.

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. In mathematics, tessellations can be generalized to higher dimensions.

A periodic tiling has a repeating pattern. 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. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space filling or honeycomb is also called a tessellation of space.

A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor or wall coverings. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher often made use of tessellations for artistic effect. Tessellations are sometimes employed for decorative effect in quilting. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs.

History
A temple mosaic from the ancient Sumerian city of Uruk IV (3400–3100 BC), showing a tessellation pattern in coloured tiles

Tessellations were used by the Sumerians (about 4000 BC) in building wall decorations formed by patterns of clay tiles.[1]

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.[2] 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.[3][4] Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov (1964),[5] and Heinrich Heesch and Otto Kienzle (1963).[6]
Etymology

In Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics.[7] The word "tessella" means "small square" (from "tessera", square, which in its turn is from the Greek word "τέσσερα" for "four"). It corresponds with the everyday term tiling which refers to applications of tessellations, often made of glazed clay.

Overview
A rhombitrihexagonal tiling: tiled floor of a church in Seville, Spain, using square, triangle and hexagon prototiles

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. These rules can be varied. A common one is that all corners should meet and that no corner of one tile can lie along the edge of another.[8] The tessellations created by bonded brickwork do not obey this rule. Among those which do, a regular tessellation has both identical[a] regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile.[9] There are only three shapes that can form such regular tessellations: the equilateral triangle, square, and regular hexagon. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps.[10]

Many other types of tessellation are possible under different constraints. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner.[11] Irregular tessellations can also be made from other shapes such as pentagons, polyominoes and in fact almost any kind of geometric shape. The artist M. C. Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects. If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, and these can be used to form physical surfaces such as church floors.[12]
Elaborate and colourful tessellations of glazed tiles at the Alhambra in Spain

More formally, a tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles, such that the tiles intersect only on their boundaries. These tiles may be polygons or any other shapes.[b] Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are congruent to the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane. A general method for identifying shapes which will tile the plane periodically without reflections is known as the Conway criterion.[13] However, mathematicians have found no general rule for determining if a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations.[14] For example, the types of convex pentagon that can tile the plane remains an unsolved problem.[15]

Mathematically, tessellations can be extended to spaces other than the Euclidean plane.[16] The Swiss geometer Ludwig Schläfli pioneered this by defining polyschemes, which mathematicians nowadays call polytopes; these are the analogues to polygons and polyhedra in spaces with more dimensions. He further defined the Schläfli symbol notation to make it easy to describe polytopes. For example, the Schläfli symbol for an equilateral triangle is {3}, while that for a square is {4}.[17] The Schläfli notation makes it possible to describe tilings compactly. For example, a tiling of regular hexagons has three six-sided polygons at each vertex, so its Schläfli symbol is {6,3}.[18]

Other methods also exist for describing polygonal tilings. When the tessellation is made of regular polygons, the most common notation is the vertex configuration, which is simply a list of the number of sides of the polygons around a vertex. The square tiling has a vertex configuration of 4.4.4.4, or 44. The tiling of regular hexagons is noted 6.6.6, or 63.[14]:59
In mathematics

Kinds of tessellations
Further information: tiling by regular polygons
The 3.4.6.4 semi-regular tessellation is made with three prototiles: a triangle, a square and a hexagon. Every vertex has a triangle, square, hexagon, square around it, in that order.

Mathematicians use some technical terms when discussing tilings. An edge is the intersection between two bordering tiles; it is often a straight line. A vertex is the point of intersection of three or more bordering tiles. Using these terms, an isogonal or vertex-transitive tiling is a tiling where every vertex point is identical; that is, the arrangement of polygons about each vertex is the same. For example, a regular tessellation of the plane with squares has a meeting of four squares at every vertex.[14]

The sides of the polygons are not necessarily identical to the edges of the tiles. An edge-to-edge tessellation is any polygonal tessellation where adjacent tiles only share one full side, i.e., no tile shares a partial side or more than one side with any other tile. In an edge-to-edge tessellation, the sides of the polygons and the edges of the tiles are the same. The familiar "brick wall" tiling is not edge-to-edge because the long side of each rectangular brick is shared with two bordering bricks.[14]

