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 the regular hexagon. Among those that 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. Common ones are that there must be no gaps between tiles, and that no corner of one tile can lie along the edge of another. Tessellation in two dimensions, also called planar tiling, is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules. Overview A rhombitrihexagonal tiling: tiled floor in the Archeological Museum of Seville, Spain, using square, triangle, and hexagon prototiles It corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. The word 'tessella' means 'small square' (from tessera, square, which in 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 Alexei Vasilievich Shubnikov and Nikolai 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. He wrote about regular and semiregular tessellations in his Harmonices Mundi he was possibly the first to explore and to explain the hexagonal structures of honeycomb and snowflakes. In 1619, Johannes Kepler made an early documented study of tessellations.

ĭecorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. Tessellations were used by the Sumerians (about 4000 BC) in building wall decorations formed by patterns of clay tiles. History A temple mosaic from the ancient Sumerian city of Uruk IV (3400–3100 BC), showing a tessellation pattern in coloured tiles Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations are sometimes employed for decorative effect in quilting.

Escher often made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the Moroccan architecture and decorative geometric 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. A tessellation of space, also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions.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 semiregular tilings with regular tiles of more than one shape and with every corner identically arranged.

In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.Ī periodic tiling has a repeating pattern. An example of non‑periodicity due to another orientation of one tile out of an infinite number of identical tiles.Ī tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps.

Triangle tessellation 128x128