Euclidean tilings by convex regular polygons

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.

Archimedean, uniform or semiregular 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 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.

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

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.

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 ( 1 − 2 n ) 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:

These four can be used in k-uniform tiling:

Dissected regular polygons

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

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

2-uniform tilings

There are twenty 2-uniform tilings of the Euclidean plane. (also called 2-isogonal tilings or demiregular tilings) 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)

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, 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.

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, 4 vertex types (2:1:1:1)

There are 94 5-uniform tilings with 4 vertex types.

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, 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.

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. Grünbaum and Shephard call these tilings uniform although it contradicts Coxeter's definition for uniformity which requires edge-to-edge regular polygons. Such isogonal tilings are actually topologically identical to the uniform tilings, with different geometric proportions.