# Tiling by regular polygons

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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

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.

p6m, *632 p4m, *442

36

63

44

## 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, both of which are shown in the following table. All other regular and semiregular tilings are achiral.

 p4m, *442 4.82 p6m, *632 3.122 p6m, *632 (3.6)2 cmm, 2*22 33.42 p6m, *632 3.4.6.4 p6m, *632 4.6.12 p6, 632 34.6 p4, 442 3.(3.4)2

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.

## Combinations of regular polygons that can meet at a vertex

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. 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.

With 3 polygons at a vertex:

• 3.7.42 (cannot appear in any tiling of regular polygons)
• 3.8.24 (cannot appear in any tiling of regular polygons)
• 3.9.18 (cannot appear in any tiling of regular polygons)
• 3.10.15 (cannot appear in any tiling of regular polygons)
• 3.122 - semi-regular, truncated hexagonal tiling
• 4.5.20 (cannot appear in any tiling of regular polygons)
• 4.6.12 - semi-regular, truncated trihexagonal tiling
• 4.82 - semi-regular, truncated square tiling
• 52.10 (cannot appear in any tiling of regular polygons)
• 63 - regular, hexagonal tiling

Below are diagrams of such vertices:

With 4 polygons at a vertex:

• 32.4.12 - does not generate a uniform tiling; can generate a 2-uniform tiling when used with the vertex type 36 (shown below).
• 3.4.3.12 - does not generate a uniform tiling; can generate a 2-uniform tiling when used with the vertex type 3.122
• 32.62 - does not generate a uniform tiling; can generate 2-uniform tilings when used with any one of the vertex types 36, 34.6 or 3.6.3.6 (two of these are shown below).
• 3.6.3.6 - semi-regular, trihexagonal tiling
• 44 - regular, square tiling
• 3.42.6 - does not generate a uniform tiling; can generate a 2-uniform tiling when used with either of the vertex types 3.4.6.4 or 3.6.3.6. In the later case, there are two inequivalent 2-uniform tilings that can be generated. (One of these later two are shown below).
• 3.4.6.4 - semi-regular, rhombitrihexagonal tiling

Below are diagrams of such vertices:

With 5 polygons at a vertex:

Below are diagrams of such vertices:

With 6 polygons at a vertex:

Below is a diagram of such a vertex:

## Other edge-to-edge tilings

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 $n$-uniform or $n$-isogonal; if there are $n$ orbits of tiles, as $n$-isohedral; if there are $n$ orbits of edges, as $n$-isotoxal.

Shown below are the twenty 2-uniform tilings. Chavey (1989) lists all those edge-to-edge tilings by regular polygons which are at most 3-uniform, 3-isohedral or 3-isotoxal.

 cmm, 2*22 4.4.4.4 3.3.3.4.4 cmm, 2*22 4.4.4.4 3.3.3.4.4 pmm, *2222 3.3.3.3.3.3 3.3.3.4.4 cmm, 2*22 3.3.3.3.3.3 3.3.3.4.4 cmm, 2*22 3.4.4.6 3.6.3.6 pmm, *2222 3.4.4.6 3.6.3.6 pmm, *2222 3.6.3.6 3.3.6.6 p4m, *442 3.12.12 3.4.3.12 p4g, 4*2 3.3.3.4.4 3.3.4.3.4 p2, 2222 3.3.3.4.4 3.3.4.3.4 p6m, *632 3.3.3.3.3.3 3.3.6.6 p6m, *632 3.3.3.3.3.3 3.3.3.3.6 p6, 632 3.3.3.3.3.3 3.3.3.3.6 cmm, 2*22 3.3.6.6 3.3.3.3.6 p6m, *632 3.3.3.3.3.3 3.3.4.3.4 p6m, *632 3.4.6.4 3.3.4.3.4 p6m, *632 4.6.12 3.4.6.4 p6m, *632 3.4.6.4 3.4.4.6 p6m, *632 3.4.6.4 3.3.3.4.4 p6m, *632 3.3.3.3.3.3 3.3.4.12

## Tilings that are not edge-to-edge

 Octagrams and squares 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.
 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. Two size triangles 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.

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; 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.

## The hyperbolic plane

These tessellations are also related to regular and semiregular polyhedra and tessellations of the hyperbolic plane. Semiregular polyhedra are made from regular polygon faces, but their angles at a point add to less than 360 degrees. Regular polygons in hyperbolic geometry have angles smaller than they do in the plane. In both these cases, that the arrangement of polygons is the same at each vertex does not mean that the polyhedron or tiling is vertex-transitive.

Some regular tilings of the hyperbolic plane (Using Poincaré disc model projection)