View source for Tessellation

{{Short description|Tiling of a plane in mathematics}}
{{multiple image
| total_width       = 220
| direction         = vertical
| image1            = Ceramic Tile Tessellations in Marrakech.jpg
| image2            = Leeuwarden - Tegeltableau Escher.jpg
| image3            = Birds on identical tiles edge-to-edge like puzzle pieces Non-periodic tiling.svg
| caption1          = [[Engineering:Zellige|Zellige]] [[Chemistry:Terracotta|terracotta]] tiles in Marrakech, forming edge‑to‑edge, regular and other tessellations
| caption2          = A wall sculpture in Leeuwarden celebrating the artistic tessellations of M. C. Escher
| caption3          = An example of non‑periodicity due to another orientation of one tile out of an infinite number of identical tiles
}}

A '''tessellation''' or '''tiling''' is the covering of a [[Surface|surface]], often a [[Plane (mathematics)|plane]], using one or more [[Geometric shape|geometric shape]]s, called ''tiles'', with no overlaps and no gaps. In [[Mathematics|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'' with [[Regular polygon|regular polygon]]al tiles all of the same shape, and ''semiregular 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|aperiodic tiling]]'' uses a small set of tile shapes that cannot form a repeating pattern (an [[Aperiodic set of prototiles|aperiodic set of prototiles]]). A ''tessellation of space'', 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 cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant [[Engineering:Pavement (architecture)|pavement]], floor, or wall coverings. Historically, tessellations were used in [[History:Ancient Rome|Ancient Rome]] and in Islamic art 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]] and in [[Hyperbolic geometry|hyperbolic geometry]], for artistic effect. Tessellations are sometimes employed for decorative effect in quilting. Tessellations form a class of [[Patterns in nature|patterns in nature]], for example in the arrays of [[Hexagonal tiling|hexagonal cells]] found in honeycombs.

== History ==
[[File:Stone-Cone Temple mosaics, Pergamon Museum.JPG|thumb|left|A temple mosaic from the ancient Sumerian city of [[Place:Uruk|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.<ref name="Pickover2009">{{cite book |last=Pickover |first=Clifford A. |title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics |year=2009 |publisher=Sterling |isbn=978-1-4027-5796-9 |page=372}}</ref>

Decorative [[Engineering:Mosaic|mosaic]] tilings made of small squared blocks called tesserae were widely employed in [[History:Classical antiquity|classical antiquity]],<ref>{{cite book |last1=Dunbabin | first1=Katherine M. D. | title=Mosaics of the Greek and Roman world | date=2006 | publisher=Cambridge University Press | page=280}}</ref> sometimes displaying geometric patterns.<ref>{{cite web | title=The Brantingham Geometric Mosaics | url=http://www.hullcc.gov.uk/museumcollections/collections/storydetail.php?irn=410 | publisher=Hull City Council | access-date=26 May 2015 | date=2008}}</ref><ref name=Field1988/>

In 1619, [[Biography:Johannes Kepler|Johannes Kepler]] made an early documented study of tessellations. He wrote about regular and semiregular tessellations in his {{Lang|la|[[Physics:Harmonices Mundi|Harmonices Mundi]]}}; he was possibly the first to explore and to explain the hexagonal structures of honeycomb and snowflakes.<ref>{{cite book |  author=Kepler, Johannes | date=1619 | title-link=Harmonices Mundi | title=Harmonices Mundi |trans-title=Harmony of the Worlds }}</ref>{{sfn|Gullberg|1997|p=395}}{{sfn|Stewart|2001|p=13}}

[[File:Roman geometric mosaic.jpg|thumb|upright=1.25<!--width for very low aspect image-->|[[History:Ancient Rome|Roman]] geometric mosaic<!--resembling [[Necker cube|Necker cube]]-->]]

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.<ref>{{cite web | url=http://public.lanl.gov/djidjev/papers/coverage_chapter.pdf | title=Dynamic Coverage Problems in Sensor Networks | publisher=[[Organization:Los Alamos National Laboratory|Los Alamos National Laboratory]] | year=2012 | access-date=6 April 2013 |author1=Djidjev, Hristo |author2=Potkonjak, Miodrag |page=2}}</ref><ref>{{cite journal | author=Fyodorov, Y. |  date=1891 | title=Simmetrija na ploskosti [Symmetry in the plane] | journal=Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva [Proceedings of the Imperial St. Petersburg Mineralogical Society] |series=2 | volume=28 | pages=245–291| language=ru}}</ref> Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include [[Biography:Alexei Vasilievich Shubnikov|Alexei Vasilievich Shubnikov]] and Nikolai Belov (1964),<ref name="ShubnikovBelov1964">{{cite book | last1=Shubnikov | first1=Alekseĭ Vasilʹevich | last2=Belov | first2=Nikolaĭ Vasilʹevich | title=Colored Symmetry | url=https://books.google.com/books?id=QKk9AAAAIAAJ | year=1964 | publisher=Macmillan}}</ref> and Heinrich Heesch and Otto Kienzle (1963).<ref>{{cite book | last1=Heesch | first1=H. | last2=Kienzle | first2=O. | title=Flächenschluss: System der Formen lückenlos aneinanderschliessender Flächteile | date=1963 | publisher=[[Company:Springer Publishing|Springer]] | language=de}}</ref>

=== Etymology ===
In Latin, ''tessella'' is a small cubical piece of [[Earth:Clay|clay]], [[Earth:Rock (geology)|stone]], or [[Chemistry:Glass|glass]] used to make mosaics.<ref>{{cite dictionary | url=http://m-w.com/dictionary/tessellate | title=Tessellate | dictionary=Merriam-Webster Online | access-date=26 May 2015}}</ref> 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 [[Engineering:Ceramic glaze|glazed]] clay.

