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A tessellation or tiling is the covering of a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Some special kinds include ''
regular tilings This article lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli sy ...
'' with
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
al tiles all of the same shape, and ''
semiregular tiling Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his '' Harmonices Mundi'' (Latin: ''The Harmony of the World'', 1619). Notation of Euc ...
s'' 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 wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformati ...
. A tiling that lacks a repeating pattern is called "non-periodic". An ''
aperiodic tiling An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- peri ...
'' uses a small set of tile shapes that cannot form a repeating pattern. A ''
tessellation of space In geometry, a honeycomb is a ''space filling'' or '' close packing'' of polyhedral or higher-dimensional ''cells'', so that there are no gaps. It is an example of the more general mathematical ''tiling'' or ''tessellation'' in any number of dim ...
'', also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. A real physical tessellation is a tiling made of materials such as
cement A cement is a binder, a chemical substance used for construction that sets, hardens, and adheres to other materials to bind them together. Cement is seldom used on its own, but rather to bind sand and gravel ( aggregate) together. Cement mi ...
ed
ceramic A ceramic is any of the various hard, brittle, heat-resistant and corrosion-resistant materials made by shaping and then firing an inorganic, nonmetallic material, such as clay, at a high temperature. Common examples are earthenware, porcelain ...
squares or hexagons. Such tilings may be decorative
patterns A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated li ...
, or may have functions such as providing durable and water-resistant pavement, floor or wall coverings. Historically, tessellations were used in
Ancient Rome In modern historiography, ancient Rome refers to Roman civilisation from the founding of the city of Rome in the 8th century BC to the collapse of the Western Roman Empire in the 5th century AD. It encompasses the Roman Kingdom (753–509 BC ...
and in
Islamic art Islamic art is a part of Islamic culture and encompasses the visual arts produced since the 7th century CE by people who lived within territories inhabited or ruled by Muslim populations. Referring to characteristic traditions across a wide ra ...
such as in the Moroccan architecture and decorative geometric tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
and in
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
, for artistic effect. Tessellations are sometimes employed for decorative effect in
quilting Quilting is the term given to the process of joining a minimum of three layers of fabric together either through stitching manually using a needle and thread, or mechanically with a sewing machine or specialised longarm quilting system. ...
. Tessellations form a class of
patterns in nature Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, ...
, for example in the arrays of hexagonal cells found in
honeycomb A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about of honey ...
s.


History

Tessellations were used by the
Sumerians Sumer () is the earliest known civilization in the historical region of southern Mesopotamia (south-central Iraq), emerging during the Chalcolithic and early Bronze Ages between the sixth and fifth millennium BC. It is one of the cradles of c ...
(about 4000 BC) in building wall decorations formed by patterns of clay tiles. Decorative
mosaic A mosaic is a pattern or image made of small regular or irregular pieces of colored stone, glass or ceramic, held in place by plaster/mortar, and covering a surface. Mosaics are often used as floor and wall decoration, and were particularly pop ...
tilings made of small squared blocks called
tessera A tessera (plural: tesserae, diminutive ''tessella'') is an individual tile, usually formed in the shape of a square, used in creating a mosaic. It is also known as an abaciscus or abaculus. Historical tesserae The oldest known tesserae ...
e were widely employed in
classical antiquity Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history between the 8th century BC and the 5th century AD centred on the Mediterranean Sea, comprising the interlocking civilizations of ...
, sometimes displaying geometric patterns. In 1619 Johannes Kepler made an early documented study of tessellations. He wrote about regular and semiregular tessellations in his ''
Harmonices Mundi ''Harmonice Mundi (Harmonices mundi libri V)''The full title is ''Ioannis Keppleri Harmonices mundi libri V'' (''The Five Books of Johannes Kepler's The Harmony of the World''). (Latin: ''The Harmony of the World'', 1619) is a book by Johannes ...
''; he was possibly the first to explore and to explain the hexagonal structures of honeycomb and snowflakes. 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. Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include
Alexei Vasilievich Shubnikov Alexei Vasilievich Shubnikov (russian: Алексей Васильевич Шубников; 29March 1887 – 27April 1970) was a Soviet crystallographer and mathematician. Shubnikov was the founding director of the Institute of Crystallograph ...
and Nikolai Belov (1964), and Heinrich Heesch and Otto Kienzle (1963).


