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 tilingscmm, 2*22(44; 33.42)1cmm, 2*22(44; 33.42)2pmm, *2222(36; 33.42)1cmm, 2*22(36; 33.42)2cmm, 2*22(3.42.6; (3.6)2)2pmm, *2222(3.42.6; (3.6)2)1pmm, *2222((3.6)2; 32.62)p4m, *442(3.12.12; 3.4.3.12)p4g, 4*2(33.42; 32.4.3.4)1pgg, 2×(33.42; 32.4.3.4)2p6m, *632(36; 32.62)p6m, *632(36; 34.6)1p6, 632(36; 34.6)2cmm, 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.Ghyka (1946) pp. 73-80 Plate XXVIINo. 124.6.123.4.6.4No. 133.4.6.43.3.3.4.4No. 13 bis.3.4.4.63.3.4.3.4No. 13 ter.3.4.4.63.3.3.4.4Plate XXIVNo. 13 quatuor.3.4.6.43.3.4.3.4No. 14 33.4236Plate XXVINo. 14 bis.3.3.4.3.43.3.3.4.436No. 14 ter.33.4236No. 153.3.4.1236Plate XXVNo. 163.3.4.123.3.4.3.436
Steinhaus's list (1969) Steinhaus gives 5 examples of non-homogeneous tessellations of regular polygons beyond the 11 regular and semiregular ones.Steinhaus, 1969, p.79-82. (All of them have 2 types of vertices, while one is 3-uniform.) 2-uniform3-uniformImage 8533.423.4.6.4Image 8632.4.3.43.4.6.4Image 873.3.4.1236Image 8933.4232.4.3.4Image 883.12.123.3.4.12
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)3.7.423.8.243.9.183.10.153.12.124.5.204.6.124.8.85.5.1063L1(demi)L2(demi)M1(demi)M2(semi)N1(demi)N2(semi)P (3)(reg)Q1(semi)Q2(semi)R(semi)S (1)(reg)3.3.4.123.4.3.123.3.6.63.6.3.63.4.4.63.4.6.4443.3.4.3.43.3.3.4.43.3.3.3.636
+ 2-uniforms124671014(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+L2L1+(1)N1+GM1+M2N2+Q1Q1+(1)N1+Q2
+ 3-uniforms3589111213 (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+Q1L1+Q1+(1)N1+Q1+Q2Q1+(1)Q1+Q2+(1)Q1+Q2+(1)N1+N2Claimed 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 pp. 35–43 Steinhaus, H. Mathematical Snapshots 3rd ed, (1969), Oxford University Press, and (1999) New York: Dover p. 65
In Search of Demiregular Tilings, Helmer Aslaksen
External links
n-uniform tilings Brian Galebach