Euclidean tilings of regular polygons

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A periodic (2-uniform) tiling of regular triangles, squares, hexagons and dodecagons

Euclidean plane tilings by regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonices Mundi (Latin: The Harmony of the World, 1619).

Regular tilings[edit]

Following Grünbaum and Shephard (section 1.3), 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.

Regular tilings (3)
p6m, *632 p4m, *442
1-uniform n11.svg 1-uniform n1.svg 1-uniform n5.svg
Vertex type 3-3-3-3-3-3.svg
36
(t=1, e=1)
Vertex type 6-6-6.svg
63
(t=1, e=1)
Vertex type 4-4-4-4.svg
44
(t=1, e=1)

Archimedean, uniform or semiregular tilings[edit]

Vertex-transitivity means that for every pair of vertices there is a symmetry operation mapping the first vertex to the second.[1]

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 semiregular tilings. Note that there are two mirror image (enantiomorphic or chiral) forms of 34.6 (snub hexagonal) tiling, both of which are shown in the following table. All other regular and semiregular tilings are achiral.

Uniform tilings (8)
p6m, *632
1-uniform n4.svg
Vertex type 3-12-12.svg
3.122
(t=2, e=2)
1-uniform n6.svg
Vertex type 3-4-6-4.svg
3.4.6.4
(t=3, e=2)
1-uniform n3.svg
Vertex type 4-6-12.svg
4.6.12
(t=3, e=3)
1-uniform n7.svg
Vertex type 3-6-3-6.svg
(3.6)2
(t=2, e=1)
p4m, *442 p4, 442 cmm, 2*22 p6, 632
1-uniform n2.svg
Vertex type 4-8-8.svg
4.82
(t=2, e=2)
1-uniform n9.svg
Vertex type 3-3-4-3-4.svg
32.4.3.4
(t=2, e=2)
1-uniform n8.svg
Vertex type 3-3-3-4-4.svg
33.42
(t=2, e=3)
1-uniform n10.svg
Vertex type 3-3-3-3-6.svg
34.6
(t=3, e=3)

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.

k-uniform tilings[edit]

3-uniform tiling #57 of 61 colored
3-uniform 57.svg
by sides, yellow triangles, red squares
3-uniform n57.png
by 4-isohedral positions, 3 colors of triangles

Such periodic tilings may be classified by the number of orbits of vertices, edges and tiles. If there are n 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.

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.[2]

k-uniform, m-Archimedean tiling counts
m
k 1 2 3 4 5 6 Total
1 11 0 0 0 0 0 11
2 0 20 0 0 0 0 20
3 0 22 39 0 0 0 61
4 0 33 85 33 0 0 151
5 0 74 149 94 15 0 332
6 0 100 284 187 92 10 673


Other types of vertices in Euclidean plane tilings[edit]

For edge-to-edge Euclidean tilings, the internal angles of the polygons meeting at a vertex must add to 360 degrees. A regular n\,\!-gon has internal angle \left(1-\frac{2}{n}\right)180 degrees. There are seventeen combinations of regular polygons whose internal angles add up to 360 degrees, each being referred to as a species of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one types of vertex.

Only eleven of these can occur in a uniform tiling of regular polygons, given in previous sections.

In particular, if three polygons meet at a vertex and one has an odd number of sides, the other two polygons must be the same. If they are not, they would have to alternate around the first polygon, which is impossible if its number of sides is odd. By that restriction these six cannot appear in any tiling of regular polygons:

3 polygons at a vertex (unuseable)
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 4 5 20.svg
4.5.20
Regular polygons meeting at vertex 3 5 5 10.svg
5.5.10

These four can be used in k-uniform tiling:

4 polygons at a vertex (mixable with other vertex types)
Valid
vertex
types
Vertex type 3-3-4-12.svg
32.4.12
Vertex type 3-4-3-12.svg
3.4.3.12
Vertex type 3-3-6-6.svg
32.62
Vertex type 3-4-4-6.svg
3.42.6
Example
2-uniform
tilings
2-uniform 13.png
with 36
2-uniform 2.png
with 3.12.12
2-uniform 11.png
with (3.6)2
2-uniform 6.png
with (3.6)2

Dissected regular polygons[edit]

Some of the k-uniform tilings can be derived by symmetrically dissecting the tiling polygons with interior edges, for example:

Dissected polygons with original edges
Triangular tiling vertfig.png Hexagonal cupola flat.png Dissected dodecagon.png
Hexagon Dodecagon
(each has 2 orientations)

Some k-uniform tilings can be derived by dissecting regular polygons with new vertices along the original edges, for example:

Dissected with 1 or 2 middle vertex
Dissected triangle-3a.png Dissected triangle-36.png Dissected triangle-3b.png Dissected square.png Dissected square-3x3.png Dissected hexagon 36a.png Dissected hexagon 36b.png Dissected hexagon 3b.png
Triangle Square Hexagon


2-uniform tilings[edit]

Shown below are the twenty 2-uniform tilings, also called 2-isogonal tilings or demiregular tilings.[3][4][5] Vertex types are listed for each. If two tilings share the same two vertex types, they are given subscripts 1,2.

