Demiregular tiling
From Wikipedia, the free encyclopedia
In geometry, the demiregular tilings are a set of Euclidean tessellations made from 2 or more regular polygon faces. Different authors have listed different sets of tilings. A more systematic approach looking at symmetry orbits are the 2-uniform tilings of which there are 20. Some of the demiregular ones are actually 3-uniform tilings.
Grünbaum and Shephard enumerated the full list of 20 2-uniform tilings in Tilings and patterns, 1987:
cmm, 2*22 (44; 33.42)1 |
cmm, 2*22 (44; 33.42)2 |
pmm, *2222 (36; 33.42)1 |
cmm, 2*22 (36; 33.42)2 |
cmm, 2*22 (3.42.6; (3.6)2)2 |
pmm, *2222 (3.42.6; (3.6)2)1 |
pmm, *2222 ((3.6)2; 32.62) |
p4m, *442 (3.12.12; 3.4.3.12) |
p4g, 4*2 (33.42; 32.4.3.4)1 |
pgg, 2× (33.42; 32.4.3.4)2 |
p6m, *632 (36; 32.62) |
p6m, *632 (36; 34.6)1 |
p6, 632 (36; 34.6)2 |
cmm, 2*22 (32.62; 34.6) |
p6m, *632 (36; 32.4.3.4) |
p6m, *632 (3.4.6.4; 32.4.3.4) |
p6m, *632 (3.4.6.4; 33.42) |
p6m, *632 (3.4.6.4; 3.42.6) |
p6m, *632 (4.6.12; 3.4.6.4) |
p6m, *632 (36; 32.4.12) |
Ghyka's list (1946)
Ghyka lists 10 of them with 2 or 3 vertex types, calling them semiregular polymorph partitions.[1]
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| Plate XXVII No. 12 4.6.12 3.4.6.4 |
No. 13 3.4.6.4 3.3.3.4.4 |
No. 13 bis. 3.4.4.6 3.3.4.3.4 |
No. 13 ter. 3.4.4.6 3.3.3.4.4 |
Plate XXIV No. 13 quatuor. 3.4.6.4 3.3.4.3.4 |
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| No. 14 33.42 36 |
Plate XXVI No. 14 bis. 3.3.4.3.4 3.3.3.4.4 36 |
No. 14 ter. 33.42 36 |
No. 15 3.3.4.12 36 |
Plate XXV No. 16 3.3.4.12 3.3.4.3.4 36 |
Steinhaus's list (1969)
Steinhaus gives 5 examples of non-homogeneous tessellations of regular polygons beyond the 11 regular and semiregular ones.[2] (All of them have 2 types of vertices, while one is 3-uniform.)
| 2-uniform | 3-uniform | |||
|---|---|---|---|---|
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| Image 85 33.42 3.4.6.4 |
Image 86 32.4.3.4 3.4.6.4 |
Image 87 3.3.4.12 36 |
Image 89 33.42 32.4.3.4 |
Image 88 3.12.12 3.3.4.12 |
