Near the end of the nineteenth century in 1891, the Russian crystallographer E. S. Fedorov proved that every tiling of the plane is constructed in accordance to one of seventeen different groups of isometries (i.e., methods of repeating tilings over the plane). His study marked the unofficial beginning of the mathematical study of tessellations, occurring only about a hundred years ago. Progress beyond this point has resulted in advanced mathematical analyses of tilings (e.g., extending tilings to more than two dimensions and to non-Euclidean geometrical systems) which can be only hinted at in the Beyond section of this site. Prominent names and dates in this research area include the following: Shubnikov and Belov (1951); and Heinrich Heesch and Otto Kienzle (1963).

See the Beyond section for information about advanced mathematical topics relating to tessellations.

In Science
Aside from being studied in mathematical or geometrical research, tessellations and tilings have been linked with x-ray crystallography. X-ray crystallography is a field of science concerned with the repeating arrangements of identical objects as found in nature, a description very similar to the geometrical definition of tessellation. Interestingly, the discoveries made in x-ray crystallography during the mid-20th century are similar to many of the discoveries the Dutch artist M. C. Escher made while formulating designs for his tessellated artwork. (See the Escher section for more details.)

The symmetry issues that are so important in tessellations have shown to be relevant to quantum mechanics, the study of particles smaller than atoms.

Several other scientific and engineering applications have been found for tessellations. For instance, tiling research has benefited the conservation of sheet material and reduction of scrap metal. How? The closer fitted that the objects to be cut out are, the less waste material produced. Since tessellations are perfectly fitted patterns of shapes, no waste material would be produced if the template resembled a tessellation.

Other fields of research associated with tiling include geology, metallurgy, biology, and cryptology (the study of using secret codes in communication).