2-uniform tilings (20)
p6m, *632 p4m, *442
2-uniform 18.png
[36; 32.4.3.4]
(t=3, e=3)
2-uniform 9.png
[3.4.6.4; 32.4.3.4]
(t=4, e=4)
2-uniform 8.png
[3.4.6.4; 33.42]
(t=4, e=4)
2-uniform 5.png
[3.4.6.4; 3.42.6]
(t=5, e=5)
2-uniform 1.png
[4.6.12; 3.4.6.4]
(t=4, e=4)
2-uniform 13.png
[36; 32.4.12]
(t=4, e=4)
2-uniform 2.png
[3.12.12; 3.4.3.12]
(t=3, e=3)
p6m, *632 p6, 632 p6, 632 cmm, 2*22 pmm, *2222 cmm, 2*22 pmm, *2222
2-uniform 10.png
[36; 32.62]
(t=2, e=3)
2-uniform 19.png
[36; 34.6]1
(t=3, e=3)
2-uniform 20.png
[36; 34.6]2
(t=5, e=7)
2-uniform 12.png
[32.62; 34.6]
(t=2, e=4)
2-uniform 11.png
[3.6.3.6; 32.62]
(t=2, e=3)
2-uniform 6.png
[3.42.6; 3.6.3.6]2
(t=3, e=4)
2-uniform 7.png
[3.42.6; 3.6.3.6]1
(t=4, e=4)
p4g, 4*2 pgg, 2× cmm, 2*22 cmm, 2*22 pmm, *2222 cmm, 2*22
2-uniform 16.png
[33.42; 32.4.3.4]1
(t=4, e=5)
2-uniform 17.png
[33.42; 32.4.3.4]2
(t=6, e=6)
2-uniform 4.png
[44; 33.42]1
(t=2, e=4)
2-uniform 3.png
[44; 33.42]2
(t=3, e=5)
2-uniform 14.png
[36; 33.42]1
(t=3, e=4)
2-uniform 15.png
[36; 33.42]2
(t=4, e=5)

3-uniform tilings[edit]

Chavey (1989) lists 61 3-uniform tilings, 39 are 3-Archimedean with 3 distinct vertex types, while 22 have 2 identical vertex types in different symmetry orbits.

3-uniform tilings with 2 vertex types (22)
3-uniform 26.svg
[(3.4.6.4)2; 3.426]
(t=6, e=6)
3-uniform 58.svg
[(36)2; 346]
(t=3, e=4)
3-uniform 59.svg
[(36)2; 346]
(t=5, e=5)
3-uniform 60.svg
[(36)2; 346]
(t=7, e=9)
3-uniform 61.svg
[36; (346)2]
(t=4, e=6)
3-uniform 57.svg
[36; (324.3.4)2]
(t=4, e=5)
3-uniform 28.svg
[(3.426)2; 3.6.3.6]
(t=6, e=8)
3-uniform 30.svg
[3.426; (3.6.3.6)2]
(t=4, e=6)
3-uniform 32.svg
[3.426; (3.6.3.6)2]
(t=5, e=6)
3-uniform 39.svg
[3262; (3.6.3.6)2]
(t=3, e=5)
3-uniform 45.svg
[(346)2; 3.6.3.6]
(t=4, e=7)
3-uniform 46.svg
[(346)2; 3.6.3.6]
(t=4, e=7)
3-uniform 10.svg
[3342; (44)2]
(t=4, e=7)
3-uniform 13.svg
[(3342)2; 44]
(t=5, e=7)
3-uniform 16.svg
[3342; (44)2]
(t=3, e=6)
3-uniform 19.svg
[(3342)2; 44]
(t=4, e=6)
3-uniform 53.svg
[(3342)2; 324.3.4]
(t=5, e=8)
3-uniform 55.svg
[3342; (324.3.4)2]
(t=6, e=9)
3-uniform 52.svg
[36; (3342)2]
(t=5, e=7)
3-uniform 51.svg
[36; (3342)2]
(t=4, e=6)
3-uniform 50.svg
[(36)2; 3342]
(t=6, e=7)
3-uniform 49.svg
[(36)2; 3342]
(t=5, e=6)
3-uniform tilings with 3 vertex types (39)
3-uniform 5.svg
[3.426; 3.6.3.6; 4.6.12]
(t=6, e=7)
3-uniform 6.svg
[36; 324.12; 4.6.12]
(t=5, e=6)
3-uniform 7.svg
[324.12; 3.4.6.4; 3.122]
(t=5, e=6)
3-uniform 8.svg
[3.4.3.12; 3.4.6.4; 3.122]
(t=5, e=6)
3-uniform 35.svg
[3342; 324.12; 3.4.6.4]
(t=6, e=8)
3-uniform 47.svg
[36; 3342; 324.12]
(t=6, e=7)
3-uniform 48.svg
[36; 324.3.4; 324.12]
(t=5, e=6)
3-uniform 56.svg
[346; 3342; 324.3.4]
(t=5, e=6)
3-uniform 24.svg
[36; 324.3.4; 3.426]
(t=5, e=6)
3-uniform 34.svg
[36; 324.3.4; 3.4.6.4]
(t=5, e=6)
3-uniform 36.svg
[36; 3342; 3.4.6.4]
(t=6, e=6)
3-uniform 37.svg
[36; 324.3.4; 3.4.6.4]
(t=6, e=6)
3-uniform 54.svg
[36; 3342; 324.3.4]
(t=4, e=5)
3-uniform 9.svg
[324.12; 3.4.3.12; 3.122]
(t=4, e=7)
3-uniform 22.svg
[3.4.6.4; 3.426; 44]
(t=3, e=4)
3-uniform 25.svg
[324.3.4; 3.4.6.4; 3.426]
(t=4, e=6)
3-uniform 23.svg
[3342; 324.3.4; 44]
(t=4, e=6)
3-uniform 11.svg
[3.426; 3.6.3.6; 44]
(t=5, e=7)
3-uniform 12.svg
[3.426; 3.6.3.6; 44]
(t=6, e=7)
3-uniform 17.svg
[3.426; 3.6.3.6; 44]
(t=4, e=5)
3-uniform 18.svg
[3.426; 3.6.3.6; 44]
(t=5, e=6)
3-uniform 27.svg
[3342; 3262; 3.426]
(t=5, e=8)
3-uniform 29.svg
[3262; 3.426; 3.6.3.6]
(t=4, e=7)
3-uniform 31.svg
[3262; 3.426; 3.6.3.6]
(t=5, e=7)
3-uniform 33.svg
[346; 3342; 3.426]
(t=5, e=7)
3-uniform 1.svg
[3262; 3.6.3.6; 63]
(t=4, e=5)
3-uniform 2.svg
[3262; 3.6.3.6; 63]
(t=2, e=4)
3-uniform 3.svg
[346; 3262; 63]
(t=2, e=5)
3-uniform 4.svg
[36; 3262; 63]
(t=2, e=3)
3-uniform 38.svg
[36; 346; 3262]
(t=5, e=8)
3-uniform 40.svg
[36; 346; 3262]
(t=3, e=5)
3-uniform 41.svg
[36; 346; 3262]
(t=3, e=6)
3-uniform 44.svg
[36; 346; 3.6.3.6]
(t=5, e=6)
3-uniform 42.svg
[36; 346; 3.6.3.6]
(t=4, e=4)
3-uniform 43.svg
[36; 346; 3.6.3.6]
(t=3, e=3)
3-uniform 14.svg
[36; 3342; 44]
(t=4, e=6)
3-uniform 15.svg
[36; 3342; 44]
(t=5, e=7)
3-uniform 20.svg
[36; 3342; 44]
(t=3, e=5)
3-uniform 21.svg
[36; 3342; 44]
(t=4, e=6)

