Euclidean tilings by convex regular polygons

Example periodic tilings
A regular tiling has one type of regular face.	A semiregular or uniform tiling has one type of vertex, but two or more types of faces.
A k-uniform tiling has k types of vertices, and two or more types of regular faces.	A non-edge-to-edge tiling can have different sized regular faces.

Euclidean plane tilings by convex 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

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

3⁶ (t=1, e=1)	6³ (t=1, e=1)	4⁴ (t=1, e=1)

Archimedean, uniform or semiregular tilings

Further information: List of convex uniform tilings

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, only one of which is shown in the following table. All other regular and semiregular tilings are achiral.

Uniform tilings (8)
p6m, *632
3.12² (t=2, e=2)	3.4.6.4 (t=3, e=2)	4.6.12 (t=3, e=3)	(3.6)² (t=2, e=1)
p4m, *442	p4g, 4*2	cmm, 2*22	p6, 632
4.8² (t=2, e=2)	3².4.3.4 (t=2, e=2)	3³.4² (t=2, e=3)	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

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

Such periodic tilings 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.

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	7	8	9	Total
	1	11	0	0	0	0	0	0	0	0	11
	2	0	20	0	0	0	0	0	0	0	20
	3	0	22	39	0	0	0	0	0	0	61
	4	0	33	85	33	0	0	0	0	0	151
	5	0	74	149	94	15	0	0	0	0	332
	6	0	100	284	187	92	10	0	0	0	673
	7	0	?	?	?	?	?	7	0	0	?
	8	0	?	?	?	?	?	20	0	0	?
	9	0	?	?	?	?	?	?	8	0	?
	10	0	?	?	?	?	?	?	27	0	?
	11	0	?	?	?	?	?	?	?	1	?

Other types of vertices in Euclidean plane tilings

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 (unusable)
3.7.42	3.8.24	3.9.18	3.10.15	4.5.20	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	3².4.12	3.4.3.12	3².6²	3.4².6
Example 2-uniform tilings	with 3⁶	with 3.12.12	with (3.6)²	with (3.6)²

Dissected regular polygons

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
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
Triangle			Square		Hexagon

2-uniform tilings

There are twenty 2-uniform tilings of the Euclidean plane. (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
[3⁶; 3².4.3.4] (t=3, e=3)	[3.4.6.4; 3².4.3.4] (t=4, e=4)	[3.4.6.4; 3³.4²] (t=4, e=4)	[3.4.6.4; 3.4².6] (t=5, e=5)	[4.6.12; 3.4.6.4] (t=4, e=4)	[3⁶; 3².4.12] (t=4, e=4)	[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
[3⁶; 3².6²] (t=2, e=3)	[3⁶; 3⁴.6]₁ (t=3, e=3)	[3⁶; 3⁴.6]₂ (t=5, e=7)	[3².6²; 3⁴.6] (t=2, e=4)	[3.6.3.6; 3².6²] (t=2, e=3)	[3.4².6; 3.6.3.6]₂ (t=3, e=4)	[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
[3³.4²; 3².4.3.4]₁ (t=4, e=5)	[3³.4²; 3².4.3.4]₂ (t=3, e=6)	[4⁴; 3³.4²]₁ (t=2, e=4)	[4⁴; 3³.4²]₂ (t=3, e=5)	[3⁶; 3³.4²]₁ (t=3, e=4)	[3⁶; 3³.4²]₂ (t=4, e=5)