Etymology

In Latin, ''tessella'' is a small cubical piece of
clay Clay is a type of fine-grained natural soil material containing clay minerals (hydrous aluminium phyllosilicates, e.g. kaolin, Al2 Si2 O5( OH)4). Clays develop plasticity when wet, due to a molecular film of water surrounding the clay par ...
,
stone In geology, rock (or stone) is any naturally occurring solid mass or aggregate of minerals or mineraloid matter. It is categorized by the minerals included, its Chemical compound, chemical composition, and the way in which it is formed. Rocks ...
or
glass Glass is a non-crystalline, often transparent, amorphous solid that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most often formed by rapid cooling ( quenching ...
used to make mosaics. The word "tessella" means "small square" (from ''tessera'', square, which in turn is from the Greek word τέσσερα for ''four''). It corresponds to the everyday term ''tiling'', which refers to applications of tessellations, often made of glazed clay.


Overview

Tessellation in two dimensions, also called planar tiling, is a topic in geometry 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. Common ones are that there must be no gaps between tiles, and that no corner of one tile can lie along the edge of another. The tessellations created by bonded brickwork do not obey this rule. Among those that do, a regular tessellation has both identical regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile. There are only three shapes that can form such regular tessellations: the equilateral
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
,
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
and the regular
hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...
. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps. 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. 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 decorate physical surfaces such as church floors. More formally, a tessellation or tiling is a cover of the Euclidean plane by a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
number of closed sets, called ''tiles'', such that the tiles intersect only on their boundaries. These tiles may be polygons or any other shapes. Many tessellations are formed from a finite number of
prototile In the mathematical theory of tessellations, a prototile is one of the shapes of a tile in a tessellation. Definition A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint in ...
s in which all tiles in the tessellation are
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
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''. The Conway criterion is a sufficient but not necessary set of rules for deciding if a given shape tiles the plane periodically without reflections: some tiles fail the criterion but still tile the plane. No general rule has been found for determining if a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations. Mathematically, tessellations can be extended to spaces other than the Euclidean plane. The Swiss
geometer A geometer is a mathematician whose area of study is geometry. Some notable geometers and their main fields of work, chronologically listed, are: 1000 BCE to 1 BCE * Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra * ...
Ludwig Schläfli Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional space ...
pioneered this by defining ''polyschemes'', which mathematicians nowadays call polytopes. These are the analogues to polygons and
polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...
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 , while that for a square is . 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 . Other methods also exist for describing polygonal tilings. When the tessellation is made of regular polygons, the most common notation is the
vertex configuration In geometry, a vertex configurationCrystallography ...
, 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.


