Demiregular tiling


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

1		cmm, 2*22

2		pmm, *2222

1		cmm, 2*22

2		cmm, 2*22

2		pmm, *2222

1		pmm, *2222

p4m, *442

p4g, 4*2

1		pgg, 2×

2		p6m, *632

p6m, *632

1		p6, 632

2		cmm, 2*22

p6m, *632

p6m, *632

p6m, *632

p6m, *632

p6m, *632

p6m, *632

-

Ghyka's list (1946)

Ghyka lists 10 of them with 2 or 3 vertex types, calling them semiregular polymorph partitions.
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
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.

Critchlow's list (1970)

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
B
C
D
E
F
G
H
J
K
-

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
L2
M1
M2
N1
N2
P
Q1
Q2
R
S

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
elongated [triangular tiling|3.3.3.4.4]
3.3.3.3.6
36

124671014







E+L2L1+N1+GM1+M2N2+Q1Q1+N1+Q2