== Overview ==
[[File:Semi-regular-floor-3464.JPG|thumb|upright|left|A [[Rhombitrihexagonal tiling|rhombitrihexagonal tiling]]: tiled floor in the Archeological Museum of Seville, Spain, using square, triangle, and hexagon prototiles]]

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.<ref>{{cite book |author1=Conway, R. |author2=Burgiel, H. |author3=Goodman-Strauss, G. | title=The Symmetries of Things | publisher=Peters | date=2008}}</ref> The tessellations created by bonded brickwork do not obey this rule. Among those that do, a regular tessellation has both identical{{efn|The mathematical term for identical shapes is "congruent" – in mathematics, "identical" means they are the same tile.}} [[Regular polygon|regular tiles]] and identical regular corners or vertices, having the same angle between adjacent edges for every tile.{{sfn|Coxeter|1973}} There are only three shapes that can form such regular tessellations: the equilateral [[Triangle|triangle]], [[Square|square]] and the regular [[Hexagon|hexagon]]. Any one of these three shapes can be duplicated infinitely to fill a [[Plane (geometry)|plane]] with no gaps.{{sfn|Gullberg|1997|p=395}}

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.<ref>{{cite book | author=Cundy and Rollett | title=Mathematical Models|title-link=Mathematical Models (Cundy and Rollett) | edition=2nd | publisher=Oxford | date=1961 | pages=61–62}}</ref> 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.{{sfn|Escher|1974|pp=11–12, 15–16}} 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.<ref>{{cite web | url=http://www.basilicasanmarco.it/basilica/architettura/il-pavimento-tessulare/?lang=en | title=Basilica di San Marco | publisher=Basilica di San Marco | work=Section: Tessellated floor | access-date=26 April 2013}}</ref>

[[File:Tassellatura alhambra.jpg|thumb|The elaborate and colourful [[Engineering:Zellige|zellige]] tessellations of glazed tiles at the Alhambra in Spain that attracted the attention of M. C. Escher]]

More formally, a tessellation or tiling is a [[Cover (topology)|cover]] of the Euclidean plane by a countable number of closed sets, called ''tiles'', such that the tiles intersect only on their [[Boundary (topology)|boundaries]]. These tiles may be polygons or any other shapes.{{efn|The tiles are usually required to be [[Homeomorphism|homeomorphic]] (topologically equivalent) to a closed disk, which means bizarre shapes with holes, dangling line segments, or infinite areas are excluded.{{sfn|Grünbaum|Shephard|1987|p=59}}}} Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are [[Congruence (geometry)|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''. The Conway criterion is a sufficient, but not necessary, set of rules for deciding whether a given shape tiles the plane periodically without reflections: some tiles fail the criterion, but still tile the plane.<ref>{{cite magazine | title=Will It Tile? Try the Conway Criterion! | author=Schattschneider, Doris |magazine=Mathematics Magazine | volume=53 | issue=4 | date=September 1980 |pages=224–233 | doi=10.2307/2689617| jstor=2689617 }}</ref> No general rule has been found for determining whether a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations.{{sfn|Grünbaum|Shephard|1987|p=59}}

Mathematically, tessellations can be extended to spaces other than the Euclidean plane.{{sfn|Gullberg|1997|p=395}} The {{wipe|Switzerland}} [[Geometry|geometer]] [[Biography:Ludwig Schläfli|Ludwig Schläfli]] pioneered this by defining ''polyschemes'', which mathematicians nowadays call polytopes. These are the analogues to polygons and [[Polyhedron|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}.<ref>{{cite book | url=https://books.google.com/books?id=iWvXsVInpgMC&pg=PP1 | title=Regular Polytopes | publisher=Methuen | author=Coxeter, H. S. M. | year=1948 |pages=14, 69, 149| isbn=978-0-486-61480-9 }}</ref> 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}.<ref name="MathWorldTessellation">{{MathWorld|title=Tessellation|id=Tessellation}}</ref>

Other methods also exist for describing polygonal tilings. When the tessellation is made of regular polygons, the most common notation is the [[Vertex configuration|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 4<sup>4</sup>. The tiling of regular hexagons is noted 6.6.6, or 6<sup>3</sup>.{{sfn|Grünbaum|Shephard|1987|p=59}}

== In mathematics ==

=== Introduction to tessellations ===
{{Further|Euclidean tilings by convex regular polygons|Uniform tiling||List of Euclidean uniform tilings}}

Mathematicians use some technical terms when discussing tilings. An ''[[Edge (geometry)|edge]]'' is the intersection between two bordering tiles; it is often a straight line. A ''[[Vertex (geometry)|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|polygon]]s about each vertex is the same.{{sfn|Grünbaum|Shephard|1987|p=59}} The [[Fundamental domain|fundamental region]] is a shape such as a rectangle that is repeated to form the tessellation.<ref name="EmmerSchattschneider2007">{{cite book |last1=Emmer |first1=Michele |last2=Schattschneider |first2=Doris |title=M.C. Escher's Legacy: A Centennial Celebration |url=https://books.google.com/books?id=5DDyBwAAQBAJ&pg=PA325 |date=8 May 2007 |publisher=Springer |location=Berlin Heidelberg |isbn=978-3-540-28849-7 |page=325}}</ref> For example, a regular tessellation of the plane with squares has a meeting of [[Square tiling|four squares at every vertex]].{{sfn|Grünbaum|Shephard|1987|p=59}}

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.{{sfn|Grünbaum|Shephard|1987|p=59}}

A ''normal tiling'' is a tessellation for which every tile is topologically equivalent to a [[Disk (mathematics)|disk]], the intersection of any two tiles is a connected set or the [[Empty set|empty set]], and all tiles are uniformly bounded. 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.<ref name=Horne2000>{{cite book |last=Horne | first=Clare E. | title=Geometric Symmetry in Patterns and Tilings | year=2000 |publisher=Woodhead Publishing |isbn=978-1-85573-492-0 |pages=172, 175}}</ref>

[[File:P5-type15-chiral_coloring.png|thumb|left|upright|An example of a non-edge‑to‑edge tiling: the 15th convex ''monohedral'' [[Pentagonal tiling|pentagonal tiling]], discovered in 2015]]

{{anchor|Monohedral}}A {{nobr|'''monohedral tiling'''}} is a tessellation in which all tiles are [[Congruence (geometry)|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; the [[Voderberg tiling]] has a unit tile that is a nonconvex enneagon.<ref name="Pickover2009" /> 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 of a regular pentagon, {{sfrac|3{{pi}}|5}}, is not a divisor of 2{{pi}}.<ref>{{cite web | url=http://www.uwgb.edu/dutchs/symmetry/radspir1.htm | title=Some Special Radial and Spiral Tilings | publisher=University of Wisconsin | date=29 July 1999 | access-date=6 April 2013 | author=Dutch, Steven}}</ref><ref name=Hirschhorn1985>{{cite journal | last=Hirschhorn | first=M. D. |author2=Hunt, D. C. | title=Equilateral convex pentagons which tile the plane | journal=Journal of Combinatorial Theory | series=Series A |year=1985 | volume=39 | issue=1 | pages=1–18 | doi=10.1016/0097-3165(85)90078-0 | doi-access=free }}</ref>

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|symmetry]] group of the tiling.<ref name=Horne2000/> 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 [[Symmetry|symmetric]], edge-to-edge tiling made up of [[Regular polygon|regular polygon]]s, all of the same shape. There are only three regular tessellations: those made up of [[Equilateral triangle|equilateral triangle]]s, squares, or regular [[Hexagon|hexagon]]s. All three of these tilings are isogonal and monohedral.<ref name=threeregular>{{MathWorld|title=Regular Tessellations|id=RegularTessellations}}</ref>

