; 3636] |
 [(346)2; (3636)3] |
 [(36)2; 346; 3342; 3636] |
 [(36)2; 346; 3342; 3636] |
 [(36)3; 346; 3636] |
 [(36)3; 346; 3636] |
 [36; (346)3; 3636] |
 [(346)4; 3636] |
 [(36)2; (346)2; 3636] |
 [(36)2; (346)2; 3636] |
 [36; 346; (3342)2; 334.12] |
 [36; (346)3; 3636] |
 [(346)4; 3636] |
 [(36)2; (346)2; 3636] |
 [(36)2; (346)2; 3636] |
 [36; 346; (3342)2; 334.12] |
 [(36)2; 3342; 33434; 334.12] |
 [36; 33434; (334.12)2; 343.12] |
 [36; (33434)3; 343.12] |
 [36; (3342)4] |
 [(36)2; (3342)3] |
 [(36)2; 3342; 33434; 334.12] |
 [36; 33434; (334.12)2; 343.12] |
 [36; (33434)3; 343.12] |
 [36; (3342)4] |
 [(36)2; (3342)3] |
 [(36)2; (3342)3] |
 [(36)2; (3342)3] |
 [(36)2; (3342)3] |
 [(36)4; 3342] |
 [(36)4; 3342] |
 [(36)2; (3342)3] |
 [(36)2; (3342)3] |
 [(36)2; (3342)3] |
 [(36)4; 3342] |
 [(36)4; 3342] |
 [36; (346)2; (3342)2] |
 [(36)2; 346; 3342; 33434] |
 [(36)2; 3342; (33434)2] |
 [36; (3342)4] |
 [36; (3342)4] |
 [36; (346)2; (3342)2] |
 [(36)2; 346; 3342; 33434] |
 [(36)2; 3342; (33434)2] |
 [36; (3342)4] |
 [36; (3342)4] |
 [(36)3; (3342)2] |
 [(36)3; (3342)2] |
 [(36)3; (3342)2] |
 [(36)3; (3342)2] |
 [(36)3; (3342)2] |
 [(36)3; (3342)2] |
 [(36)3; (3342)2] |
 [(36)3; (3342)2] |
 [(36)3; (3342)2] |
 [(36)3; (3342)2] |
 [(36)3; (3342)2] |
 [(36)2; (3342)2; 33434] |
 [(3342)3; (33434)2] |
 [(3342)4; 33434] |
 [(3342)3; (33434)2] |
 [(36)3; (3342)2] |
 [(36)2; (3342)2; 33434] |
 [(3342)3; (33434)2] |
 [(3342)4; 33434] |
 [(3342)3; (33434)2] |
 [(3342)2; (33434)3] |
 [3342; (33434)4] |
 [(36)2; 3342; (33434)2] |
 [346; (3342)2; (33434)2] |
 [(3342)2; (33434)3] |
 [(3342)2; (33434)3] |
 [3342; (33434)4] |
 [(36)2; 3342; (33434)2] |
 [346; (3342)2; (33434)2] |
 [(3342)2; (33434)3] |
 [(36)2; 3342; (33434)2] |
 [(36)2; 3342; (33434)2] |
 [36; (33434)4] |
 [36; (33434)4] |
 [(36)4; 346] |
 [(36)2; 3342; (33434)2] |
 [(36)2; 3342; (33434)2] |
 [36; (33434)4] |
 [36; (33434)4] |
 [(36)4; 346] |
 [(36)4; 346] |
 [(36)4; 346] |
 [(36)2; (346)3] |
 [(36)2; (346)3] |
 [(36)3; (346)2] |
 [(36)4; 346] |
 [(36)4; 346] |
 [(36)2; (346)3] |
 [(36)2; (346)3] |
 [(36)3; (346)2] |
 [(36)3; (346)2] |
 [(36)3; (346)2] |
 [(36)3; (346)2] |
 [36; (346)4] |
 [(36)2; (346)3] |
 [(36)3; (346)2] |
 [(36)3; (346)2] |
 [(36)3; (346)2] |
 [36; (346)4] |
 [(36)2; (346)3] |
 [(36)2; (346)3] |
 [(36)2; (346)3] | |||
 [(36)2; (346)3] |
 [(36)2; (346)3] |
all n-gonal duals
[edit]Triangles
[edit] V6.6.6 |
V4.6.12 |
V4.82 |
V3.122 |
| Regular | Semiregular | ||
|---|---|---|---|
 V6.6.6 |
V4.6.12 |
V4.82 |
V3.122 |
Quadrilaterals
[edit] V4.4.4.4 |
V3.6.3.6 |
V3.3.6.6 |
V3.4.4.6 |
V3.4.6.4 |
V3.4.3.12 |
V3.3.4.12 |
