Tessellation

A tessellated plane

A tessellation of the plane is a collection of plane figures that fill the plane with no overlaps and no gaps.

In Latin, tessella was a small cubical piece of clay, stone or glass used to make mosaics. The word "tessella" means "small square" (from "tessera", square, which in its turn is from the Greek word for "four").

One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible.

A regular tessellation is a tessellation made up of congruent regular polygons. Only three regular tessellations exist: those made up of triangles, squares, and hexagons.

Other types of tessellations are considered, depending on types of figures and types of pattern: regular vs. irregular, periodic vs. aperiodic, symmetric vs. asymmetric, fractal, etc.

Compare with the Penrose tiling, a tiling of two polygons that however create aperiodic patterns.

The tessellation is perhaps most well-known today for its use in the art of M.C. Escher.

In the subject of computer graphics, tessellation techniques are often used to manage datasets of polygons and divide them into suitable structures for rendering. Normally, at least for real-time rendering, the data is tessellated into triangles, which sometimes get referred to as triangulation.

Some geodesic domes are designed by tessellating the sphere with triangles that are as close to equilateral triangles as possible.

In computer-aided design, arbitrary 3D shapes are often too complicated to analyze directly. So they are divided (tessellated) into a mesh of small, easy-to-analyze pieces -- usually either irregular tetrahedrons, or irregular hexahedrons. The mesh is used for finite element analysis.