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 lists 10 of them with 2 or 3 vertex types, calling them semiregular polymorph partitions. [1]
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 |
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.
A (none) | B (none) | C (none) | D (none) | E (semi) | F (none) | G (semi) | H (semi) | J (none) | K (2) (reg) | |
---|---|---|---|---|---|---|---|---|---|---|
![]() 3.7.42 | ![]() 3.8.24 | ![]() 3.9.18 | ![]() 3.10.15 | ![]() 3.12.12 | ![]() 4.5.20 | ![]() 4.6.12 | ![]() 4.8.8 | ![]() 5.5.10 | ![]() 63 | |
L1 (demi) | L2 (demi) | M1 (demi) | M2 (semi) | N1 (demi) | N2 (semi) | P (3) (reg) | Q1 (semi) | Q2 (semi) | R (semi) | S (1) (reg) |
![]() 3.3.4.12 | ![]() 3.4.3.12 | ![]() 3.3.6.6 | ![]() 3.6.3.6 | ![]() 3.4.4.6 | ![]() 3.4.6.4 | ![]() 44 | ![]() 3.3.4.3.4 | ![]() 3.3.3.4.4 | ![]() 3.3.3.3.6 | ![]() 36 |
1 | 2 | 4 | 6 | 7 | 10 | 14 |
---|---|---|---|---|---|---|
![]() (3.12.12; 3.4.3.12) | ![]() (36; 32.4.12) | ![]() (4.6.12; 3.4.6.4) | ![]() ((3.6)2; 32.62) | ![]() (3.4.6.4; 32.4.3.4) | ![]() (36; 32.4.3.4) | ![]() (3.4.6.4; 3.42.6) |
E+L2 | L1+(1) | N1+G | M1+M2 | N2+Q1 | Q1+(1) | N1+Q2 |
3 | 5 | 8 | 9 | 11 | 12 | 13 |
---|---|---|---|---|---|---|
(3.3.4.3.4; 3.3.4.12, 3.4.3.12) | (36; 3.3.4.12; 3.3.4.3.4) | (3.3.4.3.4; 3.3.3.4.4, 4.3.4.6) | (36, 3.3.4.3.4) | (36; 3.3.4.3.4, 3.3.3.4.4) | (36; 3.3.4.3.4; 3.3.3.4.4) | (3.4.6.4; 3.42.6) |
L1+L2+Q1 | L1+Q1+(1) | N1+Q1+Q2 | Q1+(1) | Q1+Q2+(1) | Q1+Q2+(1) | N1+N2 |
Claimed Tilings and Duals | ||||||
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