In mathematics


Introduction to tessellations

Mathematicians use some technical terms when discussing tilings. An ''
edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed ...
'' 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
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
s about each vertex is the same. The
fundamental region Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
is a shape such as a rectangle that is repeated to form the tessellation. For example, a regular tessellation of the plane with squares has a meeting of four squares at every vertex. The sides of the polygons are not necessarily identical to the edges of the tiles. An edge-to-edge tiling 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 tiling, 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. A ''normal tiling'' is a tessellation for which every tile is topologically equivalent to a disk, the intersection of any two tiles is a connected set or the empty set, and all tiles are
uniformly bounded In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the family. ...
. This means that a single circumscribing radius and a single inscribing radius can be used for all the tiles in the whole tiling; the condition disallows tiles that are pathologically long or thin. A is a tessellation in which all tiles are
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
; 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; the Voderberg tiling has a unit tile that is a nonconvex
enneagon In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name ''nonagon'' is a prefix hybrid formation, from Latin (''nonus'', "ninth" + ''gonon''), used equivalently, attested already in the 16th century in French ''nonogo ...
. The Hirschhorn tiling, published by Michael D. Hirschhorn and D. C. Hunt in 1985, is a pentagon tiling using irregular pentagons: regular pentagons cannot tile the Euclidean plane as the
internal angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
of a regular pentagon, , is not a divisor of 2. 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. 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 Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, edge-to-edge tiling made up of
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s, all of the same shape. There are only three regular tessellations: those made up of
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
s,
squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
, or regular
hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...
s. All three of these tilings are isogonal and monohedral. 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). These can be described by their
vertex configuration In geometry, a vertex configurationCrystallography ...
; 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). Many non-edge-to-edge tilings of the Euclidean plane are possible, including the family of Pythagorean tilings, tessellations that use two (parameterised) sizes of square, each square touching four squares of the other size. An edge tessellation is one in which each tile can be reflected over an edge to take up the position of a neighbouring tile, such as in an array of equilateral or isosceles triangles.


Wallpaper groups

Tilings with
translational symmetry In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equati ...
in two independent directions can be categorized by
wallpaper group A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformati ...
s, of which 17 exist. It has been claimed that all seventeen of these groups are represented in the Alhambra palace in Granada,
Spain , image_flag = Bandera de España.svg , image_coat = Escudo de España (mazonado).svg , national_motto = ''Plus ultra'' (Latin)(English: "Further Beyond") , national_anthem = (English: "Royal March") , i ...
. Though this is disputed, the variety and sophistication of the Alhambra tilings have surprised modern researchers. 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. Orbifold notation can be used to describe wallpaper groups of the Euclidean plane.


Aperiodic tilings

Penrose tiling A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of the plane by non-overlapping polygons or other shapes, and ''aperiodic'' means that shifting any tiling with these shapes by any finite distance, without ...
s, which use two different quadrilateral prototiles, are the best known example of tiles that forcibly create non-periodic patterns. They belong to a general class of
aperiodic tiling An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- peri ...
s, which use tiles that cannot tessellate periodically. The recursive process of
substitution tiling In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The ...
is a method of generating aperiodic tilings. One class that can be generated in this way is the
rep-tile In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by ...
s; these tilings have surprising self-replicating properties.
Pinwheel tiling In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many or ...
s are non-periodic, using a rep-tile construction; the tiles appear in infinitely many orientations. It might be thought that a non-periodic pattern would be entirely without symmetry, but this is not so. Aperiodic tilings, while lacking in
translational symmetry In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equati ...
, do have symmetries of other types, by infinite repetition of any bounded patch of the tiling and in certain finite groups of rotations or reflections of those patches. A substitution rule, such as can be used to generate some Penrose patterns using assemblies of tiles called rhombs, illustrates scaling symmetry. A
Fibonacci word A Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter alphabet). The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition. It is a para ...
can be used to build an aperiodic tiling, and to study
quasicrystal A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical ...
s, which are structures with aperiodic order.
Wang tile Wang tiles (or Wang dominoes), first proposed by mathematician, logician, and philosopher Hao Wang in 1961, are a class of formal systems. They are modelled visually by square tiles with a color on each side. A set of such tiles is selected, an ...
s are squares coloured on each edge, and placed so that abutting edges of adjacent tiles have the same colour; hence they are sometimes called Wang
dominoes Dominoes is a family of tile-based games played with gaming pieces, commonly known as dominoes. Each domino is a rectangular tile, usually with a line dividing its face into two square ''ends''. Each end is marked with a number of spots (also c ...
. A suitable set of Wang dominoes can tile the plane, but only aperiodically. This is known because any
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
can be represented as a set of Wang dominoes that tile the plane if and only if the Turing machine does not halt. Since the
halting problem In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a ...
is undecidable, the problem of deciding whether a Wang domino set can tile the plane is also undecidable. Truchet tiles are square tiles decorated with patterns so they do not have
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
; in 1704, Sébastien Truchet used a square tile split into two triangles of contrasting colours. These can tile the plane either periodically or randomly.


