Wikipedia

Tessellation

"Tessellate" redirects here. For the song, seeTessellate (song).For the computer graphics technique, seeTessellation (computer graphics).
"Mathematical tiling" redirects here. For the building material, seeMathematical tile.

Atessellationortilingis the covering of asurface,often aplane,using one or moregeometric shapes,calledtiles,with no overlaps and no gaps. Inmathematics,tessellation can be generalized tohigher dimensionsand a variety of geometries.

Zelligeterracottatiles inMarrakech,forming edge‑to‑edge, regular and other tessellations
A wall sculpture inLeeuwardencelebrating the artistic tessellations ofM. C. Escher
An example of non‑periodicity due to another orientation of one tile out of an infinite number of identical tiles

Aperiodic tilinghas a repeating pattern. Some special kinds includeregular tilingswithregular polygonaltiles all of the same shape, andsemiregular tilingswith regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17wallpaper groups.A tiling that lacks a repeating pattern is called "non-periodic". Anaperiodic tilinguses a small set of tile shapes that cannot form a repeating pattern (anaperiodic set of prototiles). Atessellation 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 ascementedceramicsquares or hexagons. Such tilings may be decorativepatterns,or may have functions such as providing durable and water-resistantpavement,floor, or wall coverings. Historically, tessellations were used inAncient Romeand inIslamic artsuch as in theMoroccan architectureanddecorative geometric tilingof theAlhambrapalace. In the twentieth century, the work ofM. C. Escheroften made use of tessellations, both in ordinaryEuclidean geometryand inhyperbolic geometry,for artistic effect. Tessellations are sometimes employed for decorative effect inquilting.Tessellations form a class ofpatterns in nature,for example in the arrays ofhexagonal cellsfound inhoneycombs.

History

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

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

Decorativemosaictilings made of small squared blocks calledtesseraewere widely employed inclassical antiquity,[2]sometimes displaying geometric patterns.[3][4]

In 1619,Johannes Keplermade an early documented study of tessellations. He wrote about regular and semiregular tessellations in hisHarmonices Mundi;he was possibly the first to explore and to explain the hexagonal structures of honeycomb andsnowflakes.[5][6][7]

Romangeometric mosaic

Some two hundred years later in 1891, the Russian crystallographerYevgraf Fyodorovproved that every periodic tiling of the plane features one of seventeen different groups of isometries.[8][9]Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors includeAlexei Vasilievich ShubnikovandNikolai Belovin their bookColored Symmetry(1964),[10]andHeinrich Heeschand Otto Kienzle (1963).[11]

Etymology

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In Latin,tessellais a small cubical piece ofclay,stone,orglassused to make mosaics.[12]The word "tessella" means "small square" (fromtessera,square, which in turn is from the Greek word τέσσερα forfour). It corresponds to the everyday termtiling,which refers to applications of tessellations, often made ofglazedclay.

Overview

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Arhombitrihexagonal tiling:tiled floor in theArcheological 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 astiles,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.[13]The tessellations created bybonded brickworkdo not obey this rule. Among those that do, aregular tessellationhas both identical[a]regular tilesand identical regular corners or vertices, having the same angle between adjacent edges for every tile.[14]There are only three shapes that can form such regular tessellations: the equilateraltriangle,squareand the regularhexagon.Any one of these three shapes can be duplicated infinitely to fill aplanewith no gaps.[6]

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.[15]Irregular tessellations can also be made from other shapes such aspentagons,polyominoesand in fact almost any kind of geometric shape. The artistM. C. Escheris famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects.[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.[17]

The elaborate and colourfulzelligetessellations of glazed tiles at theAlhambrain Spain that attracted the attention ofM. C. Escher

More formally, a tessellation or tiling is acoverof the Euclidean plane by acountablenumber of closed sets, calledtiles,such that the tiles intersect only on theirboundaries.These tiles may be polygons or any other shapes.[b]Many tessellations are formed from a finite number ofprototilesin which all tiles in the tessellation arecongruentto the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said totessellateor totile the plane.TheConway criterionis 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.[19]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.[18]

Mathematically, tessellations can be extended to spaces other than the Euclidean plane.[6]TheSwissgeometerLudwig Schläflipioneered this by definingpolyschemes,which mathematicians nowadays callpolytopes.These are the analogues to polygons andpolyhedrain spaces with more dimensions. He further defined theSchläfli symbolnotation 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}.[20]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}.[21]

Other methods also exist for describing polygonal tilings. When the tessellation is made of regular polygons, the most common notation is thevertex configuration,which is simply a list of the number of sides of the polygons around a vertex. The square tiling has a vertex configuration of 4.4.4.4, or 44.The tiling of regular hexagons is noted 6.6.6, or 63.[18]

In mathematics

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Introduction to tessellations

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Further information:Euclidean tilings by convex regular polygons,Uniform tiling,andList of Euclidean uniform tilings

Mathematicians use some technical terms when discussing tilings. Anedgeis the intersection between two bordering tiles; it is often a straight line. Avertexis the point of intersection of three or more bordering tiles. Using these terms, anisogonalorvertex-transitivetiling is a tiling where every vertex point is identical; that is, the arrangement ofpolygonsabout each vertex is the same.[18]Thefundamental regionis a shape such as a rectangle that is repeated to form the tessellation.[22]For example, a regular tessellation of the plane with squares has a meeting offour squares at every vertex.[18]

The sides of the polygons are not necessarily identical to the edges of the tiles. Anedge-to-edge tilingis 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.[18]

Anormal tilingis a tessellation for which every tile istopologicallyequivalent to adisk,the intersection of any two tiles is aconnected setor theempty set,and all tiles areuniformly 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.[23]

An example of a non-edge‑to‑edge tiling: the 15th convexmonohedralpentagonal tiling,discovered in 2015

Amonohedral tilingis a tessellation in which all tiles arecongruent;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; theVoderberg tilinghas a unit tile that is a nonconvexenneagon.[1]TheHirschhorn tiling,published by Michael D. Hirschhorn and D. C. Hunt in 1985, is apentagon tilingusing irregular pentagons: regular pentagons cannot tile the Euclidean plane as theinternal angleof a regular pentagon,⁠3π/5⁠,is not a divisor of 2π.[24][25]

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 thesymmetrygroup of the tiling.[23]If a prototile admits a tiling, but no such tiling is isohedral, then the prototile is called anisohedral and formsanisohedral tilings.

Aregular tessellationis a highlysymmetric,edge-to-edge tiling made up ofregular polygons,all of the same shape. There are only three regular tessellations: those made up ofequilateral triangles,squares,or regularhexagons.All three of these tilings are isogonal and monohedral.[26]

APythagorean tilingis not an edge‑to‑edge tiling.

Asemi-regular (or Archimedean) tessellationuses 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).[27]These can be described by theirvertex configuration;for example, a semi-regular tiling using squares and regular octagons has the vertex configuration 4.82(each vertex has one square and two octagons).[28]Many non-edge-to-edge tilings of the Euclidean plane are possible, including the family ofPythagorean tilings,tessellations that use two (parameterised) sizes of square, each square touching four squares of the other size.[29]Anedge tessellationis 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.[30]

Wallpaper groups

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Main article:Wallpaper group
This tessellated, monohedral street pavement uses curved shapes instead of polygons. It belongs to wallpaper group p3.

Tilings withtranslational symmetryin two independent directions can be categorized bywallpaper groups,of which 17 exist.[31]It has been claimed that all seventeen of these groups are represented in theAlhambrapalace inGranada,Spain.Although this is disputed,[32]the variety and sophistication of the Alhambra tilings have interested modern researchers.[33]Of the three regular tilings two are in thep6mwallpaper group and one is inp4m.Tilings in 2-D with translational symmetry in just one direction may be categorized by the seven frieze groups describing the possiblefrieze patterns.[34]Orbifold notationcan be used to describe wallpaper groups of the Euclidean plane.[35]

Aperiodic tilings

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Main articles:Aperiodic tilingandList of aperiodic sets of tiles
APenrose tiling,with several symmetries, but no periodic repetitions

Penrose tilings,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 ofaperiodic tilings,which use tiles that cannot tessellate periodically. Therecursive processofsubstitution tilingis a method of generating aperiodic tilings. One class that can be generated in this way is therep-tiles;these tilings have unexpectedself-replicatingproperties.[36]Pinwheel tilingsare non-periodic, using a rep-tile construction; the tiles appear in infinitely many orientations.[37]It might be thought that a non-periodic pattern would be entirely without symmetry, but this is not so. Aperiodic tilings, while lacking intranslational 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.[38]A substitution rule, such as can be used to generate Penrose patterns using assemblies of tiles called rhombs, illustrates scaling symmetry.[39]AFibonacci wordcan be used to build an aperiodic tiling, and to studyquasicrystals,which are structures with aperiodic order.[40]

A set of 13Wang tilesthat tile the plane onlyaperiodically

Wang tilesare squares coloured on each edge, and placed so that abutting edges of adjacent tiles have the same colour; hence they are sometimes called Wangdominoes.A suitable set of Wang dominoes can tile the plane, but only aperiodically. This is known because anyTuring machinecan be represented as a set of Wang