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

3-uniform tilings with 3 vertex types (39)
[3.4²6; 3.6.3.6; 4.6.12] (t=6, e=7)	[3⁶; 3²4.12; 4.6.12] (t=5, e=6)	[3²4.12; 3.4.6.4; 3.12²] (t=5, e=6)	[3.4.3.12; 3.4.6.4; 3.12²] (t=5, e=6)	[3³4²; 3²4.12; 3.4.6.4] (t=6, e=8)
[3⁶; 3³4²; 3²4.12] (t=6, e=7)	[3⁶; 3²4.3.4; 3²4.12] (t=5, e=6)	[3⁴6; 3³4²; 3²4.3.4] (t=5, e=6)	[3⁶; 3²4.3.4; 3.4²6] (t=5, e=6)	[3⁶; 3²4.3.4; 3.4.6.4] (t=5, e=6)
[3⁶; 3³4²; 3.4.6.4] (t=6, e=6)	[3⁶; 3²4.3.4; 3.4.6.4] (t=6, e=6)	[3⁶; 3³4²; 3²4.3.4] (t=4, e=5)	[3²4.12; 3.4.3.12; 3.12²] (t=4, e=7)	[3.4.6.4; 3.4²6; 4⁴] (t=3, e=4)
[3²4.3.4; 3.4.6.4; 3.4²6] (t=4, e=6)	[3³4²; 3²4.3.4; 4⁴] (t=4, e=6)	[3.4²6; 3.6.3.6; 4⁴] (t=5, e=7)	[3.4²6; 3.6.3.6; 4⁴] (t=6, e=7)	[3.4²6; 3.6.3.6; 4⁴] (t=4, e=5)
[3.4²6; 3.6.3.6; 4⁴] (t=5, e=6)	[3³4²; 3²6²; 3.4²6] (t=5, e=8)	[3²6²; 3.4²6; 3.6.3.6] (t=4, e=7)	[3²6²; 3.4²6; 3.6.3.6] (t=5, e=7)	[3⁴6; 3³4²; 3.4²6] (t=5, e=7)
[3²6²; 3.6.3.6; 6³] (t=4, e=5)	[3²6²; 3.6.3.6; 6³] (t=2, e=4)	[3⁴6; 3²6²; 6³] (t=2, e=5)	[3⁶; 3²6²; 6³] (t=2, e=3)	[3⁶; 3⁴6; 3²6²] (t=5, e=8)
[3⁶; 3⁴6; 3²6²] (t=3, e=5)	[3⁶; 3⁴6; 3²6²] (t=3, e=6)	[3⁶; 3⁴6; 3.6.3.6] (t=5, e=6)	[3⁶; 3⁴6; 3.6.3.6] (t=4, e=4)	[3⁶; 3⁴6; 3.6.3.6] (t=3, e=3)
[3⁶; 3³4²; 4⁴] (t=4, e=6)	[3⁶; 3³4²; 4⁴] (t=5, e=7)	[3⁶; 3³4²; 4⁴] (t=3, e=5)	[3⁶; 3³4²; 4⁴] (t=4, e=6)

3-uniform tilings, 2 vertex types (2:1)

3-uniform tilings (2:1) (22)
[(3.4.6.4)2; 3.4²6] (t=6, e=6)	[(3⁶)2; 3⁴6] (t=3, e=4)	[(3⁶)2; 3⁴6] (t=5, e=5)	[(3⁶)2; 3⁴6] (t=7, e=9)	[3⁶; (3⁴6)2] (t=4, e=6)
[3⁶; (3²4.3.4)2] (t=4, e=5)	[(3.4²6)2; 3.6.3.6] (t=6, e=8)	[3.4²6; (3.6.3.6)2] (t=4, e=6)	[3.4²6; (3.6.3.6)2] (t=5, e=6)	[3²6²; (3.6.3.6)2] (t=3, e=5)
[(3⁴6)2; 3.6.3.6] (t=4, e=7)	[(3⁴6)2; 3.6.3.6] (t=4, e=7)	[3³4²; (4⁴)2] (t=4, e=7)	[(3³4²)2; 4⁴] (t=5, e=7)	[3³4²; (4⁴)2] (t=3, e=6)
[(3³4²)2; 4⁴] (t=4, e=6)	[(3³4²)2; 3²4.3.4] (t=5, e=8)	[3³4²; (3²4.3.4)2] (t=6, e=9)	[3⁶; (3³4²)2] (t=5, e=7)	[3⁶; (3³4²)2] (t=4, e=6)
[(3⁶)2; 3³4²] (t=6, e=7)	[(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 34 with 4 types of vertices.

4-uniform tilings with 4 vertex types (33)
[33434; 3²6²; 3446; 6³]	[3³4²; 3²6²; 3446; 46.12]	[33434; 3²6²; 3446; 46.12]	[3⁶; 3³4²; 33434; 334.12]	[3⁶; 33434; 334.12; 3.12²]
[3⁶; 33434; 343.12; 3.12²]	[3⁶; 3³4²; 33434; 3464]	[3⁶; 3³4²; 33434; 3464]	[3⁶; 33434; 3464; 3446]	[3⁴6; 3²6²; 3636; 6³]
[3⁴6; 3²6²; 3636; 6³]	[334.12; 343.12; 3464; 46.12]	[3³4²; 334.12; 343.12; 3.12²]	[3³4²; 334.12; 343.12; 4⁴]	[3³4²; 334.12; 343.12; 3.12²]
[3⁶; 3³4²; 33434; 4⁴]	[33434; 3²6²; 3464; 3446]	[3⁶; 3³4²; 3446; 3636]	[3⁶; 3⁴6; 3446; 3636]	[3⁶; 3⁴6; 3446; 3636]
[3⁶; 3⁴6; 3³4²; 3446]	[3⁶; 3⁴6; 3³4²; 3446]	[3⁶; 3⁴6; 3²6²; 6³]	[3⁶; 3⁴6; 3²6²; 6³]	[3⁶; 3⁴6; 3²6²; 6³]
[3⁶; 3⁴6; 3²6²; 6³]	[3⁶; 3⁴6; 3²6²; 3636]	[3³4²; 3²6²; 3446; 6³]	[3³4²; 3²6²; 3446; 6³]	[3²6²; 3446; 3636; 4⁴]
[3²6²; 3446; 3636; 4⁴]	[3²6²; 3446; 3636; 4⁴]	[3²6²; 3446; 3636; 4⁴]

4-uniform tilings, 3 vertex types (2:1:1)

There are 85 with 3 types of vertices.

