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.
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.

In Mathematics
Although tessellations have been traced back to ancient human cultures and can also be found in the natural world, they have had a relatively short history as a topic for serious mathematical and scientific study. One of the first mathematical studies of tessellations was conducted by Johannes Kepler in 1619 who wrote about the regular and semiregular tessellation, which are coverings of a plane with regular polygons. However, about two hundred years passed before new scientific progress concerning tessellations was made.
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).
The tilings in Spain were laid out by the Moors in the 14th century. They are made of coloured tiles forming patterns, some are truly symmetrical,
geometrical and beautiful. Geometrical shapes played a major role in the development of Tessellations. Some were not tessellations because they
didn't cover a surface with a repetitive design without gaps or overlaps. Tessellations have been around for many years and have evolved
through the centuries into what we see as patterns in our architectural designs of today.

Tile tilings that occur in tessellations are applied to math in the following four forms.
The following are examples used in a math forum listed in the references.

The History of TessellationIn ScienceAside 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.

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.

In MathematicsAlthough tessellations have been traced back to ancient human cultures and can also be found in the natural world, they have had a relatively short history as a topic for serious mathematical and scientific study. One of the first mathematical studies of tessellations was conducted by Johannes Kepler in 1619 who wrote about the regular and semiregular tessellation, which are coverings of a plane with regular polygons. However, about two hundred years passed before new scientific progress concerning tessellations was made.

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).

The tilings in Spain were laid out by the Moors in the 14th century. They are made of coloured tiles forming patterns, some are truly symmetrical,

geometrical and beautiful. Geometrical shapes played a major role in the development of Tessellations. Some were not tessellations because they

didn't cover a surface with a repetitive design without gaps or overlaps. Tessellations have been around for many years and have evolved

through the centuries into what we see as patterns in our architectural designs of today.

Tile tilings that occur in tessellations are applied to math in the following four forms.

The following are examples used in a math forum listed in the references.