[[File:2005-06-25 Tiles together.jpg|thumb|upright|A [[Pythagorean tiling|Pythagorean tiling]] is not an edge‑to‑edge tiling.]]
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).{{sfn|Stewart|2001|p=75}} These can be described by their [[Vertex configuration|vertex configuration]]; for example, a semi-regular tiling using squares and regular octagons has the vertex configuration 4.8<sup>2</sup> (each vertex has one square and two octagons).<ref>{{cite web | url=http://nrich.maths.org/1556 | title=Schläfli Tessellations | publisher=University of Cambridge | date=1997–2012 | access-date=26 April 2013 | author=NRICH (Millennium Maths Project)}}</ref> Many non-edge-to-edge tilings of the Euclidean plane are possible, including the family of [[Pythagorean tiling]]s, tessellations that use two (parameterised) sizes of square, each square touching four squares of the other size.<ref>{{cite book | last=Wells | first=David |  year=1991 | title=The Penguin Dictionary of Curious and Interesting Geometry | url=https://archive.org/details/penguindictionar0000well | url-access=registration | publisher=Penguin Books | location=New York | isbn=978-0-14-011813-1 | pages=[https://archive.org/details/penguindictionar0000well/page/260 260–261] | contribution=two squares tessellation}}</ref> An [[Edge tessellation|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.<ref>{{cite journal |author=Kirby, Matthew |author2=Umble, Ronald |title=Edge Tessellations and Stamp Folding Puzzles |journal=Mathematics Magazine |volume=84 |issue=4 |date=2011 |pages=283–89 |doi=10.4169/math.mag.84.4.283|s2cid=123579388 }}</ref>

=== Wallpaper groups ===
{{Main|Wallpaper group}}
[[File:Wallpaper group-p3-1.jpg|thumb|upright|This tessellated, monohedral street pavement uses curved shapes instead of polygons. It belongs to wallpaper group p3.]]
Tilings with [[Physics:Translational symmetry|translational symmetry]] in two independent directions can be categorized by [[Wallpaper group|wallpaper group]]s, of which 17 exist.<ref name=armstrong>{{cite book | last=Armstrong | first=M.A. | title=Groups and Symmetry | url=https://archive.org/details/groupssymmetry0000arms | url-access=registration | year=1988 | publisher=Springer-Verlag | location=New York | isbn=978-3-540-96675-3}}</ref> It has been claimed that all seventeen of these groups are represented in the Alhambra palace in Granada, {{wipe|Spain}}. Although this is disputed,<ref>{{cite journal |last=Grünbaum |first=Branko | journal=[[Organization:Notices of the American Mathematical Society|Notices of the American Mathematical Society]] | date=June–July 2006 | volume=53 | issue=6 | pages=670–673 | title=What symmetry groups are present in the Alhambra? | url=http://www.ams.org/notices/200606/comm-grunbaum.pdf}}</ref> the variety and sophistication of the Alhambra tilings have interested modern researchers.<ref>{{cite journal | last=Lu | first=Peter J. | author2=Steinhardt | journal=Science | date=23 February 2007 | volume=315 | issue=5815 | pages=1106–10 |doi=10.1126/science.1135491 | pmid=17322056 |title=Decagonal and quasi-crystalline tilings in medieval Islamic architecture| bibcode=2007Sci...315.1106L | s2cid=10374218 }}</ref> Of the three regular tilings two are in the ''p6m'' wallpaper group and one is in ''p4m''. Tilings in 2-D with translational symmetry in just one direction may be categorized by the seven frieze groups describing the possible frieze patterns.<ref>{{MathWorld|title=Frieze Group|id=FriezeGroup}}</ref> [[Orbifold notation]] can be used to describe wallpaper groups of the Euclidean plane.<ref>{{Cite web | last1=Huson | first1=Daniel H. | date=1991 | title=Two-Dimensional Symmetry Mutation |url=https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.8536 | publisher=Princeton University | citeseerx=10.1.1.30.8536 |via=CiteSeerX}}</ref>

=== Aperiodic tilings ===
{{Main|Aperiodic tiling|List of aperiodic sets of tiles}}
[[File:Penrose Tiling (Rhombi).svg|thumb|left|upright|A [[Penrose tiling|Penrose tiling]], with several symmetries, but no periodic repetitions]]

[[Penrose tiling]]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|aperiodic tiling]]s, which use tiles that cannot tessellate periodically. The [[Recursion|recursive process]] of substitution tiling is a method of generating aperiodic tilings. One class that can be generated in this way is the [[Rep-tile|rep-tile]]s; these tilings have unexpected self-replicating properties.{{sfn|Gardner|1989|pp=1–18}} [[Pinwheel tiling]]s are non-periodic, using a rep-tile construction; the tiles appear in infinitely many orientations.<ref>{{cite journal | author=Radin, C. | title=The Pinwheel Tilings of the Plane | journal=[[Annals of Mathematics]] | volume=139 | issue=3 | pages=661–702 | date=May 1994 | doi=10.2307/2118575 | jstor=2118575| citeseerx=10.1.1.44.9723 }}</ref> It might be thought that a non-periodic pattern would be entirely without symmetry, but this is not so. Aperiodic tilings, while lacking in [[Physics:Translational symmetry|translational symmetry]], 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.<ref name=Austin>{{cite web | last1=Austin | first1=David | title=Penrose Tiles Talk Across Miles | url=http://www.ams.org/samplings/feature-column/fcarc-penrose | publisher=American Mathematical Society | access-date=29 May 2015}}</ref> A substitution rule, such as can be used to generate Penrose patterns using assemblies of tiles called rhombs, illustrates scaling symmetry.<ref>{{cite web | last1=Harriss | first1=E. O. | title=Aperiodic Tiling | url=http://www.mathematicians.org.uk/eoh/files/Aperiodic_Tilings.pdf | publisher=University of London and EPSRC | access-date=29 May 2015 | archive-date=29 August 2017 | archive-url=https://web.archive.org/web/20170829232941/http://mathematicians.org.uk/eoh/files/Aperiodic_Tilings.pdf | url-status=dead }}</ref> A [[Fibonacci word]] can be used to build an aperiodic tiling, and to study [[Physics:Quasicrystal|quasicrystal]]s, which are structures with aperiodic order.<ref>{{cite journal | last1=Dharma-wardana | first1=M. W. C. | last2=MacDonald | first2=A. H. | last3=Lockwood | first3=D. J. | last4=Baribeau | first4=J.-M. | last5=Houghton | first5=D. C. | journal=[[Physics:Physical Review Letters|Physical Review Letters]] | pages=1761–1765 | title=Raman scattering in Fibonacci superlattices | volume=58 | issue=17 | year=1987 | doi=10.1103/physrevlett.58.1761 | pmid=10034529 | bibcode=1987PhRvL..58.1761D}}</ref>