4-uniform tilings (3:1)
[3464; (3446)2; 46.12]	[3464; 3446; (46.12)2]	[334.12; 3464; (3.12²)2]	[343.12; 3464; (3.12²)2]	[33434; 343.12; (3464)2]
[(3⁶)2; 3³4²; 334.12]	[(3464)2; 3446; 3636]	[3464; 3446; (3636)2]	[3464; (3446)2; 3636]	[(3⁶)2; 3³4²; 33434]
[(3⁶)2; 3³4²; 33434]	[3⁶; 3²6²; (6³)2]	[3⁶; 3²6²; (6³)2]	[3⁶; (3²6²)2; 6³]	[3⁶; (3²6²)2; 6³]
[3⁶; 3²6²; (6³)2]	[3⁶; 3²6²; (6³)2]	[3⁶; (3⁴6)2; 3²6²]	[3⁶; (3²6²)2; 3636]	[(3⁴6)2; 3²6²; 6³]
[(3⁴6)2; 3²6²; 6³]	[3⁴6; 3²6²; (3636)2]	[3⁴6; 3²6²; (3636)2]	[3³4²; 33434; (3464)2]	[3⁶; 33434; (3464)2]
[3⁶; (33434)2; 3464]	[3⁶; (3³4²)2; 3464]	[(3464)2; 3446; 3636]	[3⁴6; (33434)2; 3446]	[3⁶; 3³4²; (33434)2]
[3⁶; 3³4²; (33434)2]	[(3³4²)2; 33434; 4⁴]	[(3³4²)2; 33434; 4⁴]	[3464; (3446)2; 4⁴]	[33434; (334.12)2; 343.12]
[3⁶; (3²6²)2; 6³]	[3⁶; (3²6²)2; 6³]	[3⁶; 3⁴6; (3²6²)2]	[(3⁶)2; 3⁴6; 3²6²]	[(3⁶)2; 3⁴6; 3²6²]
[(3⁶)2; 3⁴6; 3636]	[3⁴6; (3²6²)2; 3636]	[3⁴6; (3²6²)2; 3636]	[(3⁴6)2; 3²6²; 3636]	[(3⁴6)2; 3²6²; 3636]
[3⁶; 3⁴6; (3636)2]	[3²6²; (3636)2; 6³]	[3²6²; (3636)2; 6³]	[(3²6²)2; 3636; 6³]	[3²6²; 3636; (6³)2]
[3⁴6; 3²6²; (6³)2]	[3⁴6; (3²6²)2; 3636]	[3²6²; 3446; (3636)2]	[3²6²; 3446; (3636)2]	[3⁴6; (3³4²)2; 3636]
[3⁴6; (3³4²)2; 3636]	[3⁴6; 3³4²; (3446)2]	[3446; 3636; (4⁴)2]	[3446; 3636; (4⁴)2]	[3446; 3636; (4⁴)2]
[3446; 3636; (4⁴)2]	[(3446)2; 3636; 4⁴]	[(3446)2; 3636; 4⁴]	[(3446)2; 3636; 4⁴]	[(3446)2; 3636; 4⁴]
[(3446)2; 3636; 4⁴]	[(3446)2; 3636; 4⁴]	[(3446)2; 3636; 4⁴]	[(3446)2; 3636; 4⁴]	[3446; (3636)2; 4⁴]
[3446; (3636)2; 4⁴]	[3446; (3636)2; 4⁴]	[3446; (3636)2; 4⁴]	[3⁶; 3³4²; (4⁴)2]	[3⁶; 3³4²; (4⁴)2]
[3⁶; (3³4²)2; 4⁴]	[3⁶; 3³4²; (4⁴)2]	[3⁶; 3³4²; (4⁴)2]	[3⁶; (3³4²)2; 4⁴]	[3⁶; (3³4²)2; 4⁴]
[3⁶; (3³4²)2; 4⁴]	[(3⁶)2; 3³4²; 4⁴]	[(3⁶)2; 3³4²; 4⁴]	[(3⁶)2; 3³4²; 4⁴]	[(3⁶)2; 3³4²; 4⁴]

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.

