User:Harry Princeton/List of k-uniform tilings

Example k-uniform tilings

1-uniform (regular)	1-uniform (semiregular)
2-uniform tiling	3-uniform tiling

A k-co-uniform tiling is a tiling of tilings of the plane by convex coregular polygons, connected edge-to-edge, with k types of dual polygons. The (1) co-uniform tiling include 3 co-regular tilings, and 8 semicoregular tilings. A co-uniform tiling can be defined by its face configuration. Higher k-co-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-co-uniform tilings, 61 3-co-uniform tilings, 151 4-co-uniform tilings, 332 5-co-uniform tilings, and 673 6-co-uniform tilings. This article lists all solutions up to k=5.

This square tiling is isogonal and isohedral, but not uniform because it is not edge-to-edge.

Other tilings of regular polygons that are not edge-to-edge allow different sized polygons, and continuous shifting positions of contact.

Classification

3-co-uniform tiling #57 of 61 colored

by sides, cyan Cairo Pentagons, green Hexagons (by the colygons)

by 3-isohedral positions, 2 shaded colors of Cairo Pentagons (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. Vertices are replaced by hedral dual facets, and regular polygons by gonal covertices (regular vertices).

k-co-uniform tilings with the same face figures can be further identified by their wallpaper group symmetry and toxality.

Enumeration

1-co-uniform tilings include 3 coregular tilings, and 8 semicoregular ones, with 2 or more types of regular covertices. There are 20 2-co-uniform tilings, 61 3-co-uniform tilings, 151 4-co-uniform tilings, 332 5-co-uniform tilings and 673 6-co-uniform tilings. Each can be grouped by the number m of distinct vertex faces, which are also called m-Catalaves tilings.

Finally, if the number of types of faces is the same as the co-uniformity (m = k below), then the tiling is said to be co-Krotenheerdt. In general, the uniformity is greater than or equal to the number of types of faces (m ≥ k), as different types of faces 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.

k-co-uniform, m-Catalaves tiling counts

		m-Catalaves
		1	2	3	4	5	6	Total
k-co-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	673
	Total	11	∞	∞	∞	∞	∞	∞

1-co-uniform tilings (coregular)

A tiling is said to be coregular 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 dual face, edge and covertex 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 as well. There must be six degree-3 covertices, four degree-4 covertices or three degree-6 covertices around each face, yielding the three coregular tessellations.

Further information: List_of_regular_polytopes_and_compounds § Euclidean_tilings

Regular tilings (3)

p6m, *632		p4m, *442
36 (t=1, e=1)	63 (t=1, e=1)	44 (t=1, e=1)

m-Catalaves and k-co-uniform 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.

If the requirement of flag-transitivity is relaxed to one of face-transitivity, while the condition that the tiling is edge-to-edge is kept, there are eight additional tilings possible, known as Catalaves, uniform or demicoregular tilings. Note that there are two mirror image (enantiomorphic or chiral) forms of V3⁴.6 (Floret Pentagonal) 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 Catalaves as referring only to the local property of the arrangement of tiles around each vertex being the same, and that as co-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 Catalaves tilings which are not uniform.

1-co-uniform tilings (semicoregular)

Co-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)
[ 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)

2-co-uniform tilings

There are twenty (20) 2-co-uniform tilings of the Euclidean plane. (also called 2-isogonal tilings or demiregular tilings) Face types are listed for each. If two tilings share the same two face 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, 22×	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-co-uniform tilings

There are 61 3-co-uniform tilings of the Euclidean plane. 39 are 3-Archimedean with 3 distinct face types, while 22 have 2 identical face types in different symmetry orbits. Chavey (1989)

3-co-uniform tilings, 3 vertex types

3-uniform tilings with 3 face 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-co-uniform tilings, 2 face types (2:1)

3-co-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-co-uniform tilings

There are 151 4-co-uniform tilings of the Euclidean plane. Brian Galebach's search reproduced Krotenheerdt's list of 33 4-uniform tilings with 4 distinct face types, as well as finding 85 of them with 3 face types, and 33 with 2 face types.

4-co-uniform tilings, 4 face types

There are 33 with 4 types of faces.

4-co-uniform tilings with 4 face 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-co-uniform tilings, 3 face types (2:1:1)

There are 85 with 3 types of faces.

4-co-uniform tilings (2:1:1)

[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-co-uniform tilings, 2 face types (2:2) and (3:1)

There are 33 with 2 types of faces, 12 with two pairs of types, and 21 with 3:1 ratio of types.

4-co-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-co-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-co-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 faces. There are 74 with 2 face figures, 149 with 3 face figures, 94 with 4 face figures, and 15 with 5 face figures.

5-co-uniform tilings, 5 face types

There are 15 5-uniform tilings with 5 unique face figure types.

5-co-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 face types (2:1:1:1)

There are 94 5-co-uniform tilings with 4 face types.

5-co-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-co-uniform tilings, 3 face types (3:1:1) and (2:2:1)

There are 149 5-co-uniform tilings, with 60 having 3:1:1 copies, and 89 having 2:2:1 copies.

5-co-uniform tilings (3:1:1)

[36; 334.12; (46.12)3]	[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-co-uniform tilings (2:2:1)

[(3446)2; (3636)2; 46.12]	[(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]	[36; (3262)2; (63)2]	[(3262)2; (3636)2; 63]
[(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]	[(346)2; (3262)2; 63]	[36; (3262)2; (63)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-co-uniform tilings, 2 face types (4:1) and (3:2)

There are 74 5-co-uniform tilings with 2 types of faces, 27 with 4:1 and 47 with 3:2 copies of each.

5-co-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-co-uniform tilings with 3 and 2 unique face figure types.

5-co-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]

Correctly Colored Tilings

When the planigons are colored according to size (over the rainbow), this 1-to-5 co-uniform mosaic results:

Higher k-co-uniform tilings

k-co-uniform tilings have been enumerated up to 6. There are 673 6-co-uniform tilings of the Euclidean plane. Brian Galebach's search reproduced Krotenheerdt's list of 10 6-co-uniform tilings with 6 distinct face types, as well as finding 92 of them with 5 face types, 187 of them with 4 face types, 284 of them with 3 face types, and 100 with 2 face types.

7-Co-Uniform Tiling

The 7-Krotenheerdt Co-Uniform Tiling 3STIrCB of all sizes. It contains all four demiregular VRPs:

3STIrCB	3STIrCB	3STIrCB	3STIrCB	3STIrCB	3STIrCB
Demi Size	2-Uniform Size	k ≥ 3 Co-Uniform Size	Co-Uniform Size	Regular Size	Large Size

Poster Size

References

• 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". 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, 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.

Category:Euclidean plane geometry Category:Regular tilings Category:Tessellation