User:Tomruen/t-isohedral tilings

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

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

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

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

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

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

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

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

(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

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

References

• 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