4-uniform tilings (2:2)
[(3464)2; (46.12)2]	[(33434)2; (3464)2]	[(33434)2; (3464)2]	[(3⁴6)2; (3636)2]	[(3⁶)2; (3⁴6)2]
[(3³4²)2; (33434)2]	[(3³4²)2; (4⁴)2]	[(3³4²)2; (4⁴)2]	[(3³4²)2; (4⁴)2]	[(3⁶)2; (3³4²)2]
[(3⁶)2; (3³4²)2]	[(3⁶)2; (3³4²)2]

4-uniform tilings (3:1)
[343.12; (3.12²)3]	[(3⁴6)3; 3636]	[3⁶; (3⁴6)3]	[(3⁶)3; 3⁴6]	[(3⁶)3; 3⁴6]
[(3³4²)3; 33434]	[3³4²; (33434)3]	[3446; (3636)3]	[3446; (3636)3]	[3²6²; (3636)3]
[3²6²; (3636)3]	[3³4²; (4⁴)3]	[3³4²; (4⁴)3]	[(3³4²)3; 4⁴]	[(3³4²)3; 4⁴]
[(3³4²)3; 4⁴]	[3⁶; (3³4²)3]	[3⁶; (3³4²)3]	[3⁶; (3³4²)3]	[(3⁶)3; 3³4²]
[(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.

5-uniform tilings, 5 types
[33434; 3²6²; 3464; 3446; 6³]	[3⁶; 3⁴6; 3²6²; 3636; 6³]	[3⁶; 3⁴6; 3³4²; 3446; 46.12]	[3⁴6; 3³4²; 33434; 3446; 4⁴]	[3⁶; 33434; 3464; 3446; 3636]
[3⁶; 3⁴6; 3464; 3446; 3636]	[33434; 334.12; 3464; 3.12.12; 46.12]	[3⁶; 3⁴6; 3446; 3636; 4⁴]	[3⁶; 3⁴6; 3446; 3636; 4⁴]	[3⁶; 3⁴6; 3446; 3636; 4⁴]
[3⁶; 3⁴6; 3446; 3636; 4⁴]	[3⁶; 3³4²; 3446; 3636; 4⁴]	[3⁶; 3⁴6; 3³4²; 3446; 4⁴]	[3⁶; 3³4²; 3²6²; 3446; 3636]	[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.