[[Image:wang tiles.svg|thumb|A set of 13 [[Wang tile|Wang tile]]s that tile the plane only [[Aperiodic tiling|aperiodically]] ]]

[[Wang tile]]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. A suitable set of Wang dominoes can tile the plane, but only aperiodically. This is known because any [[Turing machine]] 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|halting problem]] is undecidable, the problem of deciding whether a Wang domino set can tile the plane is also undecidable.<ref>{{cite journal | last=Wang | first=Hao |  issue=1 | journal=[[Engineering:Bell System Technical Journal|Bell System Technical Journal]] | pages=1–41 | title=Proving theorems by pattern recognition—II | volume=40 | year=1961 | doi=10.1002/j.1538-7305.1961.tb03975.x}}</ref><ref>{{cite magazine | last=Wang | first=Hao | author-link=Hao Wang (academic) | date=November 1965 |magazine=Scientific American | pages=98–106 | title=Games, logic and computers}}</ref><ref>{{cite journal | last=Berger | first=Robert | author-link=Robert Berger (mathematician) | journal=Memoirs of the American Mathematical Society| title=The undecidability of the domino problem | volume=66 | issue=66 | page=72 | date=1966 | doi=10.1090/memo/0066}}</ref><ref>{{cite journal | last=Robinson | first=Raphael M. | author-link=Raphael Robinson | journal=[[Inventiones Mathematicae]] | mr=0297572 | pages=177–209 | title=Undecidability and nonperiodicity for tilings of the plane | volume=12 | issue=3 | year=1971 | doi=10.1007/bf01418780| bibcode=1971InMat..12..177R | s2cid=14259496 }}</ref><ref>{{cite journal | last=Culik | first=Karel II | doi=10.1016/S0012-365X(96)00118-5 | issue=1–3 | journal=Discrete Mathematics | mr=1417576 | pages=245–251 | title=An aperiodic set of 13 Wang tiles | volume=160 | year=1996| doi-access=free }}</ref>

[[File:Truchet base tiling.svg|thumb|upright|Random Truchet tiling ]]

Truchet tiles are square tiles decorated with patterns so they do not have [[Rotational symmetry|rotational symmetry]]; in 1704, Sébastien Truchet used a square tile split into two triangles of contrasting colours. These can tile the plane either periodically or randomly.<ref>{{cite journal | last=Browne | first=Cameron | doi=10.1016/j.cag.2007.10.001 | issue=2 | journal=Computers & Graphics | pages=268–281 | title=Truchet curves and surfaces | volume=32 | year=2008}}</ref><ref>{{cite journal | last=Smith | first=Cyril Stanley |  doi=10.2307/1578535 | issue=4 | journal=Leonardo | pages=373–385 | title=The tiling patterns of Sebastian Truchet and the topology of structural hierarchy | volume=20 | year=1987| jstor=1578535 | s2cid=192944820 }}</ref>

An ''einstein tile'' is a single shape that forces aperiodic tiling. The first such tile, dubbed a "hat", was discovered in 2023 by David Smith, a hobbyist mathematician.<ref>{{Cite web |last=Conover |first=Emily |date=2023-03-24 |title=Mathematicians have finally discovered an elusive 'einstein' tile |url=https://www.sciencenews.org/article/mathematicians-discovered-einstein-tile |access-date=2023-03-25 |website=Science News |language=en-US}} with image of the pattern</ref><ref>Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; Goodman-Strauss, Chaim (March 2023). "An aperiodic monotile". arXiv:2303.10798</ref> The discovery is under professional review and, upon confirmation, will be credited as solving a longstanding [[Einstein problem|mathematical problem]].<ref>Roberts, Soibhan, ''[https://www.nytimes.com/2023/03/28/science/mathematics-tiling-einstein.html Elusive 'Einstein' Solves a Longstanding Mathematical Problem]'', the New York Times, March 28, 2023, with image of the pattern</ref>

=== Tessellations and colour ===

[[File:Torus with seven colours.svg|thumb|left|At least seven colors are required if the colours of this tiling are to form a pattern by repeating this rectangle as the [[Fundamental domain|fundamental domain]]; more generally, at least [[Four color theorem|four colours]] are needed.]]

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 color theorem|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 does not generally respect the symmetries of the tessellation. To produce a colouring that does, it is necessary to treat the colours as part of the tessellation. Here, as many as seven colours may be needed, as demonstrated in the image at right.<ref name=Hazewinkel2001>{{springer | title=Four-colour problem|id=p/f040970}}</ref>

=== Tessellations with polygons ===
[[File:Coloured Voronoi 2D.svg|thumb|upright|A [[Voronoi diagram|Voronoi tiling]], in which the cells are always convex polygons]]

Next to the various tilings by regular polygons, tilings by other polygons have also been studied.

Any triangle or [[Quadrilateral|quadrilateral]] (even [[Concave polygon|non-convex]]) can be used as a prototile to form a monohedral tessellation, often in more than one way. Copies of an arbitrary [[Quadrilateral|quadrilateral]] 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#Group p2|wallpaper group p2]]. As [[Fundamental domain|fundamental domain]] we have the quadrilateral. Equivalently, we can construct a [[Parallelogram|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.<ref>{{cite book | url=http://digital.library.wisc.edu/1711.dl/DLDecArts.GramOrnJones | title=The Grammar of Ornament | publisher=Bernard Quaritch | last=Jones | first=Owen |  year=1910 |edition=folio | orig-year=1856}}</ref>

If only one shape of tile is allowed, tilings exists with convex ''N''-gons for ''N'' equal to 3, 4, 5, and 6. For {{nowrap|1=''N'' = 5}}, see [[Pentagonal tiling]], for {{nowrap|1=''N'' = 6}}, see [[Hexagonal tiling]], for {{nowrap|1=''N'' = 7}}, see [[Heptagonal tiling]] and for {{nowrap|1=''N'' = 8}}, see [[Octagonal tiling|octagonal tiling]].

For results on tiling the plane with polyominoes, see Polyomino § Uses of polyominoes.

