List of k-uniform tilings
| File:1-uniform n1.svg 1-uniform (regular) |
File:1-uniform n6.svg 1-uniform (semiregular) |
| File:2-uniform n1.svg 2-uniform tiling |
File:3-uniform 5.svg 3-uniform tiling |
A k-uniform tiling is a tiling of tilings of the plane by convex regular polygons, connected edge-to-edge, with k types of vertices. The 1-uniform tiling include 3 regular tilings, and 8 semiregular tilings. A 1-uniform tiling can be defined by its vertex configuration. Higher k-uniform tilings are listed by their vertex figures, but are not generally uniquely identified this way. The complete lists of k-uniform tilings have been enumerated up to k=6. There are 20 2-uniform tilings, 61 3-uniform tilings, 151 4-uniform tilings, 332 5-uniform tilings, and 673 6-uniform tilings. This article lists all solutions up to k=5.
Other tilings of regular polygons that are not edge-to-edge allow different sized polygons, and continuous shifting positions of contact.
Classification
| File:3-uniform 57.svg by sides, yellow triangles, red squares (by polygons) |
File:3-uniform n57.svg by 4-isohedral positions, 3 shaded colors of triangles (by orbits) |
Such periodic tilings of convex polygons may be classified by the number of orbits of vertices, edges and tiles. If there are k orbits of vertices, a tiling is known as k-uniform or k-isogonal; if there are t orbits of tiles, as t-isohedral; if there are e orbits of edges, as e-isotoxal. k-uniform tilings with the same vertex figures can be further identified by their wallpaper group symmetry.
Enumeration
1-uniform tilings include 3 regular tilings, and 8 semiregular ones, with 2 or more types of regular polygon faces. There are 20 2-uniform tilings, 61 3-uniform tilings, 151 4-uniform tilings, 332 5-uniform tilings and 673 6-uniform tilings. Each can be grouped by the number m of distinct vertex figures, which are also called m-Archimedean tilings.[1] Finally, if the number of types of vertices is the same as the uniformity (m = k below), then the tiling is said to be Krotenheerdt. In general, the uniformity is greater than or equal to the number of types of vertices (m ≥ k), as different types of vertices necessarily have different orbits, but not vice versa. Setting m = n = k, there are 11 such tilings for n = 1; 20 such tilings for n = 2; 39 such tilings for n = 3; 33 such tilings for n = 4; 15 such tilings for n = 5; 10 such tilings for n = 6; and 7 such tilings for n = 7.
| m-Archimedean | |||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | ≥ 15 | Total | ||
| k-uniform | 1 | 11 | 0 | 11 | |||||||||||||
| 2 | 0 | 20 | 0 | 20 | |||||||||||||
| 3 | 0 | 22 | 39 | 0 | 61 | ||||||||||||
| 4 | 0 | 33 | 85 | 33 | 0 | 151 | |||||||||||
| 5 | 0 | 74 | 149 | 94 | 15 | 0 | 332 | ||||||||||
| 6 | 0 | 100 | 284 | 187 | 92 | 10 | 0 | 673 | |||||||||
| 7 | 0 | ? | ? | ? | ? | ? | 7 | 0 | ? | ||||||||
| 8 | 0 | ? | ? | ? | ? | ? | 20 | 0 | 0 | ? | |||||||
| 9 | 0 | ? | ? | ? | ? | ? | ? | 8 | 0 | 0 | ? | ||||||
| 10 | 0 | ? | ? | ? | ? | ? | ? | 27 | 0 | 0 | 0 | ? | |||||
| 11 | 0 | ? | ? | ? | ? | ? | ? | ? | 1 | 0 | 0 | 0 | ? | ||||
| 12 | 0 | ? | ? | ? | ? | ? | ? | ? | ? | 0 | 0 | 0 | 0 | ? | |||
| 13 | 0 | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | 0 | 0 | ? | ||
| 14 | 0 | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | 0 | 0 | ? | |
| ≥ 15 | 0 | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | ? | 0 | ? | |
| Total | 11 | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | 0 | ∞ | |
1-uniform tilings (regular)
A tiling is said to be regular if the symmetry group of the tiling acts transitively on the flags of the tiling, where a flag is a triple consisting of a mutually incident vertex, edge and tile of the tiling. This means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an edge-to-edge tiling by congruent regular polygons. There must be six equilateral triangles, four squares or three regular hexagons at a vertex, yielding the three regular tessellations.
