User:Harry Princeton/List of k-uniform tilings

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.

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

Classification
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.

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.

m-Catalaves and k-co-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.

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.

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.


 * 3-co-uniform tilings, 3 vertex types


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

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, 3 face types (2:1:1)

There are 85 with 3 types of faces.


 * 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.

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

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


 * 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, 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.

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

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: