A tessellated plane seen in street pavement.

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art of M. C. Escher. Tessellations are seen throughout art history, from ancient architecture to modern art. Download high resolution version (1000x960, 431 KB)Example of wallpaper group type p3. ... Download high resolution version (1000x960, 431 KB)Example of wallpaper group type p3. ... This article is about the mathematical construct. ... In geometry, two sets of points are of the same shape precisely if one can be transformed to another by dilating (i. ... This article is about the philosophical concept of Art. ... Maurits Cornelis Escher (June 17, 1898 – March 27, 1972), usually referred to as M. C. Escher, was a Dutch graphic artist. ...

In Latin, tessella was a small cubical piece of clay, stone or glass used to make mosaics.^[1] 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. For other uses, see Clay (disambiguation). ... This article is about the geological substance. ... This article is about the material. ... This article is about a decorative art. ... Roman Tessera A tessera (plural: tesserae, diminutive tessella) is an individual tile in a mosaic, usually formed in the shape of a cube. ... Composite body, painted, and glazed bottle. ...

Wallpaper groups

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. A translation slides an object by a vector a: Ta(p) = p + a. ... Example of an Egyptian design with wallpaper group p4m A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. ... The Alhambra (Arabic: الحمراء = Al-Ħamrā; literally the red fortress) is a palace and fortress complex of the Moorish monarchs of Granada in southern Spain (known as Al-Andalus when the fortress was constructed), occupying a hilly terrace on the southeastern border of the city of Granada. ... For other uses, see Granada (disambiguation). ...

Tessellations and color

When discussing a tiling that is displayed in colors, to avoid ambiguity one needs to specify whether the colors are part of the tiling or just part of its illustration. See also color in symmetry. In geometry, a torus (pl. ... Sphere symmetry group o. ...

The four color theorem states that for every tessellation of a normal Euclidean plane, with a set of four available colors, each tile can be colored in one color such that no tiles of equal color meet at a curve of positive length. Note that the coloring guaranteed by the four-color theorem will not in general respect the symmetries of the tessellation. To produce a coloring which does, as many as seven colors may be needed, as in the picture at right. Example of a four-colored map The four color theorem (also known as the four color map theorem) states that given any plane separated into regions, such as a political map of the states of a country, the regions may be colored using no more than four colors in such...

Tessellations with quadrilaterals

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 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. This article is about the geometric shape. ... Example of an Egyptian design with wallpaper group p4m A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. ...

Regular and irregular tessellations

Hexagonal tessellation of a floor

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. A semiregular tessellation uses a variety of regular polygons; there are eight of these. The arrangement of polygons at every vertex point is identical. An edge-to-edge tessellation is even less regular: the only requirement is that adjacent tiles only share full sides, i.e. no tile shares a partial side with any other tile. Other types of tessellations exist, depending on types of figures and types of pattern. There are regular versus irregular, periodic versus aperiodic, symmetric versus asymmetric, and fractal tesselations, as well as other classifications. Image File history File links Hexagonal_tessellation. ... Image File history File links Hexagonal_tessellation. ... Plane tilings by regular polygons have been widely used since antiquity. ... See also: congruence relation In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... For other uses, see Square. ... For other uses, see Hexagon (disambiguation). ... Plane tilings by regular polygons have been widely used since antiquity. ... Periodicity is the quality of occurring at regular intervals (e. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... The boundary of the Mandelbrot set is a famous example of a fractal. ...

Penrose tilings using two different polygons are the most famous example of tessellations that create aperiodic patterns. They belong to a general class of aperiodic tilings that can be constructed out of self-replicating sets of polygons by using recursion. A Penrose tiling A Penrose tiling is an aperiodic tiling of the plane discovered by Roger Penrose in 1973. ... This article or section is in need of attention from an expert on the subject. ... Self-replication is the process by which some things make copies of themselves. ... This article is about the concept of recursion. ...

A monohedral tiling is a tessellation in which all tiles are congruent. The Voderberg tiling discovered by Hans Voderberg in 1936, which is the earliest known spiral tiling. The unit tile is a bent enneagon. The Hirschhorn tiling discovered by Michael Hirschhorn in the 1970s. The unit tile is an irregular pentagon. An example of congruence. ... Year 1936 (MCMXXXVI) was a leap year starting on Wednesday (link will display the full calendar) of the Gregorian calendar. ... A regular enneagon. ... The 1970s decade refers to the years from 1970 to 1979, also called The Seventies. ... Look up pentagon in Wiktionary, the free dictionary. ...

Tessellations and computer graphics

A tessellation of a disk used to solve a finite element problem.

These rectangular bricks are connected in a tessellation, which if considered an edge-to-edge tiling, topologically identical to a hexagonal tiling, with each hexagon flattened into a rectangle with the long edges divided into two edges by the neighboring bricks.

This basketweave tiling is topologically identical to the Cairo pentagonal tiling, with one side of each rectangle counted as two edges, divided by a vertex on the two neighboring rectangles.

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 is sometimes 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. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Mathematically, the finite element method (FEM) is used for finding approximate solution of partial differential equations (PDE) as well as of integral equations such as the heat transport equation. ... Download high resolution version (1528x1528, 928 KB)Illustration of cmm type Wallpaper group (bricks on side of garage) File links The following pages link to this file: Wallpaper group Talk:Wallpaper group Categories: Public domain images ... Download high resolution version (1528x1528, 928 KB)Illustration of cmm type Wallpaper group (bricks on side of garage) File links The following pages link to this file: Wallpaper group Talk:Wallpaper group Categories: Public domain images ... In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. ... Download high resolution version (1587x1587, 940 KB)Illustration of p4g type Wallpaper group (bathroom floor tiling) File links The following pages link to this file: Wallpaper group Talk:Wallpaper group Categories: Public domain images ... Download high resolution version (1587x1587, 940 KB)Illustration of p4g type Wallpaper group (bathroom floor tiling) File links The following pages link to this file: Wallpaper group Talk:Wallpaper group Categories: Public domain images ... In geometry, the Cairo pentagonal tiling is a dual semiregular tiling of the Euclidean plane. ... This article is about the scientific discipline of computer graphics. ... Rendering has several different usages: Rendering (computer graphics) is the process of producing the pixels of an image from a higher-level description of its components. ... Polygon triangulation is a topic in computational geometry. ... CADD and CAD redirect here. ... A mesh is a collection of vertices and polygons that define the shape of an object in 3D computer graphics. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... A hexahedron is a polyhedron with six faces. ... Visualization of how a car deforms in an asymmetrical crash using finite element analysis. ... Spaceship Earth in Epcot Center at Walt Disney World is perhaps one of the most famous examples of a large scale geodesic sphere. ...

Tessellations in nature

Basaltic lava flows often display columnar jointing as a result of contraction forces causing cracks as the lava cools. The extensive crack networks that develop often produce hexagonal columns of lava. One example of such an array of columns is the Giant's Causeway in Ireland. For the cities, see Basalt, Colorado and Basalt, Idaho. ... In computer programming jargon, lava flow is a problem in which computer code, usually written under less than optimal conditions, is put into production and then built on when still in a developmental state. ... For other uses, see Column (disambiguation). ... Columnar jointed basalt in Turkey Columnar jointing in the basalt of the Giants Causeway in Ireland A joint is a generally planar fracture formed in a rock as a result of extensional stress. ... Look up Contraction in Wiktionary, the free dictionary. ... For other uses, see Giants Causeway (disambiguation). ...

Number of sides of a polygon versus number of sides at a vertex

For an infinite tiling, let a be the average number of sides of a polygon, and b the average number of sides meeting at a vertex. Then (a − 2)(b − 2) = 4. For example, we have the combinations , for the tilings in the article Tilings of regular polygons. This article needs to be cleaned up to conform to a higher standard of quality. ...

A continuation of a side in a straight line beyond a vertex is counted as a separate side. For example, the bricks in the picture are considered hexagons, and we have combination (6, 3).

Similarly, for the bathroom floor tiling we have (5, 3 1/3).

For a tiling which repeats itself, one can take the averages over the repeating part. In the general case the averages are taken as the limits for a region expanding to the whole plane. In cases like an infinite row of tiles, or tiles getting smaller and smaller outwardly, the outside is not negligible and should also be counted as a tile while taking the limit. In extreme cases the limits may not exist, or depend on how the region is expanded to infinity.

For finite tessellations and polyhedra we have For the game magazine, see Polyhedron (magazine). ...

where F is the number of faces and V the number of vertices, and χ is the Euler characteristic (for the plane and for a polyhedron without holes: 2), and, again, in the plane the outside counts as a face. In algebraic topology, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological spaces shape or structure. ...

The formula follows observing that the number of sides of a face, summed over all faces, gives twice the number of sides, which can be expressed in terms of the number of faces and the number of vertices. Similarly the number of sides at a vertex, summed over all faces, gives also twice the number of sides. From the two results the formula readily follows.

In most cases the number of sides of a face is the same as the number of vertices of a face, and the number of sides meeting at a vertex is the same as the number of faces meeting at a vertex. However, in a case like two square faces touching at a corner, the number of sides of the outer face is 8, so if the number of vertices is counted the common corner has to be counted twice. Similarly the number of sides meeting at that corner is 4, so if the number of faces at that corner is counted the face meeting the corner twice has to be counted twice.

A tile with a hole, filled with one or more other tiles, is not permissible, because the network of all sides inside and outside is disconnected. However it is allowed with a cut so that the tile with the hole touches itself. For counting the number of sides of this tile, the cut should be counted twice.

For the Platonic solids we get round numbers, because we take the average over equal numbers: for (a − 2)(b − 2) we get 1, 2, and 3. In geometry, a Platonic solid is a convex regular polyhedron. ...

From the formula for a finite polyhedron we see that in the case that while expanding to an infinite polyhedron the number of holes (each contributing −2 to the Euler characteristic) grows proportionally with the number of faces and the number of vertices, the limit of (a − 2)(b − 2) is larger than 4. For example, consider one layer of cubes, extending in two directions, with one of every 2 × 2 cubes removed. This has combination (4, 5), with (a − 2)(b − 2) = 6 = 4(1 + 2 / 10)(1 + 2 / 8), corresponding to having 10 faces and 8 vertices per hole.

Note that the result does not depend on the edges being line segments and the faces being parts of planes: mathematical rigor to deal with pathological cases aside, they can also be curves and curved surfaces.

Tessellations of other spaces

M.C.Escher, Circle Limit III (1959).

As well as tessellating the 2-dimensional Euclidean plane, it is also possible to tessellate other n-dimensional spaces by filling them with n-dimensional polytopes. Tessellations of other spaces are often referred to as honeycombs. Examples of tessellations of other spaces include: Image File history File links Escher_Circle_Limit_III.jpg‎ Circle Limit III by M. C. Escher (1959) Woodcut, second state, in yellow, green, blue, brown and black, printed from 5 blocks. ... Image File history File links Escher_Circle_Limit_III.jpg‎ Circle Limit III by M. C. Escher (1959) Woodcut, second state, in yellow, green, blue, brown and black, printed from 5 blocks. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... In geometry, a honeycomb is a name for a space-filling tessellation, just as a tiling is a tessellation of a plane or 2-dimensional surface. ...

• Tessellations of n-dimensional Euclidean space - for example, filling 3-dimensional Euclidean space with cubes to create a cubic honeycomb.

• Tessellations of n-dimensional elliptic space - for example, projecting the edges of a dodecahedron onto its circumsphere creates a tessellation of the 2-dimensional sphere with regular spherical pentagons.

• Tessellations of n-dimensional hyperbolic space - for example, M. C. Escher's Circle Limit III depicts a tessellation of the hyperbolic plane with congruent fish-like shapes. The hyperbolic plane admits a tessellation with regular p-gons meeting in q's whenever ; Circle Limit III may be understood as a tiling of octagons meeting in threes, with all sides replaced with jagged lines and each octagon then cut into four fish.

Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedrons vertices. ... In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ... Maurits Cornelis Escher (June 17, 1898 – March 27, 1972), usually referred to as M. C. Escher, was a Dutch graphic artist. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ... For other uses, see Octagon (disambiguation). ...

History

In every civilization and culture, colored tilings and patterns appear among the earliest decorations.... In particular, 2-color patterns arose -- early and frequently -- through a device known as 'counterchange'.... An early paper with remarkable counterchange designs formed by diagonally divided squares -- one-half black, one-half white -- was published by Truchet (1704).

– Branko Grünbaum and G. C. Shephard. Tilings and Patterns Branko Grünbaum is a mathematician who works mainly in geometry and is considered a founder of discrete geometry. ...

See also

In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. ... In geometry, a honeycomb is a name for a space-filling tessellation, just as a tiling is a tessellation of a plane or 2-dimensional surface. ... A jigsaw puzzle is a tiling puzzle that requires the assembly of numerous small, often oddly-shaped, interlocking and tesellating pieces. ... This table shows the 11 convex uniform tilings of the Euclidean plane, and their dual tilings. ... Mathematical ideas have been used as inspiration for a number of fiber arts including quilt making, knitting, cross-stitch, crochet, embroidery and weaving. ... This article is about a decorative art. ... A Penrose tiling A Penrose tiling is an aperiodic tiling of the plane discovered by Roger Penrose in 1973. ... A polyiamond is a counterpart to a polyomino where the polygon used as the building block is an equilateral triangle rather than a square. ... The 35 possible hexominoes. ... Quilter and Quilters redirect here. ... Self-replication is the process by which some things make copies of themselves. ... Mission, or barrel, roof tiles A tile is a manufactured piece of hard-wearing material such as ceramic, stone, porcelain, metal or even glass. ... This page is a candidate for speedy deletion. ... This article or section is in need of attention from an expert on the subject. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Trianglepoint completed on plastic canvas with varigated and solid colored yarn. ... In advanced geometry, in the most general meaning, triangulation is a subdivision of a geometric object into simplices. ... In mathematics, a uniform tessellation is a tessellation of a d-dimensional space, or a (hyper)surface, such that all its vertices are identical, i. ... This is the Voronoi diagram of a random set of points in the plane (all points lie within the image). ... Example of an Egyptian design with wallpaper group p4m A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. ... Wang tiles (or Wang dominoes), first proposed by Hao Wang in 1961, are a class of formal systems. ...

References

• Grunbaum, Branko and G. C. Shephard. Tilings and Patterns. New York: W. H. Freeman & Co., 1987. ISBN 0-7167-1193-1.
• Coxeter, H.S.M.. Regular Polytopes, Section IV : Tessellations and Honeycombs. Dover, 1973. ISBN 0-486-61480-8.

Branko Grünbaum is a mathematician who works mainly in geometry and is considered a founder of discrete geometry. ... H(arold). ... Stereographic projection of the 120-cell, a 4-dimensional regular polytope. ...

External links

Wikimedia Commons has media related to:

Tessellation

• K-12 Tessellation Lesson

• 2D Puzzles generated using a tessellation of the plane
• Tessellation Video
• Math Forum Tessellation Tutorials - make your own
• Mathematical Art of M. C. Escher - tessellations in art
• The 14 Different Types of Convex Pentagons that Tile the Plane
• Tiling Plane & Fancy at Southern Polytechnic State University
• Grotesque Geometry, Andrew Crompton
• Tessellations.org - many examples and do it yourself tutorials
• Tessellation.info A database with over 500 tessellations categorized by artist and depicted subjects
• Minesweeper Variants - A Minesweeper game using different tessellation patterns
• Semiregular pattern - This pattern can describe a collapsing cylinder
• Hyperbolic Tessellations, David E. Joyce, Clark University
• Some Special Radial and Spiral Tilings