| p6m, *632 | p4m, *442 | |
|---|---|---|
| File:1-uniform n11.svg | File:1-uniform n1.svg | File:1-uniform n5.svg |
| File:Vertex type 3-3-3-3-3-3.svg 36 (t=1, e=1) |
File:Vertex type 6-6-6.svg 63 (t=1, e=1) |
File:Vertex type 4-4-4-4.svg 44 (t=1, e=1) |
m-Archimedean and k-uniform tilings
Vertex-transitivity means that for every pair of vertices there is a symmetry operation mapping the first vertex to the second.[3] If the requirement of flag-transitivity is relaxed to one of vertex-transitivity, while the condition that the tiling is edge-to-edge is kept, there are eight additional tilings possible, known as Archimedean, uniform or demiregular tilings. Note that there are two mirror image (enantiomorphic or chiral) forms of 34.6 (snub hexagonal) tiling, only one of which is shown in the following table. All other regular and semiregular tilings are achiral. Grünbaum and Shephard distinguish the description of these tilings as Archimedean as referring only to the local property of the arrangement of tiles around each vertex being the same, and that as uniform as referring to the global property of vertex-transitivity. Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform.
1-uniform tilings (semiregular)
| p6m, *632 | |||||
|---|---|---|---|---|---|
| File:1-uniform n4.svg File:Vertex type 3-12-12.svg [ 3.122] (t=2, e=2) |
File:1-uniform n6.svg File:Vertex type 3-4-6-4.svg [ 3.4.6.4] (t=3, e=2) |
File:1-uniform n3.svg File:Vertex type 4-6-12.svg [ 4.6.12] (t=3, e=3) |
File:1-uniform n7.svg File:Vertex type 3-6-3-6.svg [ (3.6)2] (t=2, e=1) | ||
| File:1-uniform n2.svg File:Vertex type 4-8-8.svg [ 4.82] (t=2, e=2) |
File:1-uniform n9.svg File:Vertex type 3-3-4-3-4.svg [ 32.4.3.4] (t=2, e=2) |
File:1-uniform n8.svg File:Vertex type 3-3-3-4-4.svg [ 33.42] (t=2, e=3) |
File:1-uniform n10.svg File:Vertex type 3-3-3-3-6.svg [ 34.6] (t=3, e=3) | ||
2-uniform tilings
There are twenty (20) 2-uniform tilings of the Euclidean plane. (also called 2-isogonal tilings or demiregular tilings)[4][5][6] Vertex types are listed for each. If two tilings share the same two vertex types, they are given subscripts 1,2.
3-uniform tilings
There are 61 3-uniform tilings of the Euclidean plane. 39 are 3-Archimedean with 3 distinct vertex types, while 22 have 2 identical vertex types in different symmetry orbits. Chavey (1989)
3-uniform tilings, 3 vertex types
| File:3-uniform 5.svg [3.426; 3.6.3.6; 4.6.12] (t=6, e=7) |
File:3-uniform 6.svg [36; 324.12; 4.6.12] (t=5, e=6) |
File:3-uniform 7.svg [324.12; 3.4.6.4; 3.122] (t=5, e=6) |
File:3-uniform 8.svg [3.4.3.12; 3.4.6.4; 3.122] (t=5, e=6) |
File:3-uniform 35.svg [3342; 324.12; 3.4.6.4] (t=6, e=8) |
| File:3-uniform 47.svg [36; 3342; 324.12] (t=6, e=7) |
File:3-uniform 48.svg [36; 324.3.4; 324.12] (t=5, e=6) |
File:3-uniform 56.svg [346; 3342; 324.3.4] (t=5, e=6) |
File:3-uniform 24.svg [36; 324.3.4; 3.426] (t=5, e=6) |
File:3-uniform 34.svg [36; 324.3.4; 3.4.6.4] (t=5, e=6) |
| File:3-uniform 36.svg [36; 3342; 3.4.6.4] (t=6, e=6) |
