Demiregular tiling

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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 Paterns, 1987:

2-uniform tilings
cmm, 2*22
2-uniform n4.svg
(44; 33.42)1 		cmm, 2*22
2-uniform n3.svg
(44; 33.42)2 		pmm, *2222
2-uniform n14.svg
(36; 33.42)1 		cmm, 2*22
2-uniform n15.svg
(36; 33.42)2 		cmm, 2*22
2-uniform n6.svg
(3.42.6; (3.6)2)2 		pmm, *2222
2-uniform n7.svg
(3.42.6; (3.6)2)1 		pmm, *2222
2-uniform n11.svg
((3.6)2; 32.62)
p4m, *442
2-uniform n2.svg
(3.12.12; 3.4.3.12) 		p4g, 4*2
2-uniform n16.svg
(33.42; 32.4.3.4)1 		pgg, 2×
2-uniform n17.svg
(33.42; 32.4.3.4)2 		p6m, *632
2-uniform n10.svg
(36; 32.62) 		p6m, *632
2-uniform n19.svg
(36; 34.6)1 		p6, 632
2-uniform n20.svg
(36; 34.6)2 		cmm, 2*22
2-uniform n12.svg
(32.62; 34.6)
p6m, *632
2-uniform n18.svg
(36; 32.4.3.4) 		p6m, *632
2-uniform n9.svg
(3.4.6.4; 32.4.3.4) 		p6m, *632
2-uniform n8.svg
(3.4.6.4; 33.42) 		p6m, *632
2-uniform n5.svg
(3.4.6.4; 3.42.6) 		p6m, *632
2-uniform n1.svg
(4.6.12; 3.4.6.4) 		p6m, *632
2-uniform n13.svg
(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]

2-uniform n1.svg 2-uniform n8.svg 2-uniform n9.svg
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
2-uniform n13.svg 3-uniform 48.svg
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
2-uniform n8.svg 2-uniform n9.svg 2-uniform n13.svg 2-uniform n16.svg 3-uniform 9.svg
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)
Regular polygons meeting at vertex 3 3 7 42.svg
3.7.42 		Regular polygons meeting at vertex 3 3 8 24.svg
3.8.24 		Regular polygons meeting at vertex 3 3 9 18.svg
3.9.18 		Regular polygons meeting at vertex 3 3 10 15.svg
3.10.15 		Regular polygons meeting at vertex 3 3 12 12.svg
3.12.12 		Regular polygons meeting at vertex 3 4 5 20.svg
4.5.20 		Regular polygons meeting at vertex 3 4 6 12.svg
4.6.12 		Regular polygons meeting at vertex 3 4 8 8.svg
4.8.8 		Regular polygons meeting at vertex 3 5 5 10.svg
5.5.10 		Regular polygons meeting at vertex 3 6 6 6.svg
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)
Regular polygons meeting at vertex 4 3 3 4 12.svg
3.3.4.12 		Regular polygons meeting at vertex 4 3 4 3 12.svg
3.4.3.12 		Regular polygons meeting at vertex 4 3 3 6 6.svg
3.3.6.6 		Regular polygons meeting at vertex 4 3 6 3 6.svg
3.6.3.6 		Regular polygons meeting at vertex 4 3 4 4 6.svg
3.4.4.6 		Regular polygons meeting at vertex 4 3 4 6 4.svg
3.4.6.4 		Regular polygons meeting at vertex 4 4 4 4 4.svg
44 		Regular polygons meeting at vertex 5 3 3 4 3 4.svg
3.3.4.3.4 		Regular polygons meeting at vertex 5 3 3 3 4 4.svg
3.3.3.4.4 		Regular polygons meeting at vertex 5 3 3 3 3 6.svg
3.3.3.3.6 		Regular polygons meeting at vertex 6 3 3 3 3 3 3.svg
36
2-uniforms
1 2 4 6 7 10 14
2-uniform n2.svg
(3.12.12; 3.4.3.12) 		2-uniform n13.svg
(36; 32.4.12) 		2-uniform n1.svg
(4.6.12; 3.4.6.4) 		2-uniform n11.svg
((3.6)2; 32.62) 		2-uniform n9.svg
(3.4.6.4; 32.4.3.4) 		2-uniform n18.svg
(36; 32.4.3.4) 		2-uniform n5.svg
(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
Demi 1 Uniform.svg Demi 2 Uniform.svg Demi 3 Uniform.svg Demi 4 Uniform.svg Demi 5 Uniform.svg Demi 6 Uniform.svg Demi 7 Uniform.svg
Demi 1 Dual.svg Demi 2 Dual.svg Demi 3 Dual.svg Demi 4 Dual.svg Demi 5 Dual.svg Demi 6 Dual.svg Demi 7 Dual.svg

References[]

  1. ^ Ghyka (1946) pp. 73-80
  2. ^ Steinhaus, 1969, p.79-82.
  • 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. 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.
  • In Search of Demiregular Tilings, Helmer Aslaksen

External links[]

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