5-uniform tilings (2:1:1:1)
[3⁶; 33434; (3446)2; 46.12]	[3⁶; 33434; 3446; (46.12)2]	[3⁶; 33434; 3464; (46.12)2]	[3⁶; 3³4²; (334.12)2; 3464]	[3⁶; (3³4²)2; 334.12; 3464]
[3⁶; 33434; (334.12)2; 3464]	[3⁶; 33434; 334.12; (3.12.12)2]	[3⁶; 3⁴6; (3³4²)2; 334.12]	[3⁶; 33434; 343.12; (3.12.12)2]	[(3³4²)2; 334.12; 343.12; 3.12.12]
[(3³4²)2; 334.12; 343.12; 3.12.12]	[(3³4²)2; 334.12; 343.12; 4⁴]	[33434; 3²6²; (3446)2; 4⁴]	[3⁶; (3³4²)2; 33434; 4⁴]	[3⁴6; (3³4²)2; 33434; 4⁴]
[3⁶; 3³4²; (3464)2; 3446]	[3³4²; 3²6²; 3464; (3446)2]	[33434; 3²6²; 3464; (3446)2]	[3⁶; 33434; (3446)2; 3636]	[3³4²; 33434; 3464; (3446)2]
[3⁶; 33434; (3²6²)2; 3446]	[3³4²; 3²6²; (3464)2; 3446]	[33434; 3²6²; (3464)2; 3446]	[3⁴6; 3³4²; (3464)2; 3446]	[3⁶; (3³4²)2; 33434; 3464]
[3⁶; (3³4²)2; 33434; 3464]	[3⁶; 3³4²; (33434)2; 3464]	[(3⁶)2; 3³4²; 33434; 3464]	[3⁶; 3³4²; (33434)2; 3464]	[(3⁶)2; 3³4²; 33434; 334.12]
[3⁶; 33434; (334.12)2; 343.12]	[(3⁶)2; 3⁴6; 3³4²; 33434]	[(3⁶)2; 3⁴6; 3²6²; 6³]	[3⁶; (3⁴6)2; 3²6²; 6³]	[(3⁶)2; 3⁴6; 3²6²; 3636]
[3⁶; 3⁴6; (3²6²)2; 3636]	[3⁶; (3⁴6)2; 3²6²; 3636]	[(3⁶)2; 3⁴6; 3²6²; 3636]	[3⁶; 3⁴6; 3²6²; (3636)2]	[3⁶; (3⁴6)2; 3²6²; 3636]
[3⁶; (3⁴6)2; 3²6²; 3636]	[3⁶; (3⁴6)2; 3²6²; 3636]	[3⁶; 3⁴6; (3²6²)2; 3636]	[3⁶; 3⁴6; (3²6²)2; 3636]	[3⁶; 3⁴6; 3²6²; (6³)2]
[3⁶; 3⁴6; (3²6²)2; 6³]	[3⁴6; (3²6²)2; 3636; 6³]	[(3⁴6)2; 3²6²; 3636; 6³]	[(3⁶)2; 3⁴6; 3²6²; 6³]	[(3⁶)2; 3⁴6; 3²6²; 6³]
[3⁶; 3⁴6; 3²6²; (6³)2]	[3⁶; 3⁴6; 3²6²; (6³)2]	[3⁶; 3⁴6; 3²6²; (6³)2]	[3⁶; 3⁴6; (3²6²)2; 6³]	[3⁴6; (3²6²)2; 3636; 6³]
[3⁴6; (3²6²)2; 3636; 6³]	[3⁴6; (3²6²)2; 3636; 6³]	[3⁴6; 3²6²; 3636; (6³)2]	[3⁴6; (3²6²)2; 3636; 6³]	[3³4²; 3²6²; 3446; (6³)2]
[3³4²; 3²6²; 3446; (6³)2]	[3²6²; 3446; 3636; (4⁴)2]	[3²6²; 3446; 3636; (4⁴)2]	[3²6²; 3446; (3636)2; 4⁴]	[3²6²; 3446; (3636)2; 4⁴]
[3³4²; 3²6²; 3446; (4⁴)2]	[3⁴6; 3³4²; 3446; (4⁴)2]	[3²6²; 3446; 3636; (4⁴)2]	[3²6²; 3446; 3636; (4⁴)2]	[3²6²; 3446; (3636)2; 4⁴]
[3²6²; 3446; (3636)2; 4⁴]	[3³4²; 3²6²; 3446; (4⁴)2]	[3⁴6; 3³4²; 3446; (4⁴)2]	[3⁴6; (3³4²)2; 3636; 4⁴]	[3⁶; 3³4²; (3446)2; 3636]
[3⁴6; (3³4²)2; 3446; 3636]	[3⁴6; (3³4²)2; 3446; 3636]	[(3⁶)2; 3⁴6; 3446; 3636]	[3⁶; 3³4²; (3446)2; 3636]	[3⁴6; (3³4²)2; 3446; 3636]
[3⁴6; (3³4²)2; 3446; 3636]	[(3⁶)2; 3⁴6; 3446; 3636]	[(3⁶)2; 3³4²; 3446; 3636]	[3⁶; 3³4²; 3446; (3636)2]	[3⁴6; 3³4²; (3446)2; 3636]
[3⁶; 3⁴6; (3³4²)2; 3446]	[3⁴6; (3³4²)2; 3²6²; 3636]	[3⁴6; (3³4²)2; 3²6²; 3636]	[3⁶; (3⁴6)2; 3³4²; 3446]	[3⁶; (3⁴6)2; 3³4²; 3446]
[3⁶; (3⁴6)2; 3³4²; 3446]	[3⁶; 3⁴6; (3³4²)2; 3²6²]	[(3⁶)2; 3⁴6; 3³4²; 3636]	[(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.