File:3-uniform 37.svg [36; 324.3.4; 3.4.6.4] (t=6, e=6) |
File:3-uniform 54.svg [36; 3342; 324.3.4] (t=4, e=5) |
File:3-uniform 9.svg [324.12; 3.4.3.12; 3.122] (t=4, e=7) |
File:3-uniform 22.svg [3.4.6.4; 3.426; 44] (t=3, e=4) |
| File:3-uniform 25.svg [324.3.4; 3.4.6.4; 3.426] (t=4, e=6) |
File:3-uniform 23.svg [3342; 324.3.4; 44] (t=4, e=6) |
File:3-uniform 11.svg [3.426; 3.6.3.6; 44] (t=5, e=7) |
File:3-uniform 12.svg [3.426; 3.6.3.6; 44] (t=6, e=7) |
File:3-uniform 17.svg [3.426; 3.6.3.6; 44] (t=4, e=5) |
| File:3-uniform 18.svg [3.426; 3.6.3.6; 44] (t=5, e=6) |
File:3-uniform 27.svg [3342; 3262; 3.426] (t=5, e=8) |
File:3-uniform 29.svg [3262; 3.426; 3.6.3.6] (t=4, e=7) |
File:3-uniform 31.svg [3262; 3.426; 3.6.3.6] (t=5, e=7) |
File:3-uniform 33.svg [346; 3342; 3.426] (t=5, e=7) |
| File:3-uniform 1.svg [3262; 3.6.3.6; 63] (t=4, e=5) |
File:3-uniform 2.svg [3262; 3.6.3.6; 63] (t=2, e=4) |
File:3-uniform 3.svg [346; 3262; 63] (t=2, e=5) |
File:3-uniform 4.svg [36; 3262; 63] (t=2, e=3) |
File:3-uniform 38.svg [36; 346; 3262] (t=5, e=8) |
| File:3-uniform 40.svg [36; 346; 3262] (t=3, e=5) |
File:3-uniform 41.svg [36; 346; 3262] (t=3, e=6) |
File:3-uniform 44.svg [36; 346; 3.6.3.6] (t=5, e=6) |
File:3-uniform 42.svg [36; 346; 3.6.3.6] (t=4, e=4) |
File:3-uniform 43.svg [36; 346; 3.6.3.6] (t=3, e=3) |
| File:3-uniform 14.svg [36; 3342; 44] (t=4, e=6) |
File:3-uniform 15.svg [36; 3342; 44] (t=5, e=7) |
File:3-uniform 20.svg [36; 3342; 44] (t=3, e=5) |
File:3-uniform 21.svg [36; 3342; 44] (t=4, e=6) |
3-uniform tilings, 2 vertex types (2:1)
| File:3-uniform 26.svg [(3.4.6.4)2; 3.426] (t=6, e=6) |
File:3-uniform 58.svg [(36)2; 346] (t=3, e=4) |
File:3-uniform 59.svg [(36)2; 346] (t=5, e=5) |
File:3-uniform 60.svg [(36)2; 346] (t=7, e=9) |
File:3-uniform 61.svg [36; (346)2] (t=4, e=6) |
| File:3-uniform 57.svg [36; (324.3.4)2] (t=4, e=5) |
File:3-uniform 28.svg [(3.426)2; 3.6.3.6] (t=6, e=8) |
File:3-uniform 30.svg [3.426; (3.6.3.6)2] (t=4, e=6) |
File:3-uniform 32.svg [3.426; (3.6.3.6)2] (t=5, e=6) |
File:3-uniform 39.svg [3262; (3.6.3.6)2] (t=3, e=5) |
| File:3-uniform 45.svg [(346)2; 3.6.3.6] (t=4, e=7) |
File:3-uniform 46.svg [(346)2; 3.6.3.6] (t=4, e=7) |
File:3-uniform 10.svg [3342; (44)2] (t=4, e=7) |
File:3-uniform 13.svg [(3342)2; 44] (t=5, e=7) |
File:3-uniform 16.svg [3342; (44)2] (t=3, e=6) |
| File:3-uniform 19.svg [(3342)2; 44] (t=4, e=6) |
File:3-uniform 53.svg [(3342)2; 324.3.4] (t=5, e=8) |
File:3-uniform 55.svg [3342; (324.3.4)2] (t=6, e=9) |
File:3-uniform 52.svg [36; (3342)2] (t=5, e=7) |
File:3-uniform 51.svg [36; (3342)2] (t=4, e=6) |
| File:3-uniform 50.svg [(36)2; 3342] (t=6, e=7) |
File:3-uniform 49.svg [(36)2; 3342] (t=5, e=6) |
4-uniform tilings
There are 151 4-uniform tilings of the Euclidean plane. Brian Galebach's search reproduced Krotenheerdt's list of 33 4-uniform tilings with 4 distinct vertex types, as well as finding 85 of them with 3 vertex types, and 33 with 2 vertex types.
4-uniform tilings, 4 vertex types
There are 33 with 4 types of vertices.
