User:Tomruen/e-isotoxal tilings

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

Counts:

1-isotoxal: 4

2-isotoxal: 4

3-isotoxal: 10

4-isotoxal: 13

5-isotoxal: 17

6-isotoxal: 41

The 1- to 5-isotoxal edge-homogeneous tilings (40): [1]

The 6- and 7-isotoxal edge-homogeneous tilings (39): [2]

The 8- and 9-isotoxal edge-homogeneous tilings (31): [3]

The 10- to 13-isotoxal edge-homogeneous tilings (40): [4]

The 14- to 18-isotoxal edge-homogeneous tilings (41): [5]

The 19- to 29-isotoxal edge-homogeneous tilings (12): [6]

1-isotoxal tilings

36 = {3,6} (k=1, t=1, e=1)

63 = {6,3} (k=1, t=1, e=1)

(3.6)2 = r{6,3} (k=1, t=2, e=1)

44 = {4,4} (k=1, t=1, e=1)

(4.4)2 = r{4,4} (k=1, t=2, e=1)

2-isotoxal tilings

3.122 = t{6,3} (k=1, t=2, e=2)

3.4.6.4 = tr{6,3} (k=1, t=3, e=2)

4.82 = t{4,4} (k=1, t=2, e=2)

32.4.3.4 = s{4,4} (k=1, t=2, e=2)

3-isotoxal tilings

33.42 (k=1, t=2, e=3)

4.6.12 (k=1, t=3, e=3)

34.6 (k=1, t=3, e=3)

[36; 32.4.3.4] (k=2, t=3, e=3)

[3.12.12; 3.4.3.12] (k=2, t=3, e=3)

[36; 32.62] (k=2, t=2, e=3)

[36; 34.6]1 (k=2, t=3, e=3)

[3.6.3.6; 32.62] (k=2, t=2, e=3)

[36; 3262; 63] (k=3, t=2, e=3)

[36; 346; 3.6.3.6] (k=3, t=3, e=3)

4-isotoxal tilings

[3.4.6.4; 32.4.3.4] (k=2, t=4, e=4)	[3.4.6.4; 33.42] (k=2, t=4, e=4)	[4.6.12; 3.4.6.4] (k=2, t=4, e=4)	[36; 32.4.12] (k=2, t=4, e=4)
[32.62; 34.6] (k=2, t=2, e=4)	[3.42.6; 3.6.3.6]2 (k=2, t=3, e=4)	[3.42.6; 3.6.3.6]1 (k=2, t=4, e=4)	[44; 33.42]1 (k=2, t=2, e=4)	[36; 33.42]1 (k=2, t=3, e=4)

[(36)2; 346] (k=3, t=3, e=4)

[3262; 3.6.3.6; 63] (k=3, t=2, e=4)

[36; 346; 3.6.3.6] (k=3, t=4, e=4)

[3.4.6.4; 3.426; 44] (k=3, t=3, e=4)

5-isotoxal tilings

[3.4.6.4; 3.42.6] (k=2, t=5, e=5)

[33.42; 32.4.3.4]1 (k=2, t=4, e=5)

[44; 33.42]2 (k=2, t=3, e=5)

[36; 33.42]2 (k=2, t=4, e=5)

[(36)2; 346] (k=3, t=5, e=5)	[36; (324.3.4)2] (k=3, t=4, e=5)	[3262; (3.6.3.6)2] (k=3, t=3, e=5)	[36; 3342; 324.3.4] (k=3, t=4, e=5)
[3.426; 3.6.3.6; 44] (k=3, t=4, e=5)	[3262; 3.6.3.6; 63] (k=3, t=4, e=5)	[346; 3262; 63] (k=3, t=2, e=5)	[36; 346; 3262] (k=3, t=3, e=5)	[36; 3342; 44] (k=3, t=3, e=5)

6-isotoxal tilings

[(3.4.6.4)2; 3.426] (k=3, t=6, e=6)	[36; (346)2] (k=3, t=4, e=6)	[3.426; (3.6.3.6)2] (k=3, t=4, e=6)	[3.426; (3.6.3.6)2] (k=3, t=5, e=6)	[3342; (44)2] (k=3, t=3, e=6)
[(3342)2; 44] (k=3, t=4, e=6)	[36; (3342)2] (k=3, t=4, e=6)	[(36)2; 3342] (k=3, t=5, e=6)	[36; 3342; 44] (k=3, t=4, e=6)	[36; 3342; 44] (k=3, t=4, e=6)
[36; 324.12; 4.6.12] (k=3, t=5, e=6)	[324.12; 3.4.6.4; 3.122] (k=3, t=5, e=6)	[3.4.3.12; 3.4.6.4; 3.122] (k=3, t=5, e=6)	[36; 324.3.4; 324.12] (k=3, t=5, e=6)	[346; 3342; 324.3.4] (k=3, t=5, e=6)
[36; 324.3.4; 3.426] (k=3, t=5, e=6)	[36; 324.3.4; 3.4.6.4] (k=3, t=5, e=6)	[36; 3342; 3.4.6.4] (k=3, t=6, e=6)	[36; 324.3.4; 3.4.6.4] (k=3, t=6, e=6)	[324.3.4; 3.4.6.4; 3.426] (k=3, t=4, e=6)
[3342; 324.3.4; 44] (k=3, t=4, e=6)	[3.426; 3.6.3.6; 44] (k=3, t=5, e=6)	[36; 346; 3262] (k=3, t=3, e=6)	[36; 346; 3.6.3.6] (k=3, t=5, e=6)

7-isotoxal tilings

[36; 34.6]2 (k=2, t=5, e=7)

[(346)2; 3.6.3.6] (k=3, t=4, e=7)	[(346)2; 3.6.3.6] (k=3, t=4, e=7)	[3342; (44)2] (k=3, t=4, e=7)	[(3342)2; 44] (k=3, t=5, e=7)	[36; (3342)2] (k=3, t=5, e=7)
[(36)2; 3342] (k=3, t=6, e=7)	[3.426; 3.6.3.6; 4.6.12] (k=3, t=6, e=7)	[324.12; 3.4.3.12; 3.122] (k=3, t=4, e=7)	[36; 3342; 324.12] (k=3, t=6, e=7)	[3.426; 3.6.3.6; 44] (k=3, t=5, e=7)
[3.426; 3.6.3.6; 44] (k=3, t=6, e=7)	[3262; 3.426; 3.6.3.6] (k=3, t=4, e=7)	[3262; 3.426; 3.6.3.6] (k=3, t=5, e=7)	[346; 3342; 3.426] (k=3, t=5, e=7)	[36; 3342; 44] (k=3, t=5, e=7)

8-isotoxal tilings

[(3.426)2; 3.6.3.6] (k=3, t=6, e=8)

[(3342)2; 324.3.4] (k=3, t=5, e=8)

[3342; 324.12; 3.4.6.4] (k=3, t=6, e=8)

[3342; 3262; 3.426] (k=3, t=5, e=8)

[36; 346; 3262] (k=3, t=5, e=8)

9-isotoxal tilings

[(36)2; 346] (k=3, t=7, e=9)

[3342; (324.3.4)2] (k=3, t=6, 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