3/13/2023 0 Comments Types of tessellation![]() 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. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs.Įssellations were used by the Sumerians (about 4000 BC) in building wall decorations formed by patterns of clay tiles.ĭecorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. 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 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. Escher (uniform tessellations of the hyperbolic plane (in)).Ī real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons. There are also tessellations of non-Euclidean spaces, the most famous being without doubt the numerous pavements of M.C. Generally, we consider tilings by translations, that is to say that two same paving tiles are always deductible from one another by a translation (excluding rotations or symmetries). ![]() ![]() In the geometry of higher dimensions, a space-filling or honeycomb is also called a tessellation of space.Ī pavement or pavement is a partition of a space (usually a Euclidean space like the plane or three-dimensional space) by elements of a finite set, called tiles (more precisely, they are non-empty interior compacts). 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. The basic parallelogram template, however, is not the most complete way to classify regular dottings knowing the measurements of its angles and sides does not allow us to certify with certainty the geometric features of our dimming: it may happen that there is a smaller portion of the parallelogram (more precisely, a proportion of the parallelogram) with which it is possible to reconstruct all the décoration (no longer with sole translation, but using also other isometrics).Ī periodic tiling has a repeating pattern. The reason why it is useful is that it allows to compare tons of different appearance to each other. The only condition that usually arises is that they are connected, rather simply connected (that is, they are a single piece and do not have holes).Īlthough this condition may seem very restrictive, it is respected by virtually any flooring you may think. These geometric figures, (called “dowels”), are often polygons, regular or not, but may also have curved sides, or have no vertex. In flat geometry, it is called dimming (sometimes tilting or flooring) the ways to cover the plane with one or more geometric figures repeated infinitely without overlapping. The hex divide the plane with the minimum perimeter used for the covered surface portion. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.Ĭomparison of area-perimeter ratio between equilateral triangle, square and regular hexagon. A tessellation, or a Tiling 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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