| File:4-uniform 6.svg [33434; 3262; 3446; 63] |
File:4-uniform 26.svg [3342; 3262; 3446; 46.12] |
File:4-uniform 27.svg [33434; 3262; 3446; 46.12] |
File:4-uniform 131.svg [36; 3342; 33434; 334.12] |
File:4-uniform 34.svg [36; 33434; 334.12; 3.122] |
| File:4-uniform 35.svg [36; 33434; 343.12; 3.122] |
File:4-uniform 101.svg [36; 3342; 33434; 3464] |
File:4-uniform 103.svg [36; 3342; 33434; 3464] |
File:4-uniform 84.svg [36; 33434; 3464; 3446] |
File:4-uniform 9.svg [346; 3262; 3636; 63] |
| File:4-uniform 23.svg [346; 3262; 3636; 63] |
File:4-uniform 30.svg [334.12; 343.12; 3464; 46.12] |
File:4-uniform 37.svg [3342; 334.12; 343.12; 3.122] |
File:4-uniform 81.svg [3342; 334.12; 343.12; 44] |
File:4-uniform 36.svg [3342; 334.12; 343.12; 3.122] |
| File:4-uniform 82.svg [36; 3342; 33434; 44] |
File:4-uniform 85.svg [33434; 3262; 3464; 3446] |
File:4-uniform 92.svg [36; 3342; 3446; 3636] |
File:4-uniform 88.svg [36; 346; 3446; 3636] |
File:4-uniform 91.svg [36; 346; 3446; 3636] |
| File:4-uniform 96.svg [36; 346; 3342; 3446] |
File:4-uniform 98.svg [36; 346; 3342; 3446] |
File:4-uniform 5.svg [36; 346; 3262; 63] |
File:4-uniform 20.svg [36; 346; 3262; 63] |
File:4-uniform 12.svg [36; 346; 3262; 63] |
| File:4-uniform 13.svg [36; 346; 3262; 63] |
File:4-uniform 115.svg [36; 346; 3262; 3636] |
File:4-uniform 3.svg [3342; 3262; 3446; 63] |
File:4-uniform 18.svg [3342; 3262; 3446; 63] |
File:4-uniform 66.svg [3262; 3446; 3636; 44] |
| File:4-uniform 70.svg [3262; 3446; 3636; 44] |
File:4-uniform 46.svg [3262; 3446; 3636; 44] |
File:4-uniform 50.svg [3262; 3446; 3636; 44] |
4-uniform tilings, 3 vertex types (2:1:1)
There are 85 with 3 types of vertices.
4-uniform tilings, 2 vertex types (2:2) and (3:1)
There are 33 with 2 types of vertices, 12 with two pairs of types, and 21 with 3:1 ratio of types.
| File:4-uniform 29.svg [(3464)2; (46.12)2] |
File:4-uniform 106.svg [(33434)2; (3464)2] |
File:4-uniform 107.svg [(33434)2; (3464)2] |
File:4-uniform 125.svg [(346)2; (3636)2] |
File:4-uniform 150.svg [(36)2; (346)2] |
| File:4-uniform 143.svg [(3342)2; (33434)2] |
File:4-uniform 41.svg [(3342)2; (44)2] |
File:4-uniform 52.svg [(3342)2; (44)2] |
File:4-uniform 61.svg [(3342)2; (44)2] |
File:4-uniform 139.svg [(36)2; (3342)2] |
| File:4-uniform 140.svg [(36)2; (3342)2] |
File:4-uniform 141.svg [(36)2; (3342)2] |
| File:4-uniform 33.svg [343.12; (3.122)3] |
File:4-uniform 129.svg [(346)3; 3636] |
File:4-uniform 151.svg [36; (346)3] |
File:4-uniform 148.svg [(36)3; 346] |
File:4-uniform 149.svg [(36)3; 346] |
| File:4-uniform 142.svg [(3342)3; 33434] |
File:4-uniform 144.svg [3342; (33434)3] |
File:4-uniform 87.svg [3446; (3636)3] |
File:4-uniform 90.svg [3446; (3636)3] |
File:4-uniform 114.svg [3262; (3636)3] |
| File:4-uniform 117.svg [3262; (3636)3] |
File:4-uniform 38.svg [3342; (44)3] |
File:4-uniform 58.svg [3342; (44)3] |
File:4-uniform 53.svg [(3342)3; 44] |
File:4-uniform 72.svg [(3342)3; 44] |
| File:4-uniform 76.svg [(3342)3; 44] |
File:4-uniform 133.svg [36; (3342)3] |
File:4-uniform 134.svg [36; (3342)3] |
File:4-uniform 135.svg [36; (3342)3] |
File:4-uniform 136.svg [(36)3; 3342] |
| File:4-uniform 137.svg [(36)3; 3342] |
5-uniform tilings
There are 332 5-uniform tilings of the Euclidean plane. Brian Galebach's search identified 332 5-uniform tilings, with 2 to 5 types of vertices. There are 74 with 2 vertex types, 149 with 3 vertex types, 94 with 4 vertex types, and 15 with 5 vertex types.
