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Demiregular tiling
From Wikipedia, the free encyclopedia
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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.
20 2-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) |
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Ghyka's list (1946)
Ghyka lists 10 of them with 2 or 3 vertex types, calling them semiregular polymorph partitions.[1]
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.)
Critchlow's list (1970)
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Critchlow identifies 14 demi-regular tessellations, with 7 being 2-uniform, and 7 being 3-uniform.
He codes letter names for the vertex types, with superscripts to distinguish face orders. He recognizes A, B, C, D, F, and J can't be a part of continuous coverings of the whole plane.
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