5-uniform tilings (3:1:1)
[3⁶; 334.12; (46.12)3]	[(3⁶)2; (3³4²)2; 3464]	[(3³4²)2; 334.12; (3464)2]	[3⁶; (33434)2; (3464)2]	[3³4²; (33434)2; (3464)2]
[3³4²; (33434)2; (3464)2]	[3³4²; (33434)2; (3464)2]	[(33434)2; 343.12; (3464)2]	[3464; 3446; (46.12)3]	[3⁶; (334.12)3; 46.12]
[334.12; 343.12; (3.12.12)3]	[3⁶; (33434)3; 343.12]
[3²6²; 3636; (6³)3]	[3⁴6; 3²6²; (6³)3]	[3⁶; (3²6²)3; 6³]	[3⁶; (3²6²)3; 6³]	[3²6²; (3636)3; 6³]
[3446; 3636; (4⁴)3]	[3446; 3636; (4⁴)3]	[3⁶; 3³4²; (4⁴)3]	[3⁶; 3³4²; (4⁴)3]	[3446; (3636)3; 4⁴]
[3446; (3636)3; 4⁴]	[3⁶; (3³4²)3; 4⁴]	[3⁶; (3³4²)3; 4⁴]	[3⁶; (3³4²)3; 4⁴]	[(3⁶)3; 3³4²; 4⁴]
[(3⁶)3; 3³4²; 4⁴]	[3446; 3636; (4⁴)3]	[3446; 3636; (4⁴)3]	[3⁶; 3³4²; (4⁴)3]	[3⁶; 3³4²; (4⁴)3]
[(3³4²)3; 3²6²; 3446]	[3²6²; 3446; (3636)3]	[3²6²; 3446; (3636)3]	[3²6²; 3446; (3636)3]	[3²6²; 3446; (3636)3]
[3446; (3636)3; 4⁴]	[3446; (3636)3; 4⁴]	[3⁶; (3³4²)3; 4⁴]	[3⁶; (3³4²)3; 4⁴]	[3⁶; (3³4²)3; 4⁴]
[(3⁶)3; 3³4²; 4⁴]	[(3⁶)3; 3³4²; 4⁴]	[3⁶; (3³4²)3; 4⁴]	[3⁶; (3³4²)3; 4⁴]	[3⁶; (3³4²)3; 4⁴]
[(3³4²)3; 3446; 3636]	[(3³4²)3; 3446; 3636]	[3⁴6; (3³4²)3; 3446]	[(3⁶)3; 3⁴6; 3²6²]	[(3⁶)3; 3⁴6; 3²6²]
[(3⁶)3; 3⁴6; 3²6²]	[3⁴6; (3²6²)3; 3636]	[3⁴6; (3²6²)3; 3636]	[(3⁴6)3; 3²6²; 3636]	[(3⁴6)3; 3²6²; 3636]
[(3⁶)3; 3⁴6; 3²6²]	[(3⁶)3; 3⁴6; 3²6²]	[(3⁴6)3; 3²6²; 3636]	[3⁶; 3⁴6; (3636)3]	[3⁶; 3⁴6; (3636)3]
[3⁶; 3⁴6; (3636)3]	[3⁶; 3⁴6; (3636)3]	[(3⁶)3; 3⁴6; 3636]	[(3⁶)3; 3⁴6; 3636]	[3⁶; (3⁴6)3; 3636]

5-uniform tilings (2:2:1)
[(3446)2; (3636)2; 46.12]	[3⁶; (3²6²)2; (6³)2]	[(3²6²)2; (3636)2; 6³]	[(3⁴6)2; (3²6²)2; 6³]	[3⁶; (3²6²)2; (6³)2]
[(3⁶)2; (3³4²)2; 33434]	[(3⁶)2; 3³4²; (33434)2]	[3⁴6; (3³4²)2; (33434)2]	[(3⁶)2; 3³4²; (33434)2]	[(3⁶)2; 3³4²; (33434)2]
[(3²6²)2; 3636; (6³)2]	[(3446)2; 3636; (4⁴)2]	[(3446)2; 3636; (4⁴)2]	[3446; (3636)2; (4⁴)2]	[(3446)2; 3636; (4⁴)2]
[(3446)2; 3636; (4⁴)2]	[3446; (3636)2; (4⁴)2]	[3⁶; (3³4²)2; (4⁴)2]	[(3⁶)2; 3³4²; (4⁴)2]	[(3⁶)2; 3³4²; (4⁴)2]
[(3446)2; 3636; (4⁴)2]	[(3446)2; 3636; (4⁴)2]	[(3446)2; 3636; (4⁴)2]	[(3446)2; 3636; (4⁴)2]	[(3446)2; 3636; (4⁴)2]
[3⁶; (3³4²)2; (4⁴)2]	[(3⁶)2; (3³4²)2; 4⁴]	[(3446)2; 3636; (4⁴)2]	[(3446)2; 3636; (4⁴)2]	[3446; (3636)2; (4⁴)2]
[(3446)2; 3636; (4⁴)2]	[(3446)2; 3636; (4⁴)2]	[3446; (3636)2; (4⁴)2]	[3⁶; (3³4²)2; (4⁴)2]	[(3⁶)2; 3³4²; (4⁴)2]
[(3⁶)2; 3³4²; (4⁴)2]	[3⁶; (3³4²)2; (4⁴)2]	[3⁶; (3³4²)2; (4⁴)2]	[(3446)2; 3636; (4⁴)2]	[(3⁶)2; (3³4²)2; 4⁴]
[(3⁶)2; (3³4²)2; 4⁴]	[(3⁶)2; (3³4²)2; 4⁴]	[(3⁶)2; (3³4²)2; 4⁴]	[(33434)2; 3²6²; (3446)2]	[3³4²; (3²6²)2; (3446)2]
[3³4²; (3²6²)2; (3446)2]	[3²6²; (3446)2; (3636)2]	[(3²6²)2; 3446; (3636)2]	[(3²6²)2; 3446; (3636)2]	[(3464)2; (3446)2; 3636]
[3²6²; (3446)2; (3636)2]	[3²6²; (3446)2; (3636)2]	[(3⁴6)2; (3446)2; 3636]	[(3⁴6)2; (3446)2; 3636]	[(3⁴6)2; (3446)2; 3636]
[(3⁴6)2; (3446)2; 3636]	[(3³4²)2; (3446)2; 3636]	[(3³4²)2; (3446)2; 3636]	[(3⁴6)2; (3³4²)2; 3446]	[(3⁴6)2; 3³4²; (3446)2]
[(3⁶)2; (3⁴6)2; 3²6²]	[3⁶; (3⁴6)2; (3²6²)2]	[(3⁶)2; 3⁴6; (3²6²)2]
[3⁶; (3⁴6)2; (3²6²)2]	[3⁴6; (3²6²)2; (3636)2]	[(3⁴6)2; (3²6²)2; 3636]	[3⁶; (3⁴6)2; (3²6²)2]	[(3⁴6)2; 3²6²; (3636)2]
[(3⁴6)2; (3²6²)2; 3636]	[(3⁶)2; (3⁴6)2; 3²6²]	[(3⁶)2; (3⁴6)2; 3²6²]	[(3⁶)2; (3⁴6)2; 3636]	[(3⁶)2; (3⁴6)2; 3636]
[3⁶; (3⁴6)2; (3³4²)2]	[(3⁶)2; (3⁴6)2; 3²6²]	[3⁶; (3⁴6)2; (3²6²)2]	[3⁶; (3⁴6)2; (3²6²)2]	[3⁴6; (3³4²)2; (3636)2]
[3⁴6; (3³4²)2; (3636)2]	[(3⁶)2; 3⁴6; (3636)2]	[(3⁶)2; (3⁴6)2; 3636]	[(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.