5-uniform tilings, 5 vertex types
There are 15 5-uniform tilings with 5 unique vertex figure types.
| File:5-uniform 29.svg [33434; 3262; 3464; 3446; 63] |
File:5-uniform 30.svg [36; 346; 3262; 3636; 63] |
File:5-uniform 35.svg [36; 346; 3342; 3446; 46.12] |
File:5-uniform 128.svg [346; 3342; 33434; 3446; 44] |
File:5-uniform 196.svg [36; 33434; 3464; 3446; 3636] |
| File:5-uniform 197.svg [36; 346; 3464; 3446; 3636] |
File:5-uniform 43.svg [33434; 334.12; 3464; 3.12.12; 46.12] |
File:5-uniform 75.svg [36; 346; 3446; 3636; 44] |
File:5-uniform 80.svg [36; 346; 3446; 3636; 44] |
File:5-uniform 120.svg [36; 346; 3446; 3636; 44] |
| File:5-uniform 123.svg [36; 346; 3446; 3636; 44] |
File:5-uniform 124.svg [36; 3342; 3446; 3636; 44] |
File:5-uniform 125.svg [36; 346; 3342; 3446; 44] |
File:5-uniform 187.svg [36; 3342; 3262; 3446; 3636] |
File:5-uniform 199.svg [36; 346; 3342; 3262; 3446] |
5-uniform tilings, 4 vertex types (2:1:1:1)
There are 94 5-uniform tilings with 4 vertex types.
| File:5-uniform 33.svg [36; 33434; (3446)2; 46.12] |
File:5-uniform 37.svg [36; 33434; 3446; (46.12)2] |
File:5-uniform 38.svg [36; 33434; 3464; (46.12)2] |
File:5-uniform 207.svg [36; 3342; (334.12)2; 3464] |
File:5-uniform 211.svg [36; (3342)2; 334.12; 3464] |
| File:5-uniform 213.svg [36; 33434; (334.12)2; 3464] |
File:5-uniform 46.svg [36; 33434; 334.12; (3.12.12)2] |
File:5-uniform 285.svg [36; 346; (3342)2; 334.12] |
File:5-uniform 47.svg [36; 33434; 343.12; (3.12.12)2] |
File:5-uniform 48.svg [(3342)2; 334.12; 343.12; 3.12.12] |
| File:5-uniform 49.svg [(3342)2; 334.12; 343.12; 3.12.12] |
File:5-uniform 94.svg [(3342)2; 334.12; 343.12; 44] |
File:5-uniform 93.svg [33434; 3262; (3446)2; 44] |
File:5-uniform 144.svg [36; (3342)2; 33434; 44] |
File:5-uniform 145.svg [346; (3342)2; 33434; 44] |
| File:5-uniform 146.svg [36; 3342; (3464)2; 3446] |
File:5-uniform 147.svg [3342; 3262; 3464; (3446)2] |
File:5-uniform 148.svg [33434; 3262; 3464; (3446)2] |
File:5-uniform 149.svg [36; 33434; (3446)2; 3636] |
File:5-uniform 152.svg [3342; 33434; 3464; (3446)2] |
| File:5-uniform 153.svg [36; 33434; (3262)2; 3446] |
File:5-uniform 157.svg [3342; 3262; (3464)2; 3446] |
File:5-uniform 158.svg [33434; 3262; (3464)2; 3446] |
File:5-uniform 206.svg [346; 3342; (3464)2; 3446] |
File:5-uniform 209.svg [36; (3342)2; 33434; 3464] |
| File:5-uniform 210.svg [36; (3342)2; 33434; 3464] |
File:5-uniform 212.svg [36; 3342; (33434)2; 3464] |
File:5-uniform 214.svg [(36)2; 3342; 33434; 3464] |
File:5-uniform 215.svg [36; 3342; (33434)2; 3464] |
File:5-uniform 286.svg [(36)2; 3342; 33434; 334.12] |
| File:5-uniform 287.svg [36; 33434; (334.12)2; 343.12] |
File:5-uniform 297.svg [(36)2; 346; 3342; 33434] |
File:5-uniform 11.svg [(36)2; 346; 3262; 63] |
File:5-uniform 12.svg [36; (346)2; 3262; 63] |
File:5-uniform 228.svg [(36)2; 346; 3262; 3636] |
| File:5-uniform 230.svg [36; 346; (3262)2; 3636] |
File:5-uniform 246.svg [36; (346)2; 3262; 3636] |