4-uniform tilings[edit]

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 with 2 pairs of 2 types of vertices (12)
4-uniform 29.svg
[(3464)2; (46.12)2]
4-uniform 106.svg
[(33434)2; (3464)2]
4-uniform 107.svg
[(33434)2; (3464)2]
4-uniform 125.svg
[(346)2; (3636)2]
4-uniform 150.svg
[(36)2; (346)2]
4-uniform 143.svg
[(3342)2; (33434)2]
4-uniform 41.svg
[(3342)2; (44)2]
4-uniform 52.svg
[(3342)2; (44)2]
4-uniform 61.svg
[(3342)2; (44)2]
4-uniform 139.svg
[(36)2; (3342)2]
4-uniform 140.svg
[(36)2; (3342)2]
4-uniform 141.svg
[(36)2; (3342)2]
4-uniform tilings with 2 types (3 and 1) vertices (21)
4-uniform 33.svg
[343.12; (3.122)3]
4-uniform 129.svg
[(346)3; 3636]
4-uniform 151.svg
[36; (346)3]
4-uniform 148.svg
[(36)3; 346]
4-uniform 149.svg
[(36)3; 346]
4-uniform 142.svg
[(3342)3; 33434]
4-uniform 144.svg
[3342; (33434)3]
4-uniform 87.svg
[3446; (3636)3]
4-uniform 90.svg
[3446; (3636)3]
4-uniform 114.svg
[3262; (3636)3]
4-uniform 117.svg
[3262; (3636)3]
4-uniform 38.svg
[3342; (44)3]
4-uniform 58.svg
[3342; (44)3]
4-uniform 53.svg
[(3342)3; 44]
4-uniform 72.svg
[(3342)3; 44]
4-uniform 76.svg
[(3342)3; 44]
4-uniform 133.svg
[36; (3342)3]
4-uniform 134.svg
[36; (3342)3]
4-uniform 135.svg
[36; (3342)3]
4-uniform 136.svg
[(36)3; 3342]
4-uniform 137.svg
[(36)3; 3342]
4-uniform tilings with 3 vertex types (85)
4-uniform 25.svg
[3464; (3446)2; 46.12]
4-uniform 28.svg
[3464; 3446; (46.12)2]
4-uniform 31.svg
[334.12; 3464; (3.122)2]
4-uniform 32.svg
[343.12; 3464; (3.122)2]
4-uniform 108.svg
[33434; 343.12; (3464)2]
4-uniform 130.svg
[(36)2; 3342; 334.12]
4-uniform 94.svg
[(3464)2; 3446; 3636]
4-uniform 95.svg
[3464; 3446; (3636)2]
4-uniform 83.svg
[3464; (3446)2; 3636]
4-uniform 146.svg
[(36)2; 3342; 33434]
4-uniform 138.svg
[(36)2; 3342; 33434]
4-uniform 1.svg
[36; 3262; (63)2]
4-uniform 2.svg
[36; 3262; (63)2]
4-uniform 7.svg
[36; (3262)2; 63]
4-uniform 8.svg
[36; (3262)2; 63]
4-uniform 14.svg
[36; 3262; (63)2]
4-uniform 17.svg
[36; 3262; (63)2]
4-uniform 110.svg
[36; (346)2; 3262]
4-uniform 111.svg
[36; (3262)2; 3636]
4-uniform 10.svg
[(346)2; 3262; 63]
4-uniform 24.svg
[(346)2; 3262; 63]
4-uniform 118.svg
[346; 3262; (3636)2]
4-uniform 119.svg
[346; 3262; (3636)2]
4-uniform 102.svg
[3342; 33434; (3464)2]
4-uniform 105.svg
[36; 33434; (3464)2]
4-uniform 104.svg
[36; (33434)2; 3464]
4-uniform 100.svg
[36; (3342)2; 3464]
4-uniform 93.svg
[(3464)2; 3446; 3636]
4-uniform 97.svg
[346; (33434)2; 3446]
4-uniform 145.svg
[36; 3342; (33434)2]
4-uniform 147.svg
[36; 3342; (33434)2]
4-uniform 57.svg
[(3342)2; 33434; 44]
4-uniform 79.svg
[(3342)2; 33434; 44]
4-uniform 80.svg
[3464; (3446)2; 44]
4-uniform 132.svg
[33434; (334.12)2; 343.12]
4-uniform 19.svg
[36; (3262)2; 63]
4-uniform 4.svg
[36; (3262)2; 63]
4-uniform 109.svg
[36; 346; (3262)2]
4-uniform 122.svg
[(36)2; 346; 3262]
4-uniform 123.svg
[(36)2; 346; 3262]
4-uniform 128.svg
[(36)2; 346; 3636]
4-uniform 112.svg
[346; (3262)2; 3636]
4-uniform 113.svg
[346; (3262)2; 3636]
4-uniform 120.svg
[(346)2; 3262; 3636]
4-uniform 116.svg
[(346)2; 3262; 3636]
4-uniform 124.svg
[36; 346; (3636)2]
4-uniform 21.svg
[3262; (3636)2; 63]
4-uniform 22.svg
[3262; (3636)2; 63]
4-uniform 11.svg
[(3262)2; 3636; 63]
4-uniform 15.svg
[3262; 3636; (63)2]
4-uniform 16.svg
[346; 3262; (63)2]
4-uniform 121.svg
[346; (3262)2; 3636]
4-uniform 86.svg
[3262; 3446; (3636)2]
4-uniform 89.svg
[3262; 3446; (3636)2]
4-uniform 126.svg
[346; (3342)2; 3636]
4-uniform 127.svg
[346; (3342)2; 3636]
4-uniform 99.svg
[346; 3342; (3446)2]
4-uniform 39.svg
[3446; 3636; (44)2]
4-uniform 40.svg
[3446; 3636; (44)2]
4-uniform 59.svg
[3446; 3636; (44)2]
4-uniform 60.svg
[3446; 3636; (44)2]
4-uniform 44.svg
[(3446)2; 3636; 44]
4-uniform 45.svg
[(3446)2; 3636; 44]
4-uniform 48.svg
[(3446)2; 3636; 44]
4-uniform 49.svg
[(3446)2; 3636; 44]
4-uniform 68.svg
[(3446)2; 3636; 44]
4-uniform 69.svg
[(3446)2; 3636; 44]
4-uniform 64.svg
[(3446)2; 3636; 44]
4-uniform 65.svg
[(3446)2; 3636; 44]
4-uniform 47.svg
[3446; (3636)2; 44]
4-uniform 51.svg
[3446; (3636)2; 44]
4-uniform 67.svg
[3446; (3636)2; 44]
4-uniform 71.svg
[3446; (3636)2; 44]
4-uniform 43.svg
[36; 3342; (44)2]
4-uniform 63.svg
[36; 3342; (44)2]
4-uniform 54.svg
[36; (3342)2; 44]
4-uniform 42.svg
[36; 3342; (44)2]
4-uniform 62.svg
[36; 3342; (44)2]
4-uniform 77.svg
[36; (3342)2; 44]
4-uniform 78.svg
[36; (3342)2; 44]
4-uniform 73.svg
[36; (3342)2; 44]
4-uniform 55.svg
[(36)2; 3342; 44]
4-uniform 56.svg
[(36)2; 3342; 44]
4-uniform 74.svg
[(36)2; 3342; 44]
4-uniform 75.svg
[(36)2; 3342; 44]
4-uniform tilings with 4 vertex types (33)
4-uniform 6.svg
[33434; 3262; 3446; 63]
4-uniform 26.svg
[3342; 3262; 3446; 46.12]
4-uniform 27.svg
[33434; 3262; 3446; 46.12]
4-uniform 131.svg
[36; 3342; 33434; 334.12]
4-uniform 34.svg
[36; 33434; 334.12; 3.122]
4-uniform 35.svg
[36; 33434; 343.12; 3.122]
4-uniform 101.svg
[36; 3342; 33434; 3464]
4-uniform 103.svg
[36; 3342; 33434; 3464]
4-uniform 84.svg
[36; 33434; 3464; 3446]
4-uniform 9.svg
[346; 3262; 3636; 63]
4-uniform 23.svg
[346; 3262; 3636; 63]
4-uniform 30.svg
[334.12; 343.12; 3464; 46.12]
4-uniform 37.svg
[3342; 334.12; 343.12; 3.122]
4-uniform 81.svg
[3342; 334.12; 343.12; 44]
4-uniform 36.svg
[3342; 334.12; 343.12; 3.122]
4-uniform 82.svg
[36; 3342; 33434; 44]
4-uniform 85.svg
[33434; 3262; 3464; 3446]
4-uniform 92.svg
[36; 3342; 3446; 3636]
4-uniform 88.svg
[36; 346; 3446; 3636]
4-uniform 91.svg
[36; 346; 3446; 3636]
4-uniform 96.svg
[36; 346; 3342; 3446]
4-uniform 98.svg
[36; 346; 3342; 3446]
4-uniform 5.svg
[36; 346; 3262; 63]
4-uniform 20.svg
[36; 346; 3262; 63]
4-uniform 12.svg
[36; 346; 3262; 63]
4-uniform 13.svg
[36; 346; 3262; 63]
4-uniform 115.svg
[36; 346; 3262; 3636]
4-uniform 3.svg
[3342; 3262; 3446; 63]
4-uniform 18.svg
[3342; 3262; 3446; 63]
4-uniform 66.svg
[3262; 3446; 3636; 44]
4-uniform 70.svg
[3262; 3446; 3636; 44]
4-uniform 46.svg
[3262; 3446; 3636; 44]
4-uniform 50.svg
[3262; 3446; 3636; 44]

5-uniform tilings[edit]

Brian Galebach's search identified 332 5-uniform tilings.

