# Tessellation

A **Tessellation** is a regular tiling of the infinite plane by polygons, or of an infinite three dimensional space by polyhedra, or of an infinite n-dimensional space by polytopes.

Islamic art and architecture make heavy use of intricate tessellations as decoration.

## Plane Tessellations

Tessellations of the plane can be classified as follows:

**Regular Tessellations** are comprised of simple repetitions of the same regular polygon. There are three distinct regular tessellations: equilateral triangles, squares and hexagons.

**Semiregular Tessellations** are comprised of repetitions of two or more regular polygons, where the number and type of each kind of polygon surrounding each vertex is the same for all vertices in the plane. There are *eight* such semiregular tessellations. These are often referred to as Archimedean tessellations.

**Demiregular Tessellations** are comprised of repetitions of two or more regular polygons, where the number and type of each kind of polygon surrounding each vertex is different for some of the vertices in the plane. Although it can be argued that the number of such tessellations is infinite, they can be reduced to 14 specific types.

Other tessellations of the plane exist. See for example the tilings of M. C. Escher, such as the one which is illustrated here. Such tilings are not tessellations, in that they are not comprised of polygons. However, these tilings are based on one of the above named types of tessellation.

The tiling illustrated is based on a regular tessellation of squares.

A tiling is periodic if, when translated in at least two non-parallel directions the tiling 'merges' with itself. A collection of tiles is aperiodic if the collection tiles the plane, but never in a periodic fashion. Roger Penrose discovered several such tilings in the 1970's.

The one here demonstrates aspects of fivefold symmetry, and was inspired by some of the tilings of Kepler.

## Links

http://mathworld.wolfram.com/Tessellation.html