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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 ''Harmonices Mundi''.

Contents

Regular tilings

Archimedean, uniform or semiregular tilings

Combinations of regular polygons that can meet at a vertex

Other edge-to-edge tilings

Tilings that are not edge-to-edge

Beyond the plane

See also

References

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.

tile 33336.svg 34.6 Snub hexagonal tiling	tile 3636.svg 3.6.3.6 Trihexagonal tiling
tile 33344.svg 33.42 Elongated triangular tiling	tile 33434.svg 32.4.3.4 Snub square tiling
tile 3464.svg 3.4.6.4 Small rhombitrihexagonal tiling	tile 488.svg 4.82 Truncated square tiling
tile 3bb.svg 3.122 Truncated hexagonal tiling	tile 46b.svg 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 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, occures 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:

dem3366bc.png 32.62 and 36	dem3366rbc.gif 32.62 and 3.6.3.6
dem3343tbc.gif 32.4.12 and 36	dem3446bc.gif 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.

Beyond the 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)

See also

★ List of uniform tilings

★ Wythoff symbol

★ Tessellation

★ Wallpaper group

★ Regular polyhedron (the Platonic solids)

★ Semiregular polyhedron (including the Archimedean solids)

★ Hyperbolic geometry

★ Penrose tiling

References

★ Tilings and Patterns, Grünbaum, Branko; Shephard, G. C., , , W. H. Freeman and Company, 1987, ISBN 0-7167-1193-1

★ Tilings by Regular Polygons—II: A Catalog of Tilings, D. Chavey, , , Computers & Mathematics with Applications, 1989

External links

Euclidean and general tiling links:

★ Uniform Tilings Dutch, Steve

★ Semi-Regular Tilings Mitchell, K

★ Semiregular Tessellation Weisstein, Eric W
Hyperbolic tiling links:

★ The Geometry Junkyard: Hyperbolic Tiling Eppstein, David

★ Hyperbolic Planar Tessellations Hatch, Don

★ Hyperbolic Tessellations Joyce, David