Tessellations and colour

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 In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions sha ...
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 does not generally 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.


Tessellations with polygons

Next to the various tilings by regular polygons, tilings by other polygons have also been studied. Any triangle or
quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
(even non-convex) can be used as a prototile to form a monohedral tessellation, often in more than one way. Copies of an arbitrary
quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
can form a tessellation with translational symmetry and 2-fold rotational symmetry with centres at the midpoints of all sides. 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. If only one shape of tile is allowed, tilings exists with convex ''N''-gons for ''N'' equal to 3, 4, 5 and 6. For , see
Pentagonal tiling In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular tiling, regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a Pentagon#Regular ...
, for , see Hexagonal tiling, for , see Heptagonal tiling and for , see octagonal tiling. For results on tiling the plane with
polyomino A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in pop ...
es, see Polyomino § Uses of polyominoes.


Voronoi tilings

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.) The ''Voronoi cell'' for each defining point is a convex polygon. The
Delaunay triangulation In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle o ...
is a tessellation that is the
dual graph In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of . The dual graph has an edge for each pair of faces in that are separated from each other by an edge, and a self-lo ...
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. Voronoi tilings with randomly placed points can be used to construct random tilings of the plane.


Tessellations in higher dimensions

Tessellation can be extended to three dimensions. Certain
polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...
can be stacked in a regular crystal pattern to fill (or tile) three-dimensional space, including the cube (the only Platonic polyhedron to do so), the
rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahed ...
, the
truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 ...
, and triangular, quadrilateral, and hexagonal prisms, among others. Any polyhedron that fits this criterion is known as a
plesiohedron In geometry, a plesiohedron is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set. Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps. Th ...
, and may possess between 4 and 38 faces. Naturally occurring rhombic dodecahedra are found as
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
s of andradite (a kind of
garnet Garnets () are a group of silicate minerals that have been used since the Bronze Age as gemstones and abrasives. All species of garnets possess similar physical properties and crystal forms, but differ in chemical composition. The different s ...
) and
fluorite Fluorite (also called fluorspar) is the mineral form of calcium fluoride, CaF2. It belongs to the halide minerals. It crystallizes in isometric cubic habit, although octahedral and more complex isometric forms are not uncommon. The Mohs sca ...
. 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 honeycomb, which has eight
tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
and six
octahedra In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet a ...
at each polyhedron vertex. However, there are many possible semiregular honeycombs in three dimensions. Uniform honeycombs can be constructed using the
Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...
. The Schmitt-Conway biprism is a convex polyhedron with the property of tiling space only aperiodically. A Schwarz triangle is a spherical triangle that can be used to tile a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
.


Tessellations in non-Euclidean geometries

It is possible to tessellate in non-Euclidean geometries such as
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
. A uniform tiling in the hyperbolic plane (which may be regular, quasiregular or semiregular) is an edge-to-edge filling of the hyperbolic plane, with
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s as
faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
; these are vertex-transitive ( transitive on its vertices), and isogonal (there is an
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
mapping any vertex onto any other). A uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
families of compact
convex uniform honeycomb In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. Twenty-eight such honeycombs are known: * the familiar cubic honeycomb and 7 tr ...
s, generated as
Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...
s, and represented by permutations of
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
of the
Coxeter diagram Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
s for each family.