=== Voronoi tilings ===
[[Voronoi diagram|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.)<ref>{{cite journal | author=Aurenhammer, Franz | date=1991 | title=Voronoi Diagrams – A Survey of a Fundamental Geometric Data Structure | journal=ACM Computing Surveys | volume=23| issue =3 | pages=345–405 | doi=10.1145/116873.116880|s2cid= 4613674 }}</ref><ref>{{cite book |author1=Okabe, Atsuyuki |author2=Boots, Barry |author3=Sugihara, Kokichi |author4=Chiu, Sung Nok | date=2000 | title=Spatial Tessellations – Concepts and Applications of Voronoi Diagrams | edition=2nd | publisher=John Wiley | isbn=978-0-471-98635-5}}</ref> The ''Voronoi cell'' for each defining point is a convex polygon. The [[Delaunay triangulation]] is a tessellation that is the [[Dual graph|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.<ref name=George1998>{{cite book |author1=George, Paul Louis |author2=Borouchaki, Houman | title=Delaunay Triangulation and Meshing: Application to Finite Elements |year=1998 | publisher=[[Religion:Hermes|Hermes]] | isbn=978-2-86601-692-0 | pages=34–35}}</ref> Voronoi tilings with randomly placed points can be used to construct random tilings of the plane.<ref>{{cite book | last1=Moller | first1=Jesper | title=Lectures on Random Voronoi Tessellations | date=1994 | publisher=Springer | isbn=978-1-4612-2652-9 | url=https://www.springer.com/gp/book/9780387942643}}</ref>

=== Tessellations in higher dimensions ===
{{Main|Honeycomb (geometry)}}
[[File:HC R1.png|thumb|upright|Tessellating three-dimensional (3-D) space: the [[Rhombic dodecahedron|rhombic dodecahedron]] is one of the solids that can be stacked to [[Rhombic dodecahedral honeycomb|fill space exactly]].]]

Tessellation can be extended to three dimensions. Certain [[Polyhedron|polyhedra]] can be stacked in a regular [[Bravais lattice|crystal pattern]] to fill (or tile) three-dimensional space, including the [[Cube|cube]] (the only Platonic polyhedron to do so), the [[Rhombic dodecahedron|rhombic dodecahedron]], the [[Truncated octahedron|truncated octahedron]], and triangular, quadrilateral, and hexagonal [[Prism (geometry)|prisms]], among others.<ref>{{cite journal | author=Grünbaum, Branko | date=1994 | title=Uniform tilings of 3-space | journal= Geombinatorics | volume=4 | issue=2 | pages=49–56}}</ref> Any polyhedron that fits this criterion is known as a [[Plesiohedron|plesiohedron]], and may possess between 4 and 38 faces.<ref>{{cite journal | last=Engel | first=Peter | doi=10.1524/zkri.1981.154.3-4.199 | issue=3–4 | journal=Zeitschrift für Kristallographie, Kristallgeometrie, Kristallphysik, Kristallchemie | mr=598811 | pages=199–215 | title=Über Wirkungsbereichsteilungen von kubischer Symmetrie | volume=154 | year=1981| bibcode=1981ZK....154..199E }}.</ref> Naturally occurring rhombic dodecahedra are found as [[Earth:Crystal|crystal]]s of [[Chemistry:Andradite|andradite]] (a kind of [[Chemistry:Garnet|garnet]]) and [[Chemistry:Fluorite|fluorite]].<ref name="Oldershaw2003">{{cite book |last=Oldershaw | first=Cally | title=Firefly Guide to Gems | url=https://archive.org/details/fireflyguidetoge0000olde |url-access=registration |year=2003 |publisher=Firefly Books | isbn=978-1-55297-814-6 |page=[https://archive.org/details/fireflyguidetoge0000olde/page/107 107]}}</ref><ref>{{cite book | last1=Kirkaldy | first1=J. F. | title=Minerals and Rocks in Colour | date=1968 | publisher=Blandford | pages=138–139 | edition=2nd}}</ref>

[[File:SCD tile.svg|thumb|upright|left|Illustration of a Schmitt–Conway biprism, also called a Schmitt–Conway–Danzer tile]]
Tessellations in three or more dimensions are called [[Honeycomb (geometry)|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{{efn|In this context, quasiregular means that the cells are regular (solids), and the vertex figures are semiregular.}} honeycomb, which has eight [[Tetrahedron|tetrahedra]] and six [[Octahedron|octahedra]] at each polyhedron vertex. However, there are many possible [[Convex uniform honeycomb|semiregular honeycombs]] in three dimensions.<ref name="CoxeterSherk1995">{{cite book |last1=Coxeter |first1=Harold Scott Macdonald |last2=Sherk |first2=F. Arthur |author3=Canadian Mathematical Society |title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter |url=https://archive.org/details/kaleidoscopessel0000coxe |url-access=registration |date=1995 |publisher=John Wiley & Sons |isbn=978-0-471-01003-6 |page=[https://archive.org/details/kaleidoscopessel0000coxe/page/3 3] and passim}}</ref> Uniform honeycombs can be constructed using the [[Wythoff construction]].<ref>{{mathworld | urlname=WythoffConstruction | title=Wythoff construction}}</ref>

The [[Gyrobifastigium#Schmitt–Conway–Danzer biprism|Schmitt-Conway biprism]] is a convex polyhedron with the property of tiling space only aperiodically.<ref name="Senechal1996">{{cite book |last=Senechal |first=Marjorie |title=Quasicrystals and Geometry | title-link = Quasicrystals and Geometry |at =[https://books.google.com/books?id=LdQ8AAAAIAAJ&pg=PA209 p. 209] |date=26 September 1996 |publisher=CUP Archive |isbn=978-0-521-57541-6 }}</ref>

A [[Schwarz triangle]] is a [[Spherical triangle|spherical triangle]] that can be used to tile a [[Sphere|sphere]].<ref>{{Cite journal | last1=Schwarz | first1=H. A. | title=Ueber diejenigen Fälle in welchen die Gaussichen hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002155206 | year=1873 | volume=1873 | issue=75 | journal=Journal für die reine und angewandte Mathematik | issn=0075-4102 | pages=292–335 | doi=10.1515/crll.1873.75.292| s2cid=121698536 | doi-access=free }}</ref>

=== Tessellations in non-Euclidean geometries ===
[[File:Rhombitriheptagonal tiling.svg|thumb|left|upright|[[Rhombitriheptagonal tiling|Rhombitriheptagonal tiling]] in hyperbolic plane, seen in [[Poincaré disk model|Poincaré disk model]] projection]]
[[File:H3 353 CC center.png|thumb|The regular [[Icosahedral honeycomb|{3,5,3} icosahedral honeycomb]], one of four regular compact honeycombs in hyperbolic 3-space ]]

It is possible to tessellate in non-Euclidean geometries such as [[Hyperbolic geometry|hyperbolic geometry]]. A [[Uniform tilings in hyperbolic plane|uniform tiling in the hyperbolic plane]] (that may be regular, quasiregular, or semiregular) is an edge-to-edge filling of the hyperbolic plane, with [[Regular polygon|regular polygon]]s as [[Face (geometry)|faces]]; these are vertex-transitive (transitive on its [[Vertex (geometry)|vertices]]), and isogonal (there is an [[Isometry|isometry]] mapping any vertex onto any other).<ref>{{cite arXiv |last1=Margenstern |first1=Maurice |title=Coordinates for a new triangular tiling of the hyperbolic plane |date=4 January 2011 |eprint=1101.0530 |class=cs.FL }}</ref><ref>{{cite web |title=Tiling the Hyperbolic Plane with Regular Polygons |author=Zadnik, Gašper |url=http://demonstrations.wolfram.com/TilingTheHyperbolicPlaneWithRegularPolygons/  | publisher=Wolfram | access-date=27 May 2015}}</ref>

