Tiling by regular polygons.html

Plane tilings by regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in Harmonices Mundi.

1 Regular tilings
2 Archimedean, uniform or semiregular tilings
3 Combinations of regular polygons that can meet at a vertex
4 Other edge-to-edge tilings
5 Tilings that are not edge-to-edge
6 The hyperbolic plane
7 See also
8 References
9 External links

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.

3⁶ Triangular tiling	4⁴ Square tiling	6³ Hexagonal tiling

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.

3⁴.6 Snub hexagonal tiling	3⁴.6 Snub hexagonal tiling reflection	3.6.3.6 Trihexagonal tiling
3³.4² Elongated triangular tiling	3².4.3.4 Snub square tiling	3.4.6.4 Small rhombitrihexagonal tiling
4.8² Truncated square tiling	3.12² Truncated hexagonal tiling	4.6.12 Great rhombitrihexagonal tiling

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

The internal angles of the polygons meeting at a vertex must add to 360 degrees. A regular $n\,\!$ -gon has internal angle $(1-\frac{2}{n})180$ 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. 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 size. 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, great rhombitrihexagonal tiling
4.8² - semi-regular, truncated square tiling
5².10 (cannot appear in any tiling of regular polygons)
6³ - regular, hexagonal tiling

With 4 polygons at a vertex:

3².4.12 - not uniform, has two different types of vertices 3².4.12 and 3⁶
3.4.3.12 - not uniform, has two different types of vertices 3.4.3.12 and 3.3.4.3.4
3².6² - not uniform, occurs in two patterns with vertices 3².6²/3⁶ and 3².6²/3.6.3.6.
3.6.3.6 - semi-regular, trihexagonal tiling
4⁴ - regular, square tiling
3.4².6 - not uniform, has vertices 3.4².6 and 3.6.3.6.
3.4.6.4 - semi-regular, small rhombitrihexagonal tiling

With 5 polygons at a vertex:

3⁴.6 - snub hexagonal tiling
3³.4² - semi-regular, Elongated triangular tiling
3².4.3.4 - semi-regular, Snub square tiling

With 6 polygons at a vertex:

3⁶ - regular, Triangular tiling

Other edge-to-edge tilings

Any number of non-uniform (sometimes called demiregular) edge-to-edge tilings by regular polygons may be drawn. Here are four examples:

3².6² and 3⁶	3².6² and 3.6.3.6
3².4.12 and 3⁶	3.4².6 and 3.6.3.6

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. The examples above are four of the twenty 2-uniform tilings. Chavey lists all those edge-to-edge tilings by regular polygons which are at most 3-uniform, 3-isohedral or 3-isotoxal.

Tilings that are not edge-to-edge

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

Main article: Uniform tilings in 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)

References

Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1.
D. Chavey (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.
D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes