Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fill the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. The tessellation is perhaps most well-known today for its use in the art of M.C. Escher.

Related Topics:
Plane - Plane figure - Art - M.C. Escher

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In Latin, tessella was 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 tessellation, often made of glazed clay.

Related Topics:
Clay - Stone - Glass - Mosaic - Tessera - Glazed

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Tilings with translational symmetry can be categorized by wallpaper group, of which 17 exist. All seventeen of these patterns are known to exist in the Alhambra palace in Granada, Spain. Of the three regular tilings two are in the category p6m and one is in p4m.

Related Topics:
Translational symmetry - Wallpaper group - Alhambra - Granada - Spain

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Copies of an arbitrary quadrilateral can form a tessellation with 2-fold rotational centers at the midpoints of all sides, and translational symmetry with as minimal set of translation vectors a pair according to the diagonals of the quadrilateral, or equivalently, one of these and the sum or difference of the two. For an asymmetric quadrilateral this tiling belongs to wallpaper group group p2. As fundamental domain we have the quadrilateral. Equivalently, we can construct a parallelogram subtended by a minimal set of translation vectors, starting from a rotational center. We can divide this by one diagonal, and take one half (a triangle) as fundamental domain. Such a triangle has the same area as the quadrilateral and can be constructed from it by cutting and pasting.

Related Topics:
Quadrilateral - Wallpaper group group p2

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A regular tessellation is a highly symmetric tessellation made up of congruent regular polygons. Only three regular tessellations exist: those made up of equilateral triangles, squares, or hexagons. Other types of tessellations exist, depending on types of figures and types of pattern: regular vs. irregular, periodic vs. aperiodic, symmetric vs. asymmetric, fractal, etc. There is even Penrose tiling, a tessellation of two polygons that however create aperiodic patterns. A different kind of aperiodic tiling can be constructed out of self-replicating polygons by using recursion.

Related Topics:
Congruent - Regular polygon - Equilateral - Triangle - Square - Hexagon - Periodic - Symmetric - Fractal - Penrose tiling - Aperiodic - Self-replicating - Recursion

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In the subject of computer graphics, tessellation techniques are often used to manage datasets of polygons and divide them into suitable structures for rendering. Normally, at least for real-time rendering, the data is tessellated into triangles, which sometimes get referred to as triangulation. In computer-aided design, arbitrary 3D shapes are often too complicated to analyze directly. So they are divided (tessellated) into a mesh of small, easy-to-analyze pieces -- usually either irregular tetrahedrons, or irregular hexahedrons. The mesh is used for finite element analysis. Some geodesic domes are designed by tessellating the sphere with triangles that are as close to equilateral triangles as possible.

Related Topics:
Computer graphics - Rendering - Triangulation - Computer-aided design - Mesh - Tetrahedron - Hexahedron - Finite element analysis - Geodesic dome

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