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. The shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determined 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. Art refers to a diverse range of Human activities creations and expressions that are appealing to the Senses or Emotions of a human individual Maurits Cornelis Escher (17 June 1898 – 27 March 1972 usually referred to as M Tessellations are seen throughout art history, from ancient architecture to modern art.

In Latin, tessella was a small cubical piece of clay, stone or glass used to make mosaics. Clay is a naturally occurring material composed primarily of fine-grained Minerals which show plasticity through a variable range of Water content, and In Geology, rock is a naturally occurring aggregate of Minerals and/or Mineraloids The Earth's outer solid layer the ‘ Lithosphere Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many Art History Mosaics of the 4th century BC are found in the Macedonian palace-city of Aegae, and they enriched the floors of Hellenistic ^[1] The word "tessella" means "small square" (from "tessera", square, which in its turn is from the Greek word for "four"). "Abaciscus" redirects here For the Geometer moth Genus, see Abaciscus (moth. It corresponds with the everyday term tiling which refers to applications of tessellation, often made of glazed clay. Glaze is a layer or coating of a Vitreous substance which has been fired to fuse to a ceramic object to color decorate strengthen or waterproof it

Wallpaper groups

Tilings with translational symmetry can be categorized by wallpaper group, of which 17 exist. In Geometry, a translation "slides" an object by a vector a: T a (p = p + a A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the All seventeen of these patterns are known to exist in the Alhambra palace in Granada, Spain. This article is about the Alhambra in Granada Spain For other meanings see Alhambra (disambiguation. Granada is a city and the capital of the province of Granada, in the autonomous region of Andalusia, Spain. Spain () or the Kingdom of Spain (Reino de España is a country located mostly in southwestern Europe on the Iberian Peninsula. Of the three regular tilings two are in the category p6m and one is in p4m.

Tessellations and color

If this parallelogram pattern is colored before tiling it over a plane, seven colors are required to ensure each complete parallelogram has a consistent color that is distinct from that of adjacent areas. (To see why, we compare this tiling to the surface of a Torus. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar ) If we tile before coloring, only four colors are needed.

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. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or

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. 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 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.

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. In Geometry, a quadrilateral is a Polygon with four sides or edges and four vertices or corners. For an asymmetric quadrilateral this tiling belongs to wallpaper group p2. A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the 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.

Regular and irregular tessellations

Hexagonal tessellation of a floor

A regular tessellation is a highly symmetric tessellation made up of congruent regular polygons. Plane tilings by Regular polygons have been widely used since antiquity General properties These properties apply to both convex and star regular polygons Only three regular tessellations exist: those made up of equilateral triangles, squares, or hexagons. Properties The area of an equilateral triangle with sides of length a\\! Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides Regular hexagon The internal Angles of a regular hexagon (one where all sides and all angles are equal are all 120 ° and the hexagon has 720 degrees A semiregular tessellation uses a variety of regular polygons; there are eight of these. Plane tilings by Regular polygons have been widely used since antiquity 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 tessellations, as well as other classifications. Periodicity is the quality of occurring at regular intervals or periods (in Time or Space) and can occur in different contexts A Clock marks Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole"

Penrose tilings using two different polygons are the most famous example of tessellations that create aperiodic patterns. A Penrose tiling is a nonperiodic tiling generated by an aperiodic set of Prototiles named after Roger Penrose, who investigated these sets They belong to a general class of aperiodic tilings that can be constructed out of self-replicating sets of polygons by using recursion. See also Biological reproduction Self-replication is any process by which a thing might make a copy of itself Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition

A monohedral tiling is a tessellation in which all tiles are congruent. In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i The Voderberg tiling discovered by Hans Voderberg in 1936, which is the earliest known spiral tiling. Year 1936 ( MCMXXXVI) was a Leap year starting on Wednesday (link will display the full calendar of the Gregorian calendar. The unit tile is a bent enneagon. Graphs and stars The K9 Complete graph is often drawn as a regular nonagon with all 36 edges connected The Hirschhorn tiling discovered by Michael Hirschhorn in the 1970s. This article is about the Decade 1970-1979 For the Year 1970 see 1970. The unit tile is an irregular pentagon. Regular pentagons The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to

Tessellations and computer graphics

A tessellation of a disk used to solve a finite element problem. The finite element method (FEM (sometimes referred to as finite element analysis) is a numerical technique for finding approximate solutions of Partial differential

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. In Geometry, the hexagonal tiling is a Regular tiling of the Euclidean plane.

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 Geometry, the Cairo pentagonal tiling is a dual semiregular tiling of the Euclidean plane

In the subject of computer graphics, tessellation techniques are often used to manage datasets of polygons and divide them into suitable structures for rendering. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data Normally, at least for real-time rendering, the data is tessellated into triangles, which is sometimes referred to as triangulation. In Computational geometry, polygon triangulation is the decomposition of a Polygon into a set of Triangles. 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. A polygon mesh or Unstructured grid is a collection of vertices edges and faces that defines the shape of a polyhedral object in 3D computer A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. A hexahedron (plural hexahedra is a Polyhedron with six faces 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. A geodesic dome is an almost spherical shell structure based on a network of Great circles ( Geodesics lying approximately on the surface of a Sphere

Tessellations in nature

Basaltic lava flows often display columnar jointing as a result of contraction forces causing cracks as the lava cools. Basalt (bəˈsɔːlt ˈbeisɔːlt ˈbæsɔːlt is a common Extrusive Volcanic rock. Lava is molten rock expelled by a Volcano during an eruption When first expelled from a volcanic vent it is a Liquid at Temperatures A column in Structural engineering is a vertical structural element that transmits through compression, the weight of the structure above to other structural In geology the term joint refers to a fracture in rock where there has been no lateral movement in the plane of the fracture (up down or sideways of one side 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. The Giant's Causeway (or Clochán na bhFómharach is an area of about 40000 interlocking Basalt columns the result of an ancient volcanic eruption

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. Plane tilings by Regular polygons have been widely used since antiquity

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

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. What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant

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. In Geometry, polytope is a generic term that can refer to a two-dimensional Polygon, a three-dimensional Polyhedron, or any of the various generalizations Tessellations of other spaces are often referred to as honeycombs. In Geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps Examples of tessellations of other spaces include:

• Tessellations of n-dimensional Euclidean space - for example, filling 3-dimensional Euclidean space with cubes to create a cubic honeycomb. A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. Uniform colorings There is a large number of Uniform colorings derived from different symmetries

• 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. Elliptic geometry (sometimes known as Riemannian geometry) is a Non-Euclidean geometry, in which given a line L and a point A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal In Geometry, a circumscribed sphere of a Polyhedron is a Sphere that contains the polyhedron and touches each of the polyhedron's vertices

• 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. In Mathematics, hyperbolic n -space, denoted H n, is the maximally symmetric Simply connected, n -dimensional Maurits Cornelis Escher (17 June 1898 – 27 March 1972 usually referred to as M In 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. Regular octagons A regular octagon is an octagon whose sides are all the same length and whose internal angles are all the same size

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. Branko Grünbaum (born 1929 is a Croatian born Mathematician and a professor Emeritus at the University of Washington in Seattle. C. Shephard. Tilings and Patterns

See also

References

• Grunbaum, Branko and G. In Geometry, a convex uniform honeycomb is a uniform space-filling Tessellation in three-dimensional Euclidean space with non-overlapping convex uniform In Geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps For The Rolling Stones song see " Jigsaw Puzzle " A jigsaw puzzle is a tiling Puzzle that requires the 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 Art History Mosaics of the 4th century BC are found in the Macedonian palace-city of Aegae, and they enriched the floors of Hellenistic A Penrose tiling is a nonperiodic tiling generated by an aperiodic set of Prototiles named after Roger Penrose, who investigated these sets A polyiamond (also polyamond or simply iamond) is a Polyform in which the base form is an Equilateral triangle. In Recreational mathematics, a polyomino is a Polyform with the square as its base form Quilting is a Sewing method done either by hand by Sewing machine, or by a longarm quilting system See also Biological reproduction Self-replication is any process by which a thing might make a copy of itself A tile is a manufactured piece of hard-wearing material such as Ceramic, stone, metal or even Glass. Tiling puzzles are Puzzles involving two-dimensional Packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps Plane tilings by Regular polygons have been widely used since antiquity Trianglepoint is a form of embroidery in which a series of equilateral triangles are stitched in different colors to create geometric designs three-dimensional designs 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 In Mathematics, a Voronoi diagram, named after Georgy Voronoi, also called a Voronoi Tessellation, a Voronoi decomposition, or A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the Wang tiles (or Wang dominoes) first proposed by mathematician Hao Wang in 1961 are a class of Formal systems They are modelled visually by equal-sized Branko Grünbaum (born 1929 is a Croatian born Mathematician and a professor Emeritus at the University of Washington in Seattle. C. Shephard. Tilings and Patterns. New York: W. H. Freeman & Co. , 1987. ISBN 0-7167-1193-1.
• Coxeter, H.S.M.. Harold Scott MacDonald "Donald" Coxeter CC ( February 9, 1907 – March 31, 2003) is regarded as one of the great Regular Polytopes, Section IV : Tessellations and Honeycombs. Regular Polytopes is a mathematical Geometry book written by Canadian Mathematician H Dover, 1973. ISBN 0-486-61480-8.

External links

• K-12 Tessellation Lesson

• Tilings Encyclopedia - Reference for Substitution Tilings
• 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 from the artistic, not mathematical, point of view
• 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. Southern Polytechnic State University ( SPSU or Southern Tech) is "Georgia's Technology University" located just northwest of Atlanta in Joyce, Clark University
• Some Special Radial and Spiral Tilings