Tesselation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps.
A dictionary* will tell you that the word "tessellate" means to form or arrange small squares in a checkered or mosaic pattern. The word "tessellate" is derived from the Ionic version of the Greek word "tesseres," which in English means "four." The first tilings were made from square tiles.
A regular polygon has 3 or 4 or 5 or more sides and angles, all equal. A regular tessellation means a tessellation made up of congruent regular polygons. [Remember: Regular means that the sides and angles of the polygon are all equivalent (i.e., the polygon is both equiangular and equilateral). Congruent means that the polygons that you put together are all the same size and shape.]

Example of regular tessellation :

a tessellation of triangles
a tessellation of squares
a tessellation of hexagons

There are also eight semi-regular tessellations which consist of two or more regular polygons which meet at each vertex and also do not overlap or leave gaps.

There are an infinite number of tessellations which are made up of irregular shapes; these are known as non-regular tessellations.

On 3D surfaces such as the hyperbolic plane, spheres and tori, there are an infinite number of regular tessellations. For example, on the surface of a sphere, a pentagon can tessellate regularly. (The diagram above is shown as a disk on the Euclidean Plane, which leads to distortions.)

Tessellations in Art

A Roman mosaic from Fishbourne Palace, England

The original word tessellation comes from its use in art. From Ancient Greek a Tessera or Tessella is the small dice sized piece of stone used in mosaics. Therefore, as the dictionary suggests, the original tessellations were mosaics (right). Tessellations were first used in the form of mosaics in about 3000 BC in Ancient Mesopotamia. The tessellation in mosaics pertains to the actual structure of the arrangement of the small pieces of stone or tile, which is the regular tessellation of squares. Many of these mosaics not only had tessellations in their structure but the patterns were also those of tessellations.
One of the greatest practitioners of the use of tessellations in art was the Dutch graphic artist M. C. Escher (1898-1972). Although he is more famous for his drawings of the impossible, he also worked extensively on tessellations. Many of his works using tessellations consist not of a single repeated image, but of a smooth metamorphosing of one image into another. His 1938 lithograph, Sky and Water 1 (bottom right) is typical of his work. Since he started the trend many other artists have made similar tessellating art (bottom centre).