| Regular | Semiregular | ||
|---|---|---|---|
 V4.4.4.4 |
V3.6.3.6 |
V3.4.6.4 | |
| 2-uniform | |||
|
|
|
|
|
| 3-uniform | |||
|
|
|
|
|
| 3-uniform | |||
|
|
|
|
|
| 3-uniform | |||
|
|
|
|
|
| 4-uniform | |||
|
|
|
|
|
| 4-uniform | |||
|
|
|
|
|
| 4-uniform | |||
|
|
|
|
|
| 4-uniform | |||
|
|
|
|
|
| 4-uniform | |||
|
|
|
|
|
| 4-uniform | |||
|
|
|
|
|
| 4-uniform | |||
|
|
|
|
|
| 4-uniform | |||
|
|
| ||
Pentagons
[edit]V3.3.4.3.4 |
V3.3.3.4.4 |
V3.3.3.3.6 |
| Semiregular | ||
|---|---|---|
V32.4.3.4 |
V33.42 |
V34.6 |
| 2-uniform | ||
|
| |
| 3-uniform | ||
|
|
|
|
| 4-uniform | ||
|
|
|
|
Hexagons
[edit] V3.3.3.3.3.3 |
 V3.3.3.3.3.3 = 6.6.6 |
References
[edit]- Grunbaum, Branko; Shephard, Geoffrey C. (1977). "Tilings by regular polygons". Math. Mag. 50 (5): 227–247. doi:10.2307/2689529. JSTOR 2689529.
- Grünbaum, Branko; Shephard, G. C. (1978). "The ninety-one types of isogonal tilings in the plane". Trans. Am. Math. Soc. 252: 335–353. doi:10.1090/S0002-9947-1978-0496813-3. MR 0496813.
{{cite journal}}: CS1 maint: multiple names: authors list (link) - Debroey, I.; Landuyt, F. (1981). "Equitransitive edge-to-edge tilings". Geometriae Dedicata. 11 (1): 47–60. doi:10.1007/BF00183189.
- Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1.
{{cite book}}: CS1 maint: multiple names: authors list (link) - Ren, Ding; Reay, John R. (1987). "The boundary characteristic and Pick's theorem in the Archimedean planar tilings". J. Combinat. Theory A. 44 (1): 110–119. doi:10.1016/0097-3165(87)90063-X.
- Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications. 17: 147–165. doi:10.1016/0898-1221(89)90156-9.
- Order in Space: A design source book, Keith Critchlow, 1970 ISBN 978-0-670-52830-1
- Sommerville, Duncan MacLaren Young (1958). An Introduction to the Geometry of n Dimensions. Dover Publications. Chapter X: The Regular Polytopes
- Préa, P. (1997). "Distance sequences and percolation thresholds in Archimedean Tilings". Mathl. Comput. Modelling. 26 (8–10): 317–320. doi:10.1016/S0895-7177(97)00216-1.
{{cite journal}}: CS1 maint: multiple names: authors list (link) - Kovic, Jurij (2011). "Symmetry-type graphs of Platonic and Archimedean solids". Math. Commun. 16 (2): 491–507.
- Pellicer, Daniel; Williams, Gordon (2012). "Minimal covers of the Archimedean Tilings". El. J. Combinat. 19 (3): #P6. doi:10.37236/2512.
External links
[edit]- n-uniform tilings Brian Galebach
- Dutch, Steve. "Uniform Tilings". Retrieved 2006-09-09.
- Weisstein, Eric W. "Tessellation". MathWorld.
- Weisstein, Eric W. "Semiregular tessellation". MathWorld.
- Weisstein, Eric W. "Demiregular tessellation". MathWorld.