Tessella’s company name is derived from the words tesserra or tessella which refer to the tiles used to make up a mosaic picture. Shapes which tessellate cover the background without leaving gaps and without overlapping.

Tessella provides IT and mathematical problem-solving skills which fit together with your own capabilities without leaving gaps and without duplicating what you already do, thus completing the big picture. If your organization’s activities are based on science and engineering, and you experience complex problems then Tessella can help you.

The dictionary definition of tessellations

  1. The action or art of tessellating; tessellated condition; a piece of tessellated work.
  2. An arrangement or close fitting together of minute parts or distinct colours.

The dictionary definition of tessellated

  1. Composed of small blocks of variously coloured material arranged to form a pattern; formed of or ornamented with mosaic work.
  2. Combined or arranged so as to form a mosaic.
  3. Consisting of or arranged in small cubes or squares; chequered, reticulated.

Tessellations in Mathematics

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The Three Regular Tessellations on the Euclidean Plane

Shapes which tessellate cover the plane without gaps and without overlapping. There can only be three regular Tessellations on the Euclidean plane (2D plane) which are made from copies of a single regular polygon meeting at each vertex. These are of equilateral triangles, squares or regular hexagons (each shown below). There are only three because the inside angles of the polygon must be a factor of 360° so that the polygons can line up at the points leaving no gaps.

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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.)

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.

Tessellations in Art

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.

A Roman mosaic from Fishbourne Palace, England

A Roman mosaic from Fishbourne Palace, England

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).

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Sun and Moon by M. C. Escher
A tessellation of frogs on a sphere in the style of Escher
Sky and Water by M. C. Escher