A normal tiling is a tessellation for which (1) every tile is topologically equivalent to a disk, (2) the intersection of any two tiles is a single connected set or the empty set, and (3) all tiles are uniformly bounded. A uniformly bounded tile is one in which a finite circle can be circumscribed around the tile and a finite circle can be inscribed within the tile; the condition disallows tiles that are pathologically long or thin.[19]:172

A monohedral tiling is a tessellation in which all tiles are congruent; it has only one prototile. A particularly interesting type of monohedral tessellation is the spiral monohedral tiling. The first spiral monohedral tiling was discovered by Heinz Voderberg in 1936, with the Voderberg tiling having a unit tile that is a nonconvex enneagon.[1] The Hirschhorn tiling, published by Michael D. Hirschhorn and D. C. Hunt in 1985, has a unit tile that is an irregular pentagon.[20][21]

An isohedral tiling is a special variation of a monohedral tiling in which all tiles belong to the same transitivity class, that is, all tiles are transforms of the same prototile under the symmetry group of the tiling.[19]:175 If a prototile admits a tiling, but no such tiling is isohedral, then the prototile is called anisohedral and forms anisohedral tilings.

A regular tessellation is a highly symmetric, edge-to-edge tiling made up of regular polygons, all of the same shape. There are only three regular tessellations: those made up of equilateral triangles, squares, or regular hexagons. All three of these tilings are isogonal and monohedral.[10]

A semi-regular (or Archimedean) tessellation uses more than one type of regular polygon in an isogonal arrangement. There are eight semi-regular tilings (or nine if the mirror-image pair of tilings counts as two).[22] These can be described by their vertex configuration; for example, a semi-regular tiling using squares and regular octagons has the vertex configuration 4.82 (each vertex has one square and two octagons).[23]

Penrose tilings, which use two different quadrilaterals, are the best known example of tiles that forcibly create non-periodic patterns. They belong to a general class of aperiodic tilings, which use tiles that cannot tessellate periodically, though they have surprising self-replicating properties using the recursive process of substitution tiling.[24]

Voronoi or Dirichlet tilings are tessellations where each tile is defined as the set of points closest to one of the points in a discrete set of defining points. (Think of geographical regions where each region is defined as all the points closest to a given city or post office.)[25][26] The Voronoi cell for each defining point is a convex polygon. The Delaunay triangulation is a tessellation that is the dual graph of a Voronoi tessellation. Delaunay triangulations are useful in numerical simulation, in part because among all possible triangulations of the defining points, Delaunay triangulations maximize the minimum of the angles formed by the edges.[27]
This tessellated, monohedral street pavement uses curved shapes instead of polygons. It belongs to wallpaper group p3m1.
Wallpaper groups
Main article: Wallpaper group

Tilings with translational symmetry in two independent directions can be categorized by wallpaper groups, of which 17 exist.[28] It has been claimed that all seventeen of these groups are represented in the Alhambra palace in Granada, Spain. Though this is disputed,[29][30] the variety and sophistication of the Alhambra tilings have surprised modern researchers.[31] Of the three regular tilings two are in the p6m wallpaper group and one is in p4m. Tilings in 2D with translational symmetry in just one direction can be categorized by the seven frieze groups describing the possible frieze patterns.[32]
Tessellations and colour
Further information: four colour theorem
If the colours of this tiling are to form a repeat pattern, at least seven colours are required. If the colouring is allowed to be aperiodic, then at least four colours are needed. This tiling can be used on the surface of a torus.

Sometimes the colour of a tile is understood as part of the tiling, at other times arbitrary colours may be applied later. When discussing a tiling that is displayed in colours, to avoid ambiguity one needs to specify whether the colours are part of the tiling or just part of its illustration. This affects whether tiles with the same shape but different colours are considered identical, which in turn affects questions of symmetry. The four colour theorem states that for every tessellation of a normal Euclidean plane, with a set of four available colours, each tile can be coloured in one colour such that no tiles of equal colour meet at a curve of positive length. The colouring guaranteed by the four-colour theorem will not in general respect the symmetries of the tessellation. To produce a colouring which does, it is necessary to treat the colours as part of the tessellation. here, as many as seven colours may be needed, as in the picture at right.[33]
Tessellations with triangles, quadrilaterals and hexagons

Any triangle or quadrilateral (even non-convex) can be used as a prototile to form a monohedral tessellation, often in more than one way. Most hexagons with at least two parallel sides will tessellate. Copies of an arbitrary quadrilateral can form a tessellation with 2-fold rotational centres at the midpoints of all sides, and translational symmetry whose basis vectors are the diagonal 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 centre. 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.[34]
Tessellations in higher dimensions
Main article: Honeycomb (geometry)
Tessellating three-dimensional space: the rhombic dodecahedron is one of the solids that can be stacked to fill space exactly.
Illustration of a Schmitt-Conway biprism, also called a Schmitt–Conway–Danzer tile.

Tessellation can be extended to three dimensions. Certain polyhedra can be stacked in a regular crystal pattern to fill (or tile) three-dimensional space, including the cube (the only regular polyhedron to do so); the rhombic dodecahedron; and the truncated octahedron.[35] Some crystals including Andradite (a kind of Garnet) and Fluorite can take the form of rhombic dodecahedra.[36][37]

The Schmitt-Conway biprism is a convex polyhedron which has the property of tiling space only aperiodically. John Horton Conway discovered it in 1993.[35]

Tessellations in three or more dimensions are called honeycombs. In three dimensions there is just one regular honeycomb, which has eight cubes at each polyhedron vertex. Similarly, in three dimensions there is just one quasiregular[c] honeycomb, which has eight tetrahedra and six octahedra at each polyhedron vertex. However there are many possible semiregular honeycombs in three dimensions.[35]
In art
A quilt showing a regular tessellation pattern.
Roman mosaic floor panel of stone, tile and glass, from a villa near Antioch in Roman Syria. 2nd century A.D.

In architecture, tessellations have been used to create decorative motifs since ancient times. Mosaic tilings were used by the Romans, often with geometric patterns.[38] Later civilisations also used larger tiles, either plain or individually decorated. Some of the most decorative were the Moorish wall tilings of buildings such as the Alhambra[39] and the Córdoba, Andalusia mosque of La Mezquita.[40]

Tessellations frequently appeared in the graphic art of M. C. Escher; he was inspired by the Moorish use of symmetry in places such as the Alhambra when he visited Spain in 1936.[41]

Tessellated designs also often appear on textiles, either woven or stitched in or printed. In the context of quilting, tessellation refers to regular[42] and semiregular[43] of tessellation of either patch shapes or the overall design. Tessellation patterns have been used to design interlocking motifs of patch shapes.[44][45] The repeating motif is sometimes called a block design.[42]
In nature
Tessellate pattern in a Colchicum flower
Main article: Patterns in nature

The honeycomb provides a well-known example of tessellation in nature with its hexagonal cells.[46]

In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including the Fritillary[47] and some species of Colchicum are characteristically tessellate.[48]

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 Northern Ireland.[49]

Tessellated pavement, a characteristic example of which is found at Eaglehawk Neck on the Tasman Peninsula of Tasmania, is a rare sedimentary rock formation where the rock has fractured into rectangular blocks.[50]

Related topics

Jigsaw puzzle
Mathematics and fiber arts
Nesting (process) — application for minimising raw material waste in a cutting process
Polyiamond and Polyomino — figures of regular triangles and squares, often in tiling puzzles
Quilt block designs and quilt blocks
"Tessellate" - song by the British alternative indie pop quartet Alt-J (∆).
Tiling puzzle
Trianglepoint – needlepoint with polyiamonds (equilateral triangles)

Footnotes

The mathematical term for identical shapes is "congruent" - in mathematics, "identical" means they are the same tile.
The tiles are usually required to be topologically equivalent to a closed disk, which means bizarre shapes with holes, dangling line segments or infinite areas are excluded.[14]

In this context, quasiregular means that the cells are regular (solids), and the vertex figures are semiregular.

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External links

Wolfram MathWorld: Tessellation (good bibliography, drawings of regular, semiregular and demiregular tessellations)
Tilings Encyclopedia (extensive information on substitution tilings, including drawings, people, and references)
Tessellations.org (how-to guides, Escher tessellation gallery, galleries of tessellations by other artists, lesson plans, history)

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

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