5-uniform tilings
5-uniform 1.svg
[3342; 3262; 3446; (63)2]
5-uniform 2.svg
[36; (3262)2; (63)2]
5-uniform 3.svg
[36; 346; 3262; (63)2]
5-uniform 4.svg
[3262; 3636; (63)3]
5-uniform 5.svg
[346; 3262; (63)3]
5-uniform 6.svg
[36; (3262)3; 63]
5-uniform 7.svg
[36; 346; (3262)2; 63]
5-uniform 8.svg
[346; (3262)2; 3636; 63]
5-uniform 9.svg
[(3262)2; (3636)2; 63]
5-uniform 10.svg
[(346)2; 3262; 3636; 63]
5-uniform 11.svg
[(36)2; 346; 3262; 63]
5-uniform 12.svg
[36; (346)2; 3262; 63]
5-uniform 13.svg
[(346)2; (3262)2; 63]
5-uniform 14.svg
[(36)2; 346; 3262; 63]
5-uniform 15.svg
[(36)2; 346; 3262; 63]
5-uniform 16.svg
[3342; 3262; 3446; (63)2]
5-uniform 17.svg
[36; (3262)2; (63)2]
5-uniform 18.svg
[36; 346; 3262; (63)2]
5-uniform 19.svg
[(3262)2; 3636; (63)2]
5-uniform 20.svg
[36; 346; 3262; (63)2]
5-uniform 21.svg
[36; 346; 3262; (63)2]
5-uniform 22.svg
[36; (3262)3; 63]
5-uniform 23.svg
[36; 346; (3262)2; 63]
5-uniform 24.svg
[346; (3262)2; 3636; 63]
5-uniform 25.svg
[3262; (3636)3; 63]
5-uniform 26.svg
[346; (3262)2; 3636; 63]
5-uniform 27.svg
[346; (3262)2; 3636; 63]
5-uniform 28.svg
[346; 3262; 3636; (63)2]
5-uniform 29.svg
[33434; 3262; 3464; 3446; 63]
5-uniform 30.svg
[36; 346; 3262; 3636; 63]
5-uniform 31.svg
[346; (3262)2; 3636; 63]
5-uniform 32.svg
[36; 334.12; (46.12)3]
5-uniform 33.svg
[36; 33434; (3446)2; 46.12]
5-uniform 34.svg
[(3446)2; (3636)2; 46.12]
5-uniform 35.svg
[36; 346; 3342; 3446; 46.12]
5-uniform 36.svg
[(3464)4; 46.12]
5-uniform 37.svg
[36; 33434; 3446; (46.12)2]
5-uniform 38.svg
[36; 33434; 3464; (46.12)2]
5-uniform 39.svg
[(3464)2; (46.12)3]
5-uniform 40.svg
[3464; 3446; (46.12)3]
5-uniform 41.svg
[(3464)2; (46.12)3]
5-uniform 42.svg
[36; (334.12)3; 46.12]
5-uniform 43.svg
[33434; 334.12; 3464; 3.12.12; 46.12]
5-uniform 44.svg
[343.12; (3.12.12)4]
5-uniform 45.svg
[334.12; 343.12; (3.12.12)3]
5-uniform 46.svg
[36; 33434; 334.12; (3.12.12)2]
5-uniform 47.svg
[36; 33434; 343.12; (3.12.12)2]
5-uniform 48.svg
[(3342)2; 334.12; 343.12; 3.12.12]
5-uniform 49.svg
[(3342)2; 334.12; 343.12; 3.12.12]
5-uniform 50.svg
[3342; (44)4]
5-uniform 51.svg
[3446; 3636; (44)3]
5-uniform 52.svg
[3446; 3636; (44)3]
5-uniform 53.svg
[(3342)2; (44)3]
5-uniform 54.svg
[36; 3342; (44)3]
5-uniform 55.svg
[36; 3342; (44)3]
5-uniform 56.svg
[(3446)2; 3636; (44)2]
5-uniform 57.svg
[(3446)2; 3636; (44)2]
5-uniform 58.svg
[3262; 3446; 3636; (44)2]
5-uniform 59.svg
[3446; (3636)2; (44)2]
5-uniform 60.svg
[(3446)2; 3636; (44)2]
5-uniform 61.svg
[(3446)2; 3636; (44)2]
5-uniform 62.svg
[3262; 3446; 3636; (44)2]
5-uniform 63.svg
[3446; (3636)2; (44)2]
5-uniform 64.svg
[(3342)2; (44)3]
5-uniform 65.svg
[(3342)2; (44)3]
5-uniform 66.svg
[(3342)3; (44)2]
5-uniform 67.svg
[36; (3342)2; (44)2]
5-uniform 68.svg
[(36)2; 3342; (44)2]
5-uniform 69.svg
[(36)2; 3342; (44)2]
5-uniform 70.svg
[(3446)2; 3636; (44)2]
5-uniform 71.svg
[(3446)2; 3636; (44)2]
5-uniform 72.svg
[(3446)2; 3636; (44)2]
5-uniform 73.svg
[3262; 3446; (3636)2; 44]
5-uniform 74.svg
[3446; (3636)3; 44]
5-uniform 75.svg
[36; 346; 3446; 3636; 44]
5-uniform 76.svg
[(3446)2; 3636; (44)2]
5-uniform 77.svg
[(3446)2; 3636; (44)2]
5-uniform 78.svg
[3262; 3446; (3636)2; 44]
5-uniform 79.svg
[3446; (3636)3; 44]
5-uniform 80.svg
[36; 346; 3446; 3636; 44]
5-uniform 81.svg
[(3342)2; (44)3]
5-uniform 82.svg
[(3342)3; (44)2]
