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:

2-uniform tilings

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.

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. (All of them have 2 types of vertices, while one is 3-uniform.)

2-uniform				3-uniform
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'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 (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

2-uniforms

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-uniforms

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

References

• Ghyka, M. The Geometry of Art and Life, (1946), 2nd edition, New York: Dover, 1977.
• Keith Critchlow, Order in Space: A design source book, 1970, pp. 62–67
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  pp. 35–43
• Steinhaus, H. Mathematical Snapshots 3rd ed, (1969), Oxford University Press, and (1999) New York: Dover
• Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. ISBN 0-7167-1193-1. https://archive.org/details/isbn_0716711931.  p. 65
• 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. https://www.beloit.edu/computerscience/faculty/chavey/catalog/. 
• In Search of Demiregular Tilings, Helmer Aslaksen

External links

• Weisstein, Eric W.. "Demiregular tessellation". http://mathworld.wolfram.com/DemiregularTessellation.html. 
• n-uniform tilings Brian Galebach