Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are congruent to the given prototiles. These tiles may be polygons or any other shapes. More formally, a tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles, such that the tiles intersect only on their boundaries. If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, and these can be used to form physical surfaces such as church floors.Įlaborate and colourful tessellations of glazed tiles at the Alhambra in Spain Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects. Irregular tessellations can also be made from other shapes such as pentagons, polyominoes and in fact almost any kind of geometric shape. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner. Many other types of tessellation are possible under different constraints. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps. There are only three shapes that can form such regular tessellations: the equilateral triangle, square, and regular hexagon. Among those which do, a regular tessellation has both identical regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile. The tessellations created by bonded brickwork do not obey this rule. A common one is that all corners should meet and that no corner of one tile can lie along the edge of another. Tessellation or tiling in two dimensions is the branch of mathematics that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules. It corresponds with the everyday term tiling which refers to applications of tessellations, often made of glazed clay.Ī rhombitrihexagonal tiling: tiled floor of a church in Seville, Spain, using square, triangle and hexagon prototiles The word "tessella" means "small square" (from "tessera", square, which in its turn is from the Greek word "τέσσερα" for "four"). In Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. Other prominent contributors include Shubnikov and Belov (1964), and Heinrich Heesch and Otto Kienzle (1963). Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. 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. In 1619 Johannes Kepler made one of the first documented studies of tessellations when he wrote about regular and semiregular tessellation, which are coverings of a plane with regular polygons, in his Harmonices Mundi. Tessellations were used by the Sumerians (about 4000 BC) in building wall decorations formed by patterns of clay tiles. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs.Ī temple mosaic from the ancient Sumerian city of Uruk IV (3400–3100 BC), showing a tessellation pattern in coloured tiles Tessellations are sometimes employed for decorative effect in quilting. Escher often made use of tessellations for artistic effect. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative tiling of the Alhambra palace. Such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor or wall coverings. In the geometry of higher dimensions, a space filling or honeycomb is also called a tessellation of space.Ī real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons. An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern. A tiling that lacks a repeating pattern is called "non-periodic". The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semi-regular tilings with regular tiles of more than one shape and with every corner identically arranged. In mathematics, tessellations can be generalized to higher dimensions.Ī periodic tiling has a repeating pattern. A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps.
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