Generally, a tesselation (sometimes called a tiling or mosaic) is a division of space into pieces with finite area (or volume) called tiles. Tesselations and patterns have been in art for at least two thousand years. Ancient artifacts decorated with repeating patterns, tiled buildings like the Alhambra, and mosaics on the frames of artwork are all evidence of the widespread and age-old use of tesselations. Tesselations in nature include the bee's honeycomb, atomic structures of crystals, cell arrangements in life forms, patches of dried up mud and the design of spider's web. The manufacture of components, by stamping them out of sheets of metal, is made efficient by minimizing the unused portion of the metal. Thus tiling has applications to industry. Also communication theorists use "random tesselations" for image enhancement and coding. Crystallography uses tesselations of three-dimensional space to understand the structures of molecules.

The mathematical theory of...