5-uniform tilings (4:1)
[(3464)4; 46.12]	[343.12; (3.12.12)4]	[3⁶; (33434)4]	[3⁶; (33434)4]	[(3⁶)4; 3⁴6]
[(3⁶)4; 3⁴6]	[(3⁶)4; 3⁴6]	[3⁶; (3⁴6)4]	[3²6²; (3636)4]	[(3⁴6)4; 3²6²]
[(3⁴6)4; 3²6²]	[(3⁴6)4; 3636]	[3²6²; (3636)4]	[3446; (3636)4]	[3446; (3636)4]
[(3³4²)4; 33434]	[3³4²; (33434)4]
[3³4²; (4⁴)4]	[3³4²; (4⁴)4]	[(3³4²)4; 4⁴]	[(3³4²)4; 4⁴]	[(3³4²)4; 4⁴]
[3⁶; (3³4²)4]	[3⁶; (3³4²)4]	[3⁶; (3³4²)4]	[(3⁶)4; 3³4²]	[(3⁶)4; 3³4²]

There are 29 5-uniform tilings with 3 and 2 unique vertex figure types.

5-uniform tilings (3:2)
[(3464)2; (46.12)3]	[(3464)2; (46.12)3]	[(3464)3; (3446)2]	[(33434)2; (3464)3]	[(33434)3; (3464)2]
[(3⁶)2; (3⁴6)3]	[(3⁶)2; (3⁴6)3]	[(3⁶)3; (3⁴6)2]	[(3⁶)3; (3⁴6)2]	[(3⁶)3; (3⁴6)2]
[(3⁶)3; (3⁴6)2]	[(3⁶)2; (3⁴6)3]	[(3⁶)2; (3⁴6)3]	[(3⁶)2; (3⁴6)3]
[(3²6²)2; (3636)3]	[(3⁴6)3; (3636)2]	[(3⁴6)3; (3636)2]	[(3⁴6)2; (3636)3]
[(3446)3; (3636)2]	[(3446)2; (3636)3]	[(3446)3; (3636)2]	[(3446)2; (3636)3]	[(3446)2; (3636)3]
[(3³4²)3; (33434)2]	[(3³4²)3; (33434)2]	[(3³4²)2; (33434)3]	[(3³4²)2; (33434)3]
[(3³4²)2; (4⁴)3]	[(3³4²)2; (4⁴)3]	[(3³4²)2; (4⁴)3]	[(3³4²)3; (4⁴)2]	[(3³4²)2; (4⁴)3]
[(3³4²)3; (4⁴)2]	[(3³4²)2; (4⁴)3]	[(3³4²)2; (4⁴)3]	[(3³4²)3; (4⁴)2]	[(3³4²)3; (4⁴)2]
[(3⁶)2; (3³4²)3]	[(3⁶)2; (3³4²)3]	[(3⁶)2; (3³4²)3]	[(3⁶)2; (3³4²)3]	[(3⁶)3; (3³4²)2]
[(3⁶)3; (3³4²)2]	[(3⁶)3; (3³4²)2]	[(3⁶)3; (3³4²)2]	[(3⁶)3; (3³4²)2]	[(3⁶)3; (3³4²)2]

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.