5-uniform 83.svg
[(3342)4; 44]
5-uniform 84.svg
[36; (3342)3; 44]
5-uniform 85.svg
[36; (3342)3; 44]
5-uniform 86.svg
[36; (3342)2; (44)2]
5-uniform 87.svg
[36; (3342)3; 44]
5-uniform 88.svg
[(36)2; (3342)2; 44]
5-uniform 89.svg
[(36)3; 3342; 44]
5-uniform 90.svg
[(36)3; 3342; 44]
5-uniform 91.svg
[3342; 3262; 3446; (44)2]
5-uniform 92.svg
[346; 3342; 3446; (44)2]
5-uniform 93.svg
[33434; 3262; (3446)2; 44]
5-uniform 94.svg
[(3342)2; 334.12; 343.12; 44]
5-uniform 95.svg
[3342; (44)4]
5-uniform 96.svg
[3446; 3636; (44)3]
5-uniform 97.svg
[3446; 3636; (44)3]
5-uniform 98.svg
[(3342)2; (44)3]
5-uniform 99.svg
[36; 3342; (44)3]
5-uniform 100.svg
[36; 3342; (44)3]
5-uniform 101.svg
[(3446)2; 3636; (44)2]
5-uniform 102.svg
[(3446)2; 3636; (44)2]
5-uniform 103.svg
[3262; 3446; 3636; (44)2]
5-uniform 104.svg
[3446; (3636)2; (44)2]
5-uniform 105.svg
[(3446)2; 3636; (44)2]
5-uniform 106.svg
[(3446)2; 3636; (44)2]
5-uniform 107.svg
[3262; 3446; 3636; (44)2]
5-uniform 108.svg
[3446; (3636)2; (44)2]
5-uniform 109.svg
[(3342)2; (44)3]
5-uniform 110.svg
[(3342)3; (44)2]
5-uniform 111.svg
[36; (3342)2; (44)2]
5-uniform 112.svg
[(36)2; 3342; (44)2]
5-uniform 113.svg
[(36)2; 3342; (44)2]
5-uniform 114.svg
[(3342)3; (44)2]
5-uniform 115.svg
[36; (3342)2; (44)2]
5-uniform 116.svg
[36; (3342)2; (44)2]
5-uniform 117.svg
[(3446)2; 3636; (44)2]
5-uniform 118.svg
[3262; 3446; (3636)2; 44]
5-uniform 119.svg
[3446; (3636)3; 44]
5-uniform 120.svg
[36; 346; 3446; 3636; 44]
5-uniform 121.svg
[3262; 3446; (3636)2; 44]
5-uniform 122.svg
[3446; (3636)3; 44]
5-uniform 123.svg
[36; 346; 3446; 3636; 44]
5-uniform 124.svg
[36; 3342; 3446; 3636; 44]
5-uniform 125.svg
[36; 346; 3342; 3446; 44]
5-uniform 126.svg
[3342; 3262; 3446; (44)2]
5-uniform 127.svg
[346; 3342; 3446; (44)2]
5-uniform 128.svg
[346; 3342; 33434; 3446; 44]
5-uniform 129.svg
[(3342)4; 44]
5-uniform 130.svg
[36; (3342)3; 44]
5-uniform 131.svg
[36; (3342)3; 44]
5-uniform 132.svg
[36; (3342)3; 44]
5-uniform 133.svg
[(36)2; (3342)2; 44]
5-uniform 134.svg
[(36)3; 3342; 44]
5-uniform 135.svg
[(36)3; 3342; 44]
5-uniform 136.svg
[(3342)4; 44]
5-uniform 137.svg
[36; (3342)3; 44]
5-uniform 138.svg
[(36)2; (3342)2; 44]
5-uniform 139.svg
[(36)2; (3342)2; 44]
5-uniform 140.svg
[36; (3342)3; 44]
5-uniform 141.svg
[36; (3342)3; 44]
5-uniform 142.svg
[(36)2; (3342)2; 44]
5-uniform 143.svg
[346; (3342)2; 3636; 44]
5-uniform 144.svg
[36; (3342)2; 33434; 44]
5-uniform 145.svg
[346; (3342)2; 33434; 44]
5-uniform 146.svg
[36; 3342; (3464)2; 3446]
5-uniform 147.svg
[3342; 3262; 3464; (3446)2]
5-uniform 148.svg
[33434; 3262; 3464; (3446)2]
5-uniform 149.svg
[36; 33434; (3446)2; 3636]
5-uniform 150.svg
[(33434)2; 3262; (3446)2]
5-uniform 151.svg
[(3464)3; (3446)2]
5-uniform 152.svg
[3342; 33434; 3464; (3446)2]
5-uniform 153.svg
[36; 33434; (3262)2; 3446]
5-uniform 154.svg
[(3342)3; 3262; 3446]
5-uniform 155.svg
[3342; (3262)2; (3446)2]
5-uniform 156.svg
[3342; (3262)2; (3446)2]
5-uniform 157.svg
[3342; 3262; (3464)2; 3446]
5-uniform 158.svg
[33434; 3262; (3464)2; 3446]
5-uniform 159.svg
[(3446)3; (3636)2]
5-uniform 160.svg
[36; 3342; (3446)2; 3636]
5-uniform 161.svg
[3262; (3446)2; (3636)2]
5-uniform 162.svg
[(3262)2; 3446; (3636)2]
5-uniform 163.svg
[3262; 3446; (3636)3]
5-uniform 164.svg
[(3446)2; (3636)3]
5-uniform 165.svg
[3262; 3446; (3636)3]