A [[Uniform honeycombs in hyperbolic space|uniform honeycomb in hyperbolic space]] is a uniform tessellation of [[Uniform polyhedron|uniform polyhedral]] cells. In three-dimensional (3-D) hyperbolic space there are nine [[Coxeter group]] families of compact [[Convex uniform honeycomb|convex uniform honeycomb]]s, generated as [[Wythoff construction]]s, and represented by [[Permutation|permutation]]s of rings of the Coxeter diagrams for each family.<ref>{{cite book | author=Coxeter, H.S.M. | title=The Beauty of Geometry: Twelve Essays | publisher=[[Social:Dover Publications|Dover Publications]] | date=1999 | isbn=978-0-486-40919-1 | chapter=Chapter 10: Regular honeycombs in hyperbolic space | pages=212–213}}</ref>

== In art ==
{{Further|Mathematics and art}}
[[File:Mosaic floor panel - Google Art Project.jpg|thumb|upright|left|Roman mosaic floor panel of stone, tile, and glass, from a villa near Antioch in Roman Syria. second century AD]]

In architecture, tessellations have been used to create decorative motifs since ancient times. [[Engineering:Mosaic|Mosaic]] tilings often had geometric patterns.<ref name=Field1988>{{cite book | title=Geometric Patterns from Roman Mosaics | publisher=Tarquin | author=Field, Robert | year=1988 | isbn=978-0-906-21263-9}}</ref> Later civilisations also used larger tiles, either plain or individually decorated. Some of the most decorative were the Moorish wall tilings of Islamic architecture, using Girih and [[Engineering:Zellige|Zellige]] tiles in buildings such as the Alhambra<ref>{{cite web | url=http://www.math.nus.edu.sg/aslaksen/teaching/math-art-arch.shtml#Symmetry | title=Mathematics in Art and Architecture | publisher=National University of Singapore |access-date=17 May 2015}}</ref> and La Mezquita.<ref>{{cite book | last=Whittaker | first=Andrew |title=Speak the Culture: Spain | url=https://books.google.com/books?id=2rZS8SWMroUC&pg=PT153 | year=2008 | publisher=Thorogood Publishing | isbn=978-1-85418-605-8 | page=153}}</ref>

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 {{wipe|Spain}} in 1936.{{sfn|Escher|1974|pp=5, 17}} Escher made four "[[Circle Limit III|Circle Limit]]" drawings of tilings that use hyperbolic geometry.<ref>{{cite web | last1=Gersten |first1=S. M. | title=Introduction to Hyperbolic and Automatic Groups |url=http://www.math.utah.edu/~sg/Papers/banff.pdf | publisher=University of Utah | access-date=27 May 2015 | quote=Figure 1 is part of a tiling of the Euclidean plane, which we imagine as continued in all directions, and Figure 2 [Circle Limit IV] is a beautiful tesselation of the Poincaré unit disc model of the hyperbolic plane by white tiles representing angels and black tiles representing devils. An important feature of the second is that all white tiles are mutually congruent as are all black tiles; of course this is not true for the Euclidean metric, but holds for the Poincaré metric}}</ref><ref>{{cite web | last1=Leys | first1=Jos | title=Hyperbolic Escher | url=http://www.josleys.com/show_gallery.php?galid=325 | access-date=27 May 2015 | date=2015}}</ref> For his woodcut "Circle Limit IV" (1960), Escher prepared a pencil and ink study showing the required geometry.{{sfn|Escher|1974|pp=142–143}} 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."{{sfn|Escher|1974|p=16}}

[[File:Ralli Quilt.jpg|thumb|A quilt showing a regular tessellation pattern]]
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 quilts.<ref>{{cite book | last=Porter | first=Christine | title=Tessellation Quilts: Sensational Designs From Interlocking Patterns | year=2006 | publisher=F+W Media | isbn=978-0-7153-1941-3 | pages=4–8}}</ref><ref>{{cite book |last=Beyer |first=Jinny | title=Designing tessellations: the secrets of interlocking patterns |year=1999 |publisher=Contemporary Book |isbn=978-0-8092-2866-9 |pages=Ch. 7}}</ref>

Tessellations are also a main genre in origami (paper folding), where pleats are used to connect molecules, such as twist folds, together in a repeating fashion.<ref>{{Cite book |title=Origami Tessellations |last=Gjerde |first=Eric |publisher=Taylor and Francis |year=2008|isbn=978-1-568-81451-3}}</ref>

== In manufacturing ==
Tessellation is used in manufacturing industry to reduce the wastage of material (yield losses) such as [[Engineering:Sheet metal|sheet metal]] when cutting out shapes for objects such as [[Engineering:Car door|car door]]s or drink cans.<ref>{{cite web | title=Reducing yield losses: using less metal to make the same thing | url=http://www.withbotheyesopen.com/pdftransponder.php?c=13|publisher=UIT Cambridge | access-date=29 May 2015}}</ref>

Tessellation is apparent in the [[Earth:Mudcrack|mudcrack]]-like [[Physics:Fracture|cracking]] of [[Physics:Thin film|thin film]]s<ref>{{cite journal | last1 = Thouless | first1 = M. D. | year = 1990 | title = Crack Spacing in Brittle Films on Elastic Substrates | journal = J. Am. Chem. Soc. | volume = 73 | issue = 7| pages = 2144–2146 | doi = 10.1111/j.1151-2916.1990.tb05290.x }}</ref><ref>{{cite journal | last1 = Xia | first1 = Z. C. | last2 = Hutchinson | first2 = J. W. | year = 2000 | title = Crack patterns in thin films | journal = J. Mech. Phys. Solids | volume = 48 | issue = 6–7 | pages = 1107–1131 | doi = 10.1016/S0022-5096(99)00081-2 | bibcode = 2000JMPSo..48.1107X }}</ref> – with a degree of self-organisation being observed using [[Engineering:Microtechnology|micro]] and [[Engineering:Nanotechnology|nanotechnologies]].<ref>{{cite journal | last1 = Seghir | first1 = R. | last2 = Arscott | first2 = S. | year = 2015 | title = Controlled mud-crack patterning and self-organized cracking of polydimethylsiloxane elastomer surfaces | journal = Sci. Rep. | volume = 5 | page = 14787 | doi = 10.1038/srep14787 | pmid = 26437880 | bibcode = 2015NatSR...514787S | pmc = 4594096 }}</ref>