File:5-uniform 242.svg [(36)2; 346; 3262; 3636] |
File:5-uniform 245.svg [36; 346; 3262; (3636)2] |
File:5-uniform 247.svg [36; (346)2; 3262; 3636] |
| File:5-uniform 248.svg [36; (346)2; 3262; 3636] |
File:5-uniform 252.svg [36; (346)2; 3262; 3636] |
File:5-uniform 253.svg [36; 346; (3262)2; 3636] |
File:5-uniform 254.svg [36; 346; (3262)2; 3636] |
File:5-uniform 3.svg [36; 346; 3262; (63)2] |
| File:5-uniform 7.svg [36; 346; (3262)2; 63] |
File:5-uniform 8.svg [346; (3262)2; 3636; 63] |
File:5-uniform 10.svg [(346)2; 3262; 3636; 63] |
File:5-uniform 14.svg [(36)2; 346; 3262; 63] |
File:5-uniform 15.svg [(36)2; 346; 3262; 63] |
| File:5-uniform 18.svg [36; 346; 3262; (63)2] |
File:5-uniform 20.svg [36; 346; 3262; (63)2] |
File:5-uniform 21.svg [36; 346; 3262; (63)2] |
File:5-uniform 23.svg [36; 346; (3262)2; 63] |
File:5-uniform 24.svg [346; (3262)2; 3636; 63] |
| File:5-uniform 26.svg [346; (3262)2; 3636; 63] |
File:5-uniform 27.svg [346; (3262)2; 3636; 63] |
File:5-uniform 28.svg [346; 3262; 3636; (63)2] |
File:5-uniform 31.svg [346; (3262)2; 3636; 63] |
File:5-uniform 16.svg [3342; 3262; 3446; (63)2] |
| File:5-uniform 1.svg [3342; 3262; 3446; (63)2] |
File:5-uniform 58.svg [3262; 3446; 3636; (44)2] |
File:5-uniform 62.svg [3262; 3446; 3636; (44)2] |
File:5-uniform 73.svg [3262; 3446; (3636)2; 44] |
File:5-uniform 78.svg [3262; 3446; (3636)2; 44] |
| File:5-uniform 91.svg [3342; 3262; 3446; (44)2] |
File:5-uniform 92.svg [346; 3342; 3446; (44)2] |
File:5-uniform 103.svg [3262; 3446; 3636; (44)2] |
File:5-uniform 107.svg [3262; 3446; 3636; (44)2] |
File:5-uniform 118.svg [3262; 3446; (3636)2; 44] |
| File:5-uniform 121.svg [3262; 3446; (3636)2; 44] |
File:5-uniform 126.svg [3342; 3262; 3446; (44)2] |
File:5-uniform 127.svg [346; 3342; 3446; (44)2] |
File:5-uniform 143.svg [346; (3342)2; 3636; 44] |
File:5-uniform 160.svg [36; 3342; (3446)2; 3636] |
| File:5-uniform 167.svg [346; (3342)2; 3446; 3636] |
File:5-uniform 168.svg [346; (3342)2; 3446; 3636] |
File:5-uniform 169.svg [(36)2; 346; 3446; 3636] |
File:5-uniform 171.svg [36; 3342; (3446)2; 3636] |
File:5-uniform 176.svg [346; (3342)2; 3446; 3636] |
| File:5-uniform 177.svg [346; (3342)2; 3446; 3636] |
File:5-uniform 178.svg [(36)2; 346; 3446; 3636] |
File:5-uniform 186.svg [(36)2; 3342; 3446; 3636] |
File:5-uniform 188.svg [36; 3342; 3446; (3636)2] |
File:5-uniform 190.svg [346; 3342; (3446)2; 3636] |
| File:5-uniform 198.svg [36; 346; (3342)2; 3446] |
File:5-uniform 240.svg [346; (3342)2; 3262; 3636] |
File:5-uniform 241.svg [346; (3342)2; 3262; 3636] |
File:5-uniform 200.svg [36; (346)2; 3342; 3446] |
File:5-uniform 202.svg [36; (346)2; 3342; 3446] |
| File:5-uniform 203.svg [36; (346)2; 3342; 3446] |
File:5-uniform 224.svg [36; 346; (3342)2; 3262] |
File:5-uniform 277.svg [(36)2; 346; 3342; 3636] |
File:5-uniform 278.svg [(36)2; 346; 3342; 3636] |
5-uniform tilings, 3 vertex types (3:1:1) and (2:2:1)
There are 149 5-uniform tilings, with 60 having 3:1:1 copies, and 89 having 2:2:1 copies.