Tilings that are not edge-to-edge

Convex regular polygons can also form plane tilings that are not edge-to-edge. Such tilings can be considered edge-to-edge as nonregular polygons with adjacent colinear edges.

There are seven families of isogonal 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. Two of the families are generated from shifted square, either progessive or zig-zagging positions. Grunbaum and Shephard call these tilings uniform although it contradicts Coxeter's definition for uniformity which requires edge-to-edge regular polygons.^[6] Such isogonal tilings are actually topologically identical to the uniform tilings, with different geometric proportions.

Periodic isogonal tilings by non-edge-to-edge convex regular polygons
1	2	3	4	5	6	7
Rows of squares with horizontal offsets		Rows of triangles with horizontal offsets	A tiling by squares	Three hexagons surround each triangle	Six triangles surround every hexagon.	Three size triangles
cmm (2*22)	p2 (2222)	cmm (2*22)	p4m (*442)	p6 (632)		p3 (333)
Hexagonal tiling		Square tiling	Truncated square tiling	Truncated hexagonal tiling	Hexagonal tiling	Trihexagonal tiling

References

↑ Critchlow, p.60-61
↑ k-uniform tilings by regular polygons Nils Lenngren, 2009
↑ Critchlow, p.62-67
↑ Tilings and Patterns, Grünbaum and Shephard 1986, pp. 65-67
↑ In Search of Demiregular Tilings
↑ Tilings by regular polygons p.236

Grunbaum, Branko; Shephard, Geoffrey C. (1977). "Tilings by regular polygons". Math. Mag. 50 (5): 227–247. doi:10.2307/2689529.
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 496813.
Debroey, I.; Landuyt, F. (1981). "Equitransitive edge-to-edge tilings". Geometriae Dedicata 11 (1): 47–60. doi:10.1007/BF00183189.
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". J. Combinat. Theory 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". El. J. Combinat 19 (3): #P6.
Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–57

External links

Euclidean and general tiling links:

Dutch, Steve. "Uniform Tilings". 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.

Tessellation

Periodic

Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling & honeycomb Coloring Convex Kisrhombille Wallpaper group Wythoff

Aperiodic

Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pentagonal Pinwheel Quaquaversal Rep-tile & Self-tiling Sphinx Truchet

Other

Anisohedral & Isohedral Architectonic & catoptric Circle Limit III Computer graphics Honeycomb Isotoxal Packing Problems Domino Wang Heesch's Squaring Prototile Conway criterion Girih Regular Division of the Plane Regular grid Substitution Voronoi Voderberg

By vertex type

Spherical	2ⁿ 3³.n V3³.n 4².n V4².n

Regular	2^∞ 3⁶ 4⁴ 6³

Semi- regular	3².4.3.4 V3².4.3.4 3³.4² 3³.∞ 3⁴.6 V3⁴.6 3.4.6.4 (3.6)² 3.12² 4².∞ 4.6.12 4.8²

Hyper- bolic	3².4.3.5 3².4.3.6 3².4.3.7 3².4.3.8 3².4.3.∞ 3².5.3.5 3².5.3.6 3².6.3.6 3².6.3.8 3².7.3.7 3².8.3.8 3³.4.3.4 3².∞.3.∞ 3⁴.7 3⁴.8 3⁴.∞ 3⁵.4 3⁷ 3⁸ 3^∞ (3.4)³ (3.4)⁴ 3.4.6².4 3.4.7.4 3.4.8.4 3.4.∞.4 3.6.4.6 (3.7)² (3.8)² 3.14² 3.16² (3.∞)² 3.∞² 4².5.4 4².6.4 4².7.4 4².8.4 4².∞.4 4⁵ 4⁶ 4⁷ 4⁸ 4^∞ (4.5)² (4.6)² 4.6.12 4.6.14 V4.6.14 4.6.16 V4.6.16 4.6.∞ (4.7)² (4.8)² 4.8.10 V4.8.10 4.8.12 4.8.14 4.8.16 4.8.∞ 4.10² 4.10.12 4.12² 4.12.16 4.14² 4.16² 4.∞² (4.∞)² 5⁴ 5⁵ 5⁶ 5^∞ 5.4.6.4 (5.6)² 5.8² 5.10² 5.12² (5.∞)² 6⁴ 6⁵ 6⁶ 6⁸ 6.4.8.4 (6.8)² 6.8² 6.10² 6.12² 6.16² 7³ 7⁴ 7⁷ 7.6² 7.8² 7.14² 8³ 8⁴ 8⁶ 8⁸ 8¹² 8.6² 8.16² ∞³ ∞⁴ ∞⁵ ∞^∞ ∞.6² ∞.8²

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