![]() ![]() 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. Let us know how you felt also by "reacting" and commenting below.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. I hope this was a worthwhile blog post to read. How do you think tessellations can become an important part of life? I hope you learned some information today, but I wanna ask you this. Since these are regular hexagons, each interior angle of each hexagon are 120 degrees, and all the angles in one of the hexagons equal 720 degrees. It uses regular hexagons to form this natural mosaic around the surface area of the hive. Pentagons have a total angle measure of 540 degrees, hexagons have a total measure of 720 degrees, and quadrilaterals have a total angle measure of 360.įinally, A honeycomb is a perfect example of a natural tessellation. In this shell, we see 3 irregular hexagons surrounded by pentagons, also surrounded by many quadrilaterals. A turtle shell shows a special tessellation (at least for Kristian) since they use multiple, different shapes, instead of seeing the same shape over and over again. ![]()
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