| File:5-uniform 34.svg [(3446)2; (3636)2; 46.12] |
File:5-uniform 208.svg [(36)2; (3342)2; 3464] |
File:5-uniform 217.svg [(3342)2; 334.12; (3464)2] |
File:5-uniform 218.svg [36; (33434)2; (3464)2] |
File:5-uniform 220.svg [3342; (33434)2; (3464)2] |
| File:5-uniform 221.svg [3342; (33434)2; (3464)2] |
File:5-uniform 222.svg [3342; (33434)2; (3464)2] |
File:5-uniform 223.svg [(33434)2; 343.12; (3464)2] |
File:5-uniform 2.svg [36; (3262)2; (63)2] |
File:5-uniform 9.svg [(3262)2; (3636)2; 63] |
| File:5-uniform 307.svg [(36)2; (3342)2; 33434] |
File:5-uniform 313.svg [(36)2; 3342; (33434)2] |
File:5-uniform 314.svg [346; (3342)2; (33434)2] |
File:5-uniform 316.svg [(36)2; 3342; (33434)2] |
File:5-uniform 317.svg [(36)2; 3342; (33434)2] |
| File:5-uniform 19.svg [(3262)2; 3636; (63)2] |
File:5-uniform 56.svg [(3446)2; 3636; (44)2] |
File:5-uniform 57.svg [(3446)2; 3636; (44)2] |
File:5-uniform 59.svg [3446; (3636)2; (44)2] |
File:5-uniform 60.svg [(3446)2; 3636; (44)2] |
| File:5-uniform 61.svg [(3446)2; 3636; (44)2] |
File:5-uniform 63.svg [3446; (3636)2; (44)2] |
File:5-uniform 67.svg [36; (3342)2; (44)2] |
File:5-uniform 68.svg [(36)2; 3342; (44)2] |
File:5-uniform 69.svg [(36)2; 3342; (44)2] |
| File:5-uniform 70.svg [(3446)2; 3636; (44)2] |
File:5-uniform 71.svg [(3446)2; 3636; (44)2] |
File:5-uniform 72.svg [(3446)2; 3636; (44)2] |
File:5-uniform 76.svg [(3446)2; 3636; (44)2] |
File:5-uniform 77.svg [(3446)2; 3636; (44)2] |
| File:5-uniform 86.svg [36; (3342)2; (44)2] |
File:5-uniform 88.svg [(36)2; (3342)2; 44] |
File:5-uniform 101.svg [(3446)2; 3636; (44)2] |
File:5-uniform 102.svg [(3446)2; 3636; (44)2] |
File:5-uniform 104.svg [3446; (3636)2; (44)2] |
| File:5-uniform 105.svg [(3446)2; 3636; (44)2] |
File:5-uniform 106.svg [(3446)2; 3636; (44)2] |
File:5-uniform 108.svg [3446; (3636)2; (44)2] |
File:5-uniform 111.svg [36; (3342)2; (44)2] |
File:5-uniform 112.svg [(36)2; 3342; (44)2] |
| File:5-uniform 113.svg [(36)2; 3342; (44)2] |
File:5-uniform 115.svg [36; (3342)2; (44)2] |
File:5-uniform 116.svg [36; (3342)2; (44)2] |
File:5-uniform 117.svg [(3446)2; 3636; (44)2] |
File:5-uniform 133.svg [(36)2; (3342)2; 44] |
| File:5-uniform 138.svg [(36)2; (3342)2; 44] |
File:5-uniform 139.svg [(36)2; (3342)2; 44] |
File:5-uniform 142.svg [(36)2; (3342)2; 44] |
File:5-uniform 150.svg [(33434)2; 3262; (3446)2] |
File:5-uniform 155.svg [3342; (3262)2; (3446)2] |
| File:5-uniform 156.svg [3342; (3262)2; (3446)2] |
File:5-uniform 161.svg [3262; (3446)2; (3636)2] |
File:5-uniform 162.svg [(3262)2; 3446; (3636)2] |
File:5-uniform 172.svg [(3262)2; 3446; (3636)2] |
File:5-uniform 179.svg [(3464)2; (3446)2; 3636] |
| File:5-uniform 180.svg [3262; (3446)2; (3636)2] |
File:5-uniform 182.svg [3262; (3446)2; (3636)2] |
File:5-uniform 189.svg [(346)2; (3446)2; 3636] |
File:5-uniform 191.svg [(346)2; (3446)2; 3636] |
File:5-uniform 192.svg [(346)2; (3446)2; 3636] |
| File:5-uniform 193.svg [(346)2; (3446)2; 3636] |
File:5-uniform 194.svg [(3342)2; (3446)2; 3636] |
File:5-uniform 195.svg [(3342)2; (3446)2; 3636] |
File:5-uniform 201.svg [(346)2; (3342)2; 3446] |
File:5-uniform 205.svg [(346)2; 3342; (3446)2] |
| File:5-uniform 229.svg [(36)2; (346)2; 3262] |
File:5-uniform 231.svg [36; (346)2; (3262)2] |
File:5-uniform 232.svg [(36)2; 346; (3262)2] |
File:5-uniform 13.svg [(346)2; (3262)2; 63] |
File:5-uniform 17.svg [36; (3262)2; (63)2] |
| File:5-uniform 234.svg [36; (346)2; (3262)2] |
File:5-uniform 235.svg [346; (3262)2; (3636)2] |
File:5-uniform 236.svg [(346)2; (3262)2; 3636] |
File:5-uniform 237.svg [36; (346)2; (3262)2] |
File:5-uniform 250.svg [(346)2; 3262; (3636)2] |
| File:5-uniform 255.svg [(346)2; (3262)2; 3636] |
File:5-uniform 258.svg [(36)2; (346)2; 3262] |
File:5-uniform 259.svg [(36)2; (346)2; 3262] |
File:5-uniform 283.svg [(36)2; (346)2; 3636] |
File:5-uniform 284.svg [(36)2; (346)2; 3636] |
| File:5-uniform 296.svg [36; (346)2; (3342)2] |
File:5-uniform 260.svg [(36)2; (346)2; 3262] |
File:5-uniform 264.svg [36; (346)2; (3262)2] |
File:5-uniform 265.svg [36; (346)2; (3262)2] |
File:5-uniform 269.svg [346; (3342)2; (3636)2] |
| File:5-uniform 270.svg [346; (3342)2; (3636)2] |
File:5-uniform 271.svg [(36)2; 346; (3636)2] |
File:5-uniform 274.svg [(36)2; (346)2; 3636] |
File:5-uniform 298.svg [(36)2; 3342; (33434)2] |
5-uniform tilings, 2 vertex types (4:1) and (3:2)
There are 74 5-uniform tilings with 2 types of vertices, 27 with 4:1 and 47 with 3:2 copies of each.