5-uniform 166.svg
[3446; (3636)4]
5-uniform 167.svg
[346; (3342)2; 3446; 3636]
5-uniform 168.svg
[346; (3342)2; 3446; 3636]
5-uniform 169.svg
[(36)2; 346; 3446; 3636]
5-uniform 170.svg
[(3446)3; (3636)2]
5-uniform 171.svg
[36; 3342; (3446)2; 3636]
5-uniform 172.svg
[(3262)2; 3446; (3636)2]
5-uniform 173.svg
[3262; 3446; (3636)3]
5-uniform 174.svg
[3262; 3446; (3636)3]
5-uniform 175.svg
[3446; (3636)4]
5-uniform 176.svg
[346; (3342)2; 3446; 3636]
5-uniform 177.svg
[346; (3342)2; 3446; 3636]
5-uniform 178.svg
[(36)2; 346; 3446; 3636]
5-uniform 179.svg
[(3464)2; (3446)2; 3636]
5-uniform 180.svg
[3262; (3446)2; (3636)2]
5-uniform 181.svg
[(3446)2; (3636)3]
5-uniform 182.svg
[3262; (3446)2; (3636)2]
5-uniform 183.svg
[(3446)2; (3636)3]
5-uniform 184.svg
[(3342)3; 3446; 3636]
5-uniform 185.svg
[(3342)3; 3446; 3636]
5-uniform 186.svg
[(36)2; 3342; 3446; 3636]
5-uniform 187.svg
[36; 3342; 3262; 3446; 3636]
5-uniform 188.svg
[36; 3342; 3446; (3636)2]
5-uniform 189.svg
[(346)2; (3446)2; 3636]
5-uniform 190.svg
[346; 3342; (3446)2; 3636]
5-uniform 191.svg
[(346)2; (3446)2; 3636]
5-uniform 192.svg
[(346)2; (3446)2; 3636]
5-uniform 193.svg
[(346)2; (3446)2; 3636]
5-uniform 194.svg
[(3342)2; (3446)2; 3636]
5-uniform 195.svg
[(3342)2; (3446)2; 3636]
5-uniform 196.svg
[36; 33434; 3464; 3446; 3636]
5-uniform 197.svg
[36; 346; 3464; 3446; 3636]
5-uniform 198.svg
[36; 346; (3342)2; 3446]
5-uniform 199.svg
[36; 346; 3342; 3262; 3446]
5-uniform 200.svg
[36; (346)2; 3342; 3446]
5-uniform 201.svg
[(346)2; (3342)2; 3446]
5-uniform 202.svg
[36; (346)2; 3342; 3446]
5-uniform 203.svg
[36; (346)2; 3342; 3446]
5-uniform 204.svg
[346; (3342)3; 3446]
5-uniform 205.svg
[(346)2; 3342; (3446)2]
5-uniform 206.svg
[346; 3342; (3464)2; 3446]
5-uniform 207.svg
[36; 3342; (334.12)2; 3464]
5-uniform 208.svg
[(36)2; (3342)2; 3464]
5-uniform 209.svg
[36; (3342)2; 33434; 3464]
5-uniform 210.svg
[36; (3342)2; 33434; 3464]
5-uniform 211.svg
[36; (3342)2; 334.12; 3464]
5-uniform 212.svg
[36; 3342; (33434)2; 3464]
5-uniform 213.svg
[36; 33434; (334.12)2; 3464]
5-uniform 214.svg
[(36)2; 3342; 33434; 3464]
5-uniform 215.svg
[36; 3342; (33434)2; 3464]
5-uniform 216.svg
[(33434)2; (3464)3]
5-uniform 217.svg
[(3342)2; 334.12; (3464)2]
5-uniform 218.svg
[36; (33434)2; (3464)2]
5-uniform 219.svg
[(33434)3; (3464)2]
5-uniform 220.svg
[3342; (33434)2; (3464)2]
5-uniform 221.svg
[3342; (33434)2; (3464)2]
5-uniform 222.svg
[3342; (33434)2; (3464)2]
5-uniform 223.svg
[(33434)2; 343.12; (3464)2]
5-uniform 224.svg
[36; 346; (3342)2; 3262]
5-uniform 225.svg
[(36)3; 346; 3262]
5-uniform 226.svg
[(36)3; 346; 3262]
5-uniform 227.svg
[(36)3; 346; 3262]
5-uniform 228.svg
[(36)2; 346; 3262; 3636]
5-uniform 229.svg
[(36)2; (346)2; 3262]
5-uniform 230.svg
[36; 346; (3262)2; 3636]
5-uniform 231.svg
[36; (346)2; (3262)2]
5-uniform 232.svg
[(36)2; 346; (3262)2]
5-uniform 233.svg
[346; (3262)3; 3636]
5-uniform 234.svg
[36; (346)2; (3262)2]
5-uniform 235.svg
[346; (3262)2; (3636)2]
5-uniform 236.svg
[(346)2; (3262)2; 3636]
5-uniform 237.svg
[36; (346)2; (3262)2]
5-uniform 238.svg
[(3262)2; (3636)3]
5-uniform 239.svg
[3262; (3636)4]
5-uniform 240.svg
[346; (3342)2; 3262; 3636]
5-uniform 241.svg
[346; (3342)2; 3262; 3636]
5-uniform 242.svg
[(36)2; 346; 3262; 3636]
5-uniform 243.svg
[346; (3262)3; 3636]
5-uniform 244.svg
[3262; (3636)4]