== In nature ==

[[File:Buckfast bee.jpg|thumb|left|A honeycomb is a natural tessellated structure.]]
The honeycomb is a well-known example of tessellation in nature with its hexagonal cells.<ref>{{cite journal | last1=Ball | first1=Philip | title=How honeycombs can build themselves | url=http://www.nature.com/news/how-honeycombs-can-build-themselves-1.13398 | journal=Nature | year=2013 | doi=10.1038/nature.2013.13398 | s2cid=138195687 | access-date=7 November 2014}}</ref>

In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including the [[Biology:Fritillaria|fritillary]],<ref>{{cite book | title=Shorter Oxford English dictionary | edition=6th | year=2007 | publisher=Oxford University Press | location=United Kingdom |isbn=978-0-19-920687-2 |page=3804}}</ref> and some species of ''Colchicum'', are characteristically tessellate.<ref>{{cite journal | last1=Purdy | first1=Kathy | title=Colchicums: autumn's best-kept secret | journal=American Gardener | date=2007 | issue=September/October | pages=18–22}}</ref>

Many [[Patterns in nature|patterns in nature]] are formed by cracks in sheets of materials. These patterns can be described by Gilbert tessellations,<ref>{{cite arXiv | last1=Schreiber | first1=Tomasz | last2=Soja | first2=Natalia | eprint=1005.0023 | title=Limit theory for planar Gilbert tessellations | year=2010| class=math.PR }}</ref> also known as random crack networks.<ref>{{cite journal | last1=Gray | first1=N. H. | last2=Anderson | first2=J. B. | last3=Devine | first3=J. D. | last4=Kwasnik | first4=J. M. | doi=10.1007/BF01031092 | issue=6 | journal=Mathematical Geology | pages=617–626 | title=Topological properties of random crack networks | volume=8 | year=1976| s2cid=119949515 }}</ref> The Gilbert tessellation is a mathematical model for the formation of [[Earth:Mudcrack|mudcrack]]s, needle-like [[Earth:Crystal|crystal]]s, and similar structures. The model, named after [[Biography:Edgar Gilbert|Edgar Gilbert]], allows cracks to form starting from being 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.<ref>{{cite book | last=Gilbert | first=E. N. |  editor-last=Noble | editor-first=B. | contribution=Random plane networks and needle-shaped crystals | location=New York | publisher=Macmillan | title=Applications of Undergraduate Mathematics in Engineering | year=1967}}</ref> [[Earth:Basalt|Basalt]]ic lava flows often display [[Earth:Columnar basalt|columnar jointing]] as a result of [[Physics:Thermal expansion#Contraction effects (negative thermal expansion)|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 [[Earth:Giant's Causeway|Giant's Causeway]] in Northern Ireland.<ref>{{cite journal |last1=Weaire |first1=D. |last2=Rivier |first2=N. |title=Soap, cells and statistics: Random patterns in two dimensions |journal=Contemporary Physics |date=1984|volume=25 |issue=1 |pages=59–99 | author-link=Denis Weaire |doi=10.1080/00107518408210979 |bibcode=1984ConPh..25...59W}}</ref> [[Earth:Tessellated pavement|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.<ref>{{cite book | author=Branagan, D.F. | date=1983 | title=Tesselated pavements |editor1=Young, R.W. |editor2=Nanson, G.C. | pages=11–20 | series=Aspects of Australian sandstone landscapes. Special Publication No. 1, Australian and New Zealand Geomorphology |location=Wollongong, NSW | publisher=University of Wollongong | isbn=978-0-864-18001-8 |oclc=12650092}}</ref>

[[File:Colchicum - unknown species.jpg|thumb|Tessellate pattern in a ''[[Biology:Colchicum|Colchicum]]'' flower]]

Other natural patterns occur in [[Chemistry:Foam|foam]]s; these are packed according to [[Plateau's laws]], which require [[Minimal surface|minimal surface]]s. Such foams present a problem in how to pack cells as tightly as possible: in 1887, [[Biography:Lord Kelvin|Lord Kelvin]] proposed a packing using only one solid, the [[Bitruncated cubic honeycomb|bitruncated cubic honeycomb]] with very slightly curved faces. In 1993, Denis Weaire and Robert Phelan proposed the [[Weaire–Phelan structure]], which uses less surface area to separate cells of equal volume than Kelvin's foam.<ref>{{cite book | author=Ball, Philip | title=Shapes | publisher=[[Organization:Oxford University Press|Oxford University Press]] | date=2009 | pages=73–76 | isbn=978-0-199-60486-9}}</ref>

== In puzzles and recreational mathematics ==
[[File:Tangram set 00.jpg|thumb|upright|Traditional [[Tangram|tangram]] [[Dissection puzzle|dissection puzzle]] ]]

Tessellations have given rise to many types of tiling puzzle, from traditional jigsaw puzzles (with irregular pieces of wood or cardboard)<ref>{{cite web | last1=McAdam | first1=Daniel | title=History of Jigsaw Puzzles | url=http://www.jigsaw-puzzle.org/jigsaw-puzzle-history.html | publisher=American Jigsaw Puzzle Society | access-date=28 May 2015 |archive-url=https://web.archive.org/web/20140211155914/http://www.jigsaw-puzzle.org/jigsaw-puzzle-history.html |archive-date=11 February 2014}}</ref> and the [[Tangram|tangram]],<ref>{{cite book | title=The Tao of Tangram | last=Slocum | first=Jerry | year=2001 |publisher=Barnes & Noble | isbn=978-1-4351-0156-2 | page=9}}</ref> to more modern puzzles that often have a mathematical basis. For example, polyiamonds and polyominoes are figures of regular triangles and squares, often used in tiling puzzles.<ref>{{cite book | last1=Golomb | first1=Solomon W. | title=Polyominoes| title-link= Polyominoes: Puzzles, Patterns, Problems, and Packings | date=1994 | publisher=Princeton University Press | edition=2nd | isbn=978-0-691-02444-8}}</ref><ref>{{cite book | last1=Martin | first1=George E.|title=Polyominoes: A guide to puzzles and problems in tiling | url=https://archive.org/details/polyominoesguide00mart_0 | url-access=registration | date=1991 | publisher=Mathematical Association of America| isbn=978-0-88385-501-0}}</ref> Authors such as [[Biography:Henry Dudeney|Henry Dudeney]] and [[Biography:Martin Gardner|Martin Gardner]] have made many uses of tessellation in recreational mathematics. For example, Dudeney invented the [[Hinged dissection|hinged dissection]],<ref>{{cite book | last=Frederickson | first=Greg N. | date=2002 | title=Hinged Dissections: Swinging and Twisting | url=https://archive.org/details/hingeddissection0000fred | url-access=registration | publisher=Cambridge University Press |isbn=978-0-521-81192-7}}</ref> while Gardner wrote about the "[[Rep-tile|rep-tile]]", a shape that can be dissected into smaller copies of the same shape.<ref>{{cite magazine | author=Gardner, Martin | date=May 1963 | title=On 'Rep-tiles,' Polygons that can make larger and smaller copies of themselves |magazine=Scientific American | volume=208 | issue=May | pages=154–164}}</ref><ref name="Gardner2006">{{cite book | last=Gardner | first=Martin |author-link=Martin Gardner | title=Aha! A Two Volume Collection: Aha! Gotcha Aha! Insight | url=https://books.google.com/books?id=3HzwNVMQOZkC&pg=RA1-PA48 | date=14 December 2006 | publisher=MAA | isbn=978-0-88385-551-5 | page=48}}</ref> Inspired by Gardner's articles in Scientific American, the amateur mathematician [[Biography:Marjorie Rice|Marjorie Rice]] found four new tessellations with pentagons.<ref>{{cite news | url=https://www.nytimes.com/2015/10/12/opinion/the-importance-of-recreational-math.html?ref=opinion&_r=0 | title=The Importance of Recreational Math | last1=Suri | first1=Mani | date=12 October 2015 | website=New York Times}}</ref><ref>{{cite journal |last1=Schattschneider |first1=Doris |title=Tiling the Plane with Congruent Pentagons |url=http://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1979/0025570x.di021103.02p0247f.pdf |publisher=MAA |journal=Mathematics Magazine |date=1978 |volume=51 |issue=1 |pages=29–44 |doi=10.2307/2689644|jstor=2689644 }}</ref> [[Squaring the square]] is the problem of tiling an integral square (one whose sides have integer length) using only other integral squares.<ref>{{cite web | last1=Tutte | first1=W. T. | title=Squaring the Square | url=http://www.squaring.net/history_theory/brooks_smith_stone_tutte_II.html | website=Squaring.net | access-date=29 May 2015}}</ref><ref>{{cite magazine | last1=Gardner | first1=Martin| last2=Tutte | first2=William T. | title=Mathematical Games |magazine=Scientific American | date=November 1958}}</ref> 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.<ref>{{cite journal | first1=Frederick V. | last1=Henle | first2=James M. | last2=Henle | url=http://maven.smith.edu/~jhenle/stp/stp.pdf | title=Squaring the plane | journal=[[American Mathematical Monthly]] | volume=115 | issue=1 | year=2008 | pages=3–12 | doi=10.1080/00029890.2008.11920491 | jstor=27642387 | s2cid=26663945 | url-status=dead | archive-url=https://web.archive.org/web/20060620150515/http://maven.smith.edu/%7Ejhenle/stp/stp.pdf | archive-date=20 June 2006 }}</ref>

== Examples ==
<gallery mode="nolines">
File:Tile 3,6.svg|[[Triangular tiling]], one of the three regular tilings of the plane
File:Academ Periodic tiling where eighteen triangles encircle each hexagon.svg|Snub hexagonal tiling, a semiregular tiling 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 enneagons 
File:Uniform tiling 433-t0.png|[[Alternated octagonal tiling|Alternated octagonal or tritetragonal tiling]] is a uniform tiling of the [[Hyperbolic geometry|hyperbolic plane]] 
File:Capital I tiling-4color.svg|[[Topology|Topological]] square tiling, isohedrally distorted into I shapes 
</gallery>

== See also ==
* [[Astronomy:Discrete global grid|Discrete global grid]]
* [[Honeycomb (geometry)]]
* [[Space partitioning]]

== Explanatory footnotes ==
{{Notelist}}

== References ==
{{Reflist}}

== Sources ==
* {{Cite book |last=Coxeter |first=H. S. M. |year=1973 |title-link=Regular Polytopes (book) |title=Regular Polytopes |contribution = Section IV : Tessellations and Honeycombs |publisher=[[Social:Dover Publications|Dover Publications]] |isbn=978-0-486-61480-9 }}
* {{Cite book |last=Escher |first=M. C. |date=1974 |editor=J. L. Locher |title=The World of M. C. Escher |url=https://archive.org/details/worldofmcescher00esch |url-access=registration |edition=New Concise NAL |publisher=Abrams |isbn=978-0-451-79961-6}}
* {{Cite book |last=Gardner |first=Martin |year=1989 |title=Penrose Tiles to Trapdoor Ciphers |publisher={{wipe|Cambridge University Press}} |isbn=978-0-88385-521-8}}
* {{Cite book |last1=Grünbaum |first1=Branko |last2=Shephard |first2=G. C. |date=1987 |title=Tilings and Patterns |url=https://archive.org/details/isbn_0716711931 |url-access=registration |publisher=W. H. Freeman |isbn=978-0-7167-1193-3}}
* {{Cite book |last=Gullberg |first=Jan |year=1997 |title=Mathematics From the Birth of Numbers |url=https://archive.org/details/mathematicsfromb1997gull |url-access=registration |publisher=Norton |isbn=978-0-393-04002-9}}
* {{Cite book |last=Stewart |first=Ian |year=2001 |title=What Shape Is a Snowflake? |publisher=Weidenfeld and Nicolson |isbn=978-0-297-60723-6}}

== External links ==
{{Commons}}
<!-- Please don't put links to your commercial website here: they'll be removed, and you could be blocked! -->
* [https://ab.inf.uni-tuebingen.de/software/tegula Tegula] (open-source software for exploring two-dimensional tilings of the plane, sphere and hyperbolic plane; includes databases containing millions of tilings)
* [http://mathworld.wolfram.com/Tessellation.html Wolfram MathWorld: Tessellation] (good bibliography, drawings of regular, semiregular and demiregular tessellations)
* Dirk Frettlöh and [[Biography:Edmund Harriss|Edmund Harriss]]. "[http://tilings.math.uni-bielefeld.de/ Tilings Encyclopedia]" (extensive information on substitution tilings, including drawings, people, and references)
* [http://www.tessellations.org/ Tessellations.org] (how-to guides, Escher tessellation gallery, galleries of tessellations by other artists, lesson plans, history)
* {{cite web | author=Eppstein, David |  title=The Geometry Junkyard: Hyperbolic Tiling | url=http://www.ics.uci.edu/~eppstein/junkyard/hypertile.html}} (list of web resources including articles and galleries)
<!-- PLEASE DO NOT ATTEMPT TO LINK YOUR SOFTWARE SITE HERE - YOU COULD BE BLOCKED FOR ADVERTISING! -->

{{Mathematical art}}
{{Tessellation}}
{{Patterns in nature}}

[[Category:Symmetry]]

{{Sourceattribution|Tessellation}}