There are 29 5-uniform tilings with 3 and 2 unique vertex figure types.
Higher k-uniform tilings
k-uniform tilings have been enumerated up to 6. There are 673 6-uniform tilings of the Euclidean plane. Brian Galebach's search reproduced Krotenheerdt's list of 10 6-uniform tilings with 6 distinct vertex types, as well as finding 92 of them with 5 vertex types, 187 of them with 4 vertex types, 284 of them with 3 vertex types, and 100 with 2 vertex types.
References
- ↑ k-uniform tilings by regular polygons Archived 2015-06-30 at the Wayback Machine Nils Lenngren, 2009
- ↑ "n-Uniform Tilings". probabilitysports.com. Retrieved 2019-06-21.
- ↑ Critchlow, p.60-61
- ↑ Critchlow, p.62-67
- ↑ Tilings and Patterns, Grünbaum and Shephard 1986, pp. 65-67
- ↑ "In Search of Demiregular Tilings" (PDF). Archived from the original (PDF) on 2016-05-07. Retrieved 2015-06-04.
- Grünbaum, Branko; Shephard, Geoffrey C. (1977). "Tilings by regular polygons". Math. Mag. 50 (5): 227–247. doi:10.2307/2689529. JSTOR 2689529.
- Grünbaum, Branko; Shephard, G. C. (1978). "The ninety-one types of isogonal tilings in the plane". Trans. Am. Math. Soc. 252: 335–353. doi:10.1090/S0002-9947-1978-0496813-3. MR 0496813.
- Debroey, I.; Landuyt, F. (1981). "Equitransitive edge-to-edge tilings". Geometriae Dedicata. 11 (1): 47–60. doi:10.1007/BF00183189. S2CID 122636363.
- Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1.
- Ren, Ding; Reay, John R. (1987). "The boundary characteristic and Pick's theorem in the Archimedean planar tilings". Journal of Combinatorial Theory. Series A. 44 (1): 110–119. doi:10.1016/0097-3165(87)90063-X.
- 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.
- Order in Space: A design source book, Keith Critchlow, 1970 ISBN 978-0-670-52830-1
- Sommerville, Duncan MacLaren Young (1958). An Introduction to the Geometry of n Dimensions. Dover Publications. Chapter X: The Regular Polytopes
- Préa, P. (1997). "Distance sequences and percolation thresholds in Archimedean Tilings". Mathl. Comput. Modelling. 26 (8–10): 317–320. doi:10.1016/S0895-7177(97)00216-1.
- Kovic, Jurij (2011). "Symmetry-type graphs of Platonic and Archimedean solids". Math. Commun. 16 (2): 491–507.
- Pellicer, Daniel; Williams, Gordon (2012). "Minimal Covers of the Archimedean Tilings, Part 1". The Electronic Journal of Combinatorics. 19 (3): #P6. doi:10.37236/2512.
- Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–57
External links
Euclidean and general tiling links:
- n-uniform tilings, Brian Galebach
- Dutch, Steve. "Uniform Tilings". Archived from the original on 2006-09-09. Retrieved 2006-09-09.
- Mitchell, K. "Semi-Regular Tilings". Retrieved 2006-09-09.
- Weisstein, Eric W. "Tessellation". MathWorld.
- Weisstein, Eric W. "Semiregular tessellation". MathWorld.
- Weisstein, Eric W. "Demiregular tessellation". MathWorld.
