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Tessellation

January 15, 2012 by staff 

TessellationTessellation, Tessellation is the process of creating a two-dimensional plane using the repetition of a geometric shape with no overlaps and no gaps. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art of M. C. Escher, who was inspired by studying the Moorish use of symmetry in the Alhambra tiles during a visit in 1922. Tessellations are seen throughout art history, from ancient architecture to modern art.

In Latin, tessella is 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”). It corresponds with the everyday term tiling which refers to applications of tessellations, often made of glazed clay. Examples of tessellations in the real world include honeycombs and pavement tilings

In 1619 Johannes Kepler did one of the first documented studies of tessellations when he wrote about the regular and semiregular tessellation, which are coverings of a plane with regular polygons. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries. Fyodorov’s work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov (1951); and Heinrich Heesch and Otto Kienzle (1963).
If this parallelogram pattern is colored before tiling it over a plane, seven colors are required to ensure each complete parallelogram has a consistent color that is distinct from that of adjacent areas. (This tiling can be compared to the surface of a torus.) Coloring after tiling, only four colors are needed.
When discussing a tiling that is displayed in colors, to avoid ambiguity one needs to specify whether the colors are part of the tiling or just part of its illustration. See also symmetry.

The four color theorem states that for every tessellation of a normal Euclidean plane, with a set of four available colors, each tile can be colored in one color such that no tiles of equal color meet at a curve of positive length. Note that the coloring guaranteed by the four-color theorem will not in general respect the symmetries of the tessellation. To produce a coloring which does, as many as seven colors may be needed, as in the picture at right.

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