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User:Tomruen/t-isohedral tilings

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k-uniform tilings, edge-to-edge tilings of regular polygons, can be regrouped as t-isohedral tilings. 1, 2 and 3-uniform tilings are grouped below.

Face figures and notation

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A face figure defines the edge-to-edge connectivity of a t-isohedral tiling. The notation n:a1.a2...an implies a regular n-gon surrounded by regular faces in sequence of sides a1, a2, ...an.

Regular face figures
n:mn 3 4 6 8 12
3
3:33

3:43

3:63

3:123
4
4:34

4:44

4:84
6
6:36

6:46

6:66
Quasiregular face figures
4
4:(3.4)2

4:(3.6)2

4:(6.12)2
6
6:(4.12)3

6:(3.6)3
8
8:(4.8)4
12
12:(3.12)6

12:(4.6)6
Other face figures
3
3:32.4

3:3.42

3:32.6

3:3.62

3:42.6

3:4.62

3:4.122
4
4:3.43

4:3.42.6

4:3.4.6.4
6
6:32.64

6:3.6.3.63

6:(3.3.6)2

6:(3.6.6)2

6:3.65

6:35.6

6:(32.4)2

6:(3.42)2
12

1-isohedral tilings

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p6m, (*632) p4m, (*442)

36
[3:33]
(k=1, t=1, e=1)

63
[6:66]
(k=1, t=1, e=1)

44
[4:44]
(k=1, t=1, e=1)

2-isohedral tilings

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p6m, (*632) p4m, (*442) p4, (442)

(k=2, t=2, e=1)
[3:63; 6:36]

(k=1, t=2, e=2)
[3:63; 12:(3.12)6]

(k=1, t=2, e=2)
[4:84; 8:(4.8)4]

(k=1, t=2, e=2)
[3:3.42; 4:34]
cmm, (2*22) p6m, (*632) pmm, (*2222) cmm, (2*22)

(k=1, t=2, e=3)
[3:32.4; 4:(3.4)2]

(k=2, t=2, e=3)
[3:32.6; 6:(3.6)3]

(k=2, t=2, e=3)
[3:3.62; 6:(3.3.6)2]

(k=2, t=2, e=4)
[3:32.6; 6:(3.3.6)2]

(k=2, t=2, e=4)
[3:32.4; 4:3.44]
cmm, (2*22) p6m, (*632)

(k=3, t=2, e=4)
[3:3.62; 6:3264]

(k=3, t=2, e=5)
[3:32.6; 6:(3.6.6)2]

(k=3, t=2, e=3)
[3:32.6; 6:(3.6.6)2]

(k=4, t=2, e=5)
[3:32.6; 6:3.65]

3-isohedral tilings

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p6m, (*632) p6, (632)

(k=1, t=3, e=2)
[3:43; 4:(3.6)2; 6:46]
,,

(k=1, t=3, e=3)
[4:(6.12)2; 6:(4.12)3; 12:(4.6)6]
,,

(k=2, t=3, e=3)
[3:32.4; 3:43; 4:34]

(k=2, t=3, e=3)
[3:33; 3:32.6; 6:36]

(k=1, t=3, e=3)
[3:33; 3:32.6; 6:36]
,
p4m, (*442) pgg, (2*22) pmm, (*2222) pgg, (2*22)

(k=2, t=3, e=3)
[3:4.122; 4:34; 12:(32.12)4]

(k=2, t=3, e=4)
[3:4.62; 4:3.4.6.4; 6:(32.4)2]

(k=2, t=3, e=4)
[3:33; 3:32.4; 4:(3.4)2]

(k=2, t=3, e=5)
[3:32.4; 3:3.42; 4:44]

(k=3, t=3, e=4)
[3:33; 3:32.6; 6:36]

(k=3, t=3, e=5)
[3:3.62; 3:63; 6:35.6]

(k=3, t=3, e=6)
[3:32.4; 4:3.43; 4:44]

(k=3, t=3, e=4)
[3:42.6; 4:3.42.6; 6:(3.42)2]

(k=3, t=3, e=5)
[3:33; 3:32.6; 6:(32.6)2]

(k=3, t=3, e=6)
[3:33; 3:32.6; 6:(32.6)2]

(k=3, t=3, e=3)
[3:33; 3:32.6; 6:36]

(k=3, t=3, e=5)
[3:33; 3:32.4; 4:3.43]

4-isohedral tilings

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p6m, (*632)

(k=2, t=4, e=4)

(k=2, t=4, e=4)

(k=2, t=4, e=4)

(k=2, t=4, e=4)
pmm, (*2222) p4m, (*442)

(k=2, t=4, e=4)

(k=2, t=4, e=5)

(k=2, t=4, e=5)

(k=3, t=4, e=6)

(k=3, t=4, e=6)

(k=3, t=4, e=7)

(k=3, t=4, e=7)

(k=3, t=4, e=7)

(k=3, t=4, e=6)

(k=3, t=4, e=6)

(k=3, t=4, e=7)

(k=3, t=4, e=5)

(k=3, t=4, e=5)

(k=3, t=4, e=7)

(k=3, t=4, e=6)

(k=3, t=4, e=6)

(k=3, t=4, e=5)

(k=3, t=4, e=4)

(k=3, t=4, e=6)

(k=3, t=4, e=6)

(k=3, t=4, e=5)

5-isohedral tilings

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p6m, (*632) p6, (632)

(k=2, t=5, e=5)
[3:42.6; 4:(4.6)2; 4:3.4.3.6; 6:46; 6:(3.4)3]
File:Face figure 3 446.svgFile:Face figure 4 4646.svgFile:Face figure 4 3436.svgFile:Face figure 6 343434.svg

(k=2, t=5, e=7)
[]

(k=3, t=5, e=5)
]
(k=3, t=5, e=6)

(k=3, t=5, e=7)

(k=3, t=5, e=8)

(k=3, t=5, e=7)

(k=3, t=5, e=6)
<
(k=3, t=5, e=6)

(k=3, t=5, e=6)

(k=3, t=5, e=6)

(k=3, t=5, e=6)

(k=3, t=5, e=6)

(k=3, t=5, e=6)

(k=3, t=5, e=6)

(k=3, t=5, e=7)

(k=3, t=5, e=6)

(k=3, t=5, e=8)

(k=3, t=5, e=7)

(k=3, t=5, e=7)

(k=3, t=5, e=8)

(k=3, t=5, e=6)

(k=3, t=5, e=7)

6-isohedral tilings

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(k=3, t=6, e=6)

(k=3, t=6, e=8)

(k=3, t=6, e=7)

(k=3, t=6, e=7)

(k=3, t=6, e=9)

(k=3, t=6, e=8)

(k=3, t=6, e=7)

(k=3, t=6, e=6)

(k=3, t=6, e=6)

(k=3, t=6, e=7)

7-isohedral tilings

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(k=3, t=7, e=9)

References

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  • 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.
  • n-uniform tilings Brian Galebach