5-uniform 245.svg
[36; 346; 3262; (3636)2]
5-uniform 246.svg
[36; (346)2; 3262; 3636]
5-uniform 247.svg
[36; (346)2; 3262; 3636]
5-uniform 248.svg
[36; (346)2; 3262; 3636]
5-uniform 249.svg
[(346)3; 3262; 3636]
5-uniform 250.svg
[(346)2; 3262; (3636)2]
5-uniform 251.svg
[(346)3; 3262; 3636]
5-uniform 252.svg
[36; (346)2; 3262; 3636]
5-uniform 253.svg
[36; 346; (3262)2; 3636]
5-uniform 254.svg
[36; 346; (3262)2; 3636]
5-uniform 255.svg
[(346)2; (3262)2; 3636]
5-uniform 256.svg
[(36)3; 346; 3262]
5-uniform 257.svg
[(36)3; 346; 3262]
5-uniform 258.svg
[(36)2; (346)2; 3262]
5-uniform 259.svg
[(36)2; (346)2; 3262]
5-uniform 260.svg
[(36)2; (346)2; 3262]
5-uniform 261.svg
[(346)3; 3262; 3636]
5-uniform 262.svg
[(346)4; 3262]
5-uniform 263.svg
[(346)4; 3262]
5-uniform 264.svg
[36; (346)2; (3262)2]
5-uniform 265.svg
[36; (346)2; (3262)2]
5-uniform 266.svg
[36; 346; (3636)3]
5-uniform 267.svg
[(346)3; (3636)2]
5-uniform 268.svg
[36; 346; (3636)3]
5-uniform 269.svg
[346; (3342)2; (3636)2]
5-uniform 270.svg
[346; (3342)2; (3636)2]
5-uniform 271.svg
[(36)2; 346; (3636)2]
5-uniform 272.svg
[36; 346; (3636)3]
5-uniform 273.svg
[36; 346; (3636)3]
5-uniform 274.svg
[(36)2; (346)2; 3636]
5-uniform 275.svg
[(346)3; (3636)2]
5-uniform 276.svg
[(346)2; (3636)3]
5-uniform 277.svg
[(36)2; 346; 3342; 3636]
5-uniform 278.svg
[(36)2; 346; 3342; 3636]
5-uniform 279.svg
[(36)3; 346; 3636]
5-uniform 280.svg
[(36)3; 346; 3636]
5-uniform 281.svg
[36; (346)3; 3636]
5-uniform 282.svg
[(346)4; 3636]
5-uniform 283.svg
[(36)2; (346)2; 3636]
5-uniform 284.svg
[(36)2; (346)2; 3636]
5-uniform 285.svg
[36; 346; (3342)2; 334.12]
5-uniform 286.svg
[(36)2; 3342; 33434; 334.12]
5-uniform 287.svg
[36; 33434; (334.12)2; 343.12]
5-uniform 288.svg
[36; (33434)3; 343.12]
5-uniform 289.svg
[36; (3342)4]
5-uniform 290.svg
[(36)2; (3342)3]
5-uniform 291.svg
[(36)2; (3342)3]
5-uniform 292.svg
[(36)2; (3342)3]
5-uniform 293.svg
[(36)2; (3342)3]
5-uniform 294.svg
[(36)4; 3342]
5-uniform 295.svg
[(36)4; 3342]
5-uniform 296.svg
[36; (346)2; (3342)2]
5-uniform 297.svg
[(36)2; 346; 3342; 33434]
5-uniform 298.svg
[(36)2; 3342; (33434)2]
5-uniform 299.svg
[36; (3342)4]
5-uniform 300.svg
[36; (3342)4]
5-uniform 301.svg
[(36)3; (3342)2]
5-uniform 302.svg
[(36)3; (3342)2]
5-uniform 303.svg
[(36)3; (3342)2]
5-uniform 304.svg
[(36)3; (3342)2]
5-uniform 305.svg
[(36)3; (3342)2]
5-uniform 306.svg
[(36)3; (3342)2]
5-uniform 307.svg
[(36)2; (3342)2; 33434]
5-uniform 308.svg
[(3342)3; (33434)2]
5-uniform 309.svg
[(3342)4; 33434]
5-uniform 310.svg
[(3342)3; (33434)2]
5-uniform 311.svg
[(3342)2; (33434)3]
5-uniform 312.svg
[3342; (33434)4]
5-uniform 313.svg
[(36)2; 3342; (33434)2]
5-uniform 314.svg
[346; (3342)2; (33434)2]
5-uniform 315.svg
[(3342)2; (33434)3]
5-uniform 316.svg
[(36)2; 3342; (33434)2]
5-uniform 317.svg
[(36)2; 3342; (33434)2]
5-uniform 318.svg
[36; (33434)4]
5-uniform 319.svg
[36; (33434)4]
5-uniform 320.svg
[(36)4; 346]
5-uniform 321.svg
[(36)4; 346]
5-uniform 322.svg
[(36)4; 346]
5-uniform 323.svg
[(36)2; (346)3]
5-uniform 324.svg
[(36)2; (346)3]
5-uniform 325.svg
[(36)3; (346)2]
5-uniform 326.svg
[(36)3; (346)2]
5-uniform 327.svg
[(36)3; (346)2]
5-uniform 328.svg
[(36)3; (346)2]
5-uniform 329.svg
[36; (346)4]
5-uniform 330.svg
[(36)2; (346)3]
5-uniform 331.svg
[(36)2; (346)3]
5-uniform 332.svg
[(36)2; (346)3]