In art

In architecture, tessellations have been used to create decorative motifs since ancient times.
Mosaic A mosaic is a pattern or image made of small regular or irregular pieces of colored stone, glass or ceramic, held in place by plaster/mortar, and covering a surface. Mosaics are often used as floor and wall decoration, and were particularly pop ...
tilings often had geometric patterns. Later civilisations also used larger tiles, either plain or individually decorated. Some of the most decorative were the Moorish wall tilings of
Islamic architecture Islamic architecture comprises the architectural styles of buildings associated with Islam. It encompasses both secular and religious styles from the early history of Islam to the present day. The Islamic world encompasses a wide geographic ...
, using
Girih ''Girih'' ( fa, گره, "knot", also written ''gereh'') are decorative Islamic geometric patterns used in architecture and handicraft objects, consisting of angled lines that form an interlaced strapwork pattern. ''Girih'' decoration is beli ...
and
Zellige ''Zellij'' ( ar, الزليج, translit=zillīj; also spelled zillij or zellige) is a style of mosaic tilework made from individually hand-chiseled tile pieces. The pieces were typically of different colours and fitted together to form various pa ...
tiles in buildings such as the Alhambra and La Mezquita. 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 , image_flag = Bandera de España.svg , image_coat = Escudo de España (mazonado).svg , national_motto = ''Plus ultra'' (Latin)(English: "Further Beyond") , national_anthem = (English: "Royal March") , i ...
in 1936. Escher made four " Circle Limit" drawings of tilings that use hyperbolic geometry. For his
woodcut Woodcut is a relief printing technique in printmaking. An artist carves an image into the surface of a block of wood—typically with gouges—leaving the printing parts level with the surface while removing the non-printing parts. Areas tha ...
"Circle Limit IV" (1960), Escher prepared a pencil and ink study showing the required geometry. Escher explained that "No single component of all the series, which from infinitely far away rise like rockets perpendicularly from the limit and are at last lost in it, ever reaches the boundary line." Tessellated designs often appear on textiles, whether woven, stitched in or printed. Tessellation patterns have been used to design interlocking motifs of patch shapes in
quilt A quilt is a multi-layered textile, traditionally composed of two or more layers of fabric or fiber. Commonly three layers are used with a filler material. These layers traditionally include a woven cloth top, a layer of padding, batting or w ...
s. Tessellations are also a main genre in
origami ) is the Japanese art of paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of paper into a f ...
(paper folding), where pleats are used to connect molecules such as twist folds together in a repeating fashion.


In manufacturing

Tessellation is used in
manufacturing industry Manufacturing is the creation or production of goods with the help of equipment, labor, machines, tools, and chemical or biological processing or formulation. It is the essence of secondary sector of the economy. The term may refer to a ran ...
to reduce the wastage of material (yield losses) such as
sheet metal Sheet metal is metal formed into thin, flat pieces, usually by an industrial process. Sheet metal is one of the fundamental forms used in metalworking, and it can be cut and bent into a variety of shapes. Thicknesses can vary significantly; ex ...
when cutting out shapes for objects like car doors or drinks cans. Tessellation is apparent in the mudcrack-like cracking of
thin film A thin film is a layer of material ranging from fractions of a nanometer ( monolayer) to several micrometers in thickness. The controlled synthesis of materials as thin films (a process referred to as deposition) is a fundamental step in many ...
s – with a degree of self-organisation being observed using micro and
nanotechnologies Nanotechnology, also shortened to nanotech, is the use of matter on an atomic, molecular, and supramolecular scale for industrial purposes. The earliest, widespread description of nanotechnology referred to the particular technological goal o ...
.


In nature

The
honeycomb A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about of honey ...
is a well-known example of tessellation in nature with its hexagonal cells. In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including the
fritillary ''Fritillaria'' (fritillaries) is a genus of spring flowering herbaceous bulbous perennial plants in the lily family (Liliaceae). The type species, ''Fritillaria meleagris'', was first described in Europe in 1571, while other species from the ...
and some species of ''Colchicum'' are characteristically tessellate. Many
patterns in nature Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, ...
are formed by cracks in sheets of materials. These patterns can be described by Gilbert tessellations, also known as random crack networks. The Gilbert tessellation is a mathematical model for the formation of mudcracks, needle-like
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
s, and similar structures. The model, named after
Edgar Gilbert Edgar Nelson Gilbert (July 25, 1923 – June 15, 2013) was an American mathematician and coding theorist, a longtime researcher at Bell Laboratories whose accomplishments include the Gilbert–Varshamov bound in coding theory, the Gilbert–Ell ...
, allows cracks to form starting from randomly scattered over the plane; each crack propagates in two opposite directions along a line through the initiation point, its slope chosen at random, creating a tessellation of irregular convex polygons.
Basalt Basalt (; ) is an aphanitic (fine-grained) extrusive igneous rock formed from the rapid cooling of low-viscosity lava rich in magnesium and iron (mafic lava) exposed at or very near the surface of a rocky planet or moon. More than 90 ...
ic lava flows often display
columnar jointing Columnar jointing is a geological structure where sets of intersecting closely spaced fractures, referred to as joints, result in the formation of a regular array of polygonal prisms, or columns. Columnar jointing occurs in many types of igneo ...
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 The Giant's Causeway is an area of about 40,000 interlocking basalt columns, the result of an ancient volcanic fissure eruption. It is located in County Antrim on the north coast of Northern Ireland, about three miles (5 km) northeast of ...
in Northern Ireland.
Tessellated pavement In geology and geomorphology, a tessellated pavement is a relatively flat rock surface that is subdivided into more or less regular rectangles, blocks approaching rectangles, or irregular or regular polygons by fractures, frequently systematic jo ...
, a characteristic example of which is found at Eaglehawk Neck on the
Tasman Peninsula The Tasman Peninsula, officially Turrakana / Tasman Peninsula, is a peninsula located in south-east Tasmania, Australia, approximately by the Arthur Highway, south-east of Hobart. The Tasman Peninsula lies south and west of Forestier Peninsula ...
of
Tasmania ) , nickname = , image_map = Tasmania in Australia.svg , map_caption = Location of Tasmania in AustraliaCoordinates: , subdivision_type = Country , subdi ...
, is a rare sedimentary rock formation where the rock has fractured into rectangular blocks. Other natural patterns occur in
foam Foams are materials formed by trapping pockets of gas in a liquid or solid. A bath sponge and the head on a glass of beer are examples of foams. In most foams, the volume of gas is large, with thin films of liquid or solid separating the ...
s; these are packed according to
Plateau's laws Plateau's laws describe the structure of soap films. These laws were formulated in the 19th century by the Belgian physicist Joseph Plateau from his experimental observations. Many patterns in nature are based on foams obeying these laws. Laws ...
, which require
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
s. Such foams present a problem in how to pack cells as tightly as possible: in 1887,
Lord Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did important ...
proposed a packing using only one solid, the bitruncated cubic honeycomb with very slightly curved faces. In 1993, Denis Weaire and Robert Phelan proposed the
Weaire–Phelan structure In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution ...
, which uses less surface area to separate cells of equal volume than Kelvin's foam.


In puzzles and recreational mathematics

Tessellations have given rise to many types of tiling puzzle, from traditional
jigsaw puzzle A jigsaw puzzle is a tiling puzzle that requires the assembly of often irregularly shaped interlocking and mosaiced pieces, each of which typically has a portion of a picture. When assembled, the puzzle pieces produce a complete picture. In t ...
s (with irregular pieces of wood or cardboard) and the
tangram The tangram () is a dissection puzzle consisting of seven flat polygons, called ''tans'', which are put together to form shapes. The objective is to replicate a pattern (given only an outline) generally found in a puzzle book using all seven pi ...
to more modern puzzles which often have a mathematical basis. For example,
polyiamond A polyiamond (also polyamond or simply iamond, or sometimes triangular polyomino) is a polyform whose base form is an equilateral triangle. The word ''polyiamond'' is a back-formation from ''diamond'', because this word is often used to describe ...
s and
polyomino A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in pop ...
es are figures of regular triangles and squares, often used in tiling puzzles. Authors such as
Henry Dudeney Henry Ernest Dudeney (10 April 1857 – 23 April 1930) was an English author and mathematician who specialised in logic puzzles and mathematical games. He is known as one of the country's foremost creators of mathematical puzzles. Early life ...
and
Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lew ...
have made many uses of tessellation in
recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
. For example, Dudeney invented the
hinged dissection In geometry, a hinged dissection, also known as a swing-hinged dissection or Dudeney dissection, is a kind of geometric dissection in which all of the pieces are connected into a chain by "hinged" points, such that the rearrangement from one f ...
, while Gardner wrote about the
rep-tile In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by ...
, a shape that can be dissected into smaller copies of the same shape. Inspired by Gardner's articles in
Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it ...
, the amateur mathematician Marjorie Rice found four new tessellations with pentagons.
Squaring the square Squaring the square is the problem of tiling an integral square using only other integral squares. (An integral square is a square whose sides have integer length.) The name was coined in a humorous analogy with squaring the circle. Squaring the sq ...
is the problem of tiling an integral square (one whose sides have integer length) using only other integral squares. An extension is squaring the plane, tiling it by squares whose sizes are all natural numbers without repetitions; James and Frederick Henle proved that this was possible.


Examples

File:Tile 3,6.svg,
Triangular tiling In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilate ...
, one of the three
regular tiling Euclidean Plane (mathematics), plane Tessellation, tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler, Kepler in his ''Harmonices Mundi'' (Latin langua ...
s of the plane. File:Academ Periodic tiling where eighteen triangles encircle each hexagon.svg,
Snub hexagonal tiling In geometry, the snub hexagonal tiling (or ''snub trihexagonal tiling'') is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol ''sr''. The snub tetrahexagonal tiling is a ...
, a
semiregular tiling Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his '' Harmonices Mundi'' (Latin: ''The Harmony of the World'', 1619). Notation of Euc ...
of the plane File:Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg, Floret pentagonal tiling, dual to a semiregular tiling and one of 15 monohedral pentagon tilings. File:A variation on a tiling in the Alhambra of Spain.svg, All tiling elements are  identical pseudo‑triangles by disregarding their colors and ornaments. File:Voderberg.png, The Voderberg tiling, a spiral, monohedral tiling made of
enneagon In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name ''nonagon'' is a prefix hybrid formation, from Latin (''nonus'', "ninth" + ''gonon''), used equivalently, attested already in the 16th century in French ''nonogo ...
s. File:Uniform tiling 433-t0.png, Alternated octagonal or tritetragonal tiling is a uniform tiling of the hyperbolic plane. File:Capital I tiling-4color.svg,
Topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
square tiling, isohedrally distorted into I shapes.


See also

*
Discrete global grid A discrete global grid (DGG) is a mosaic that covers the entire Earth's surface. Mathematically it is a space partitioning: it consists of a set of non-empty regions that form a partition of the Earth's surface. In a usual grid-modeling strate ...
* Space partitioning *
Honeycomb (geometry) In geometry, a honeycomb is a ''space filling'' or '' close packing'' of polyhedral or higher-dimensional ''cells'', so that there are no gaps. It is an example of the more general mathematical ''tiling'' or ''tessellation'' in any number of dim ...


Footnotes


References


Sources

* * * * * *


External links


Tegula
(open-source software for exploring two-dimensional tilings of the plane, sphere and hyperbolic plane; includes databases containing millions of tilings)

(good bibliography, drawings of regular, semiregular and demiregular tessellations) *Dirk Frettlöh and Edmund Harriss.
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) * (list of web resources including articles and galleries) {{Good article Symmetry Mosaic