Tilings that are not edge-to-edge[edit]

Regular polygons can also form plane tilings that are not edge-to-edge. Such tilings may also be known as uniform if they are vertex-transitive (isogonal); there are eight families of such uniform tilings, each family having a real-valued parameter determining the overlap between sides of adjacent tiles or the ratio between the edge lengths of different tiles.

Periodic tilings with star polygons
Academ Squares and regular star octagons in a periodic tiling.svg
Octagrams and squares
Academ Periodic tiling by star dodecagons and equilateral triangles.svg
Dodecagrams and equilateral triangles

Two tilings by regular polygons of two kinds. Two elements of the same kind are congruent.
Every element which is not convex is a stellation of a regular polygon with stripes.

Periodic tilings by regular polygons
Gyrated hexagonal tiling2.png
Six triangles surround every hexagon.
No pair of triangles has a common boundary, if their sides have a length lower than
the side length of hexagons.
Distorted trihexagonal tiling.png
Two size triangles
Distorted truncated square tiling.png
A tiling by squares of two different sizes, manifestly periodic by overlaying an appropriate grid. The present grid divides every large tile into four congruent polygons: possible puzzle pieces to prove the Pythagorean theorem.
Hexagon hexagram tiling2.png
Regular octagrams with small and large squares

See also[edit]

References[edit]

  1. ^ Critchlow, p.60-61
  2. ^ http://www2.math.uu.se/research/pub/Lenngren1.pdf k-uniform tilings by regular polygons] Nils Lenngren, 2009
  3. ^ Critchlow, p.62-67
  4. ^ Tilings and Patterns, Grünbaum and Shephard 1986, pp. 65-67
  5. ^ In Search of Demiregular Tilings

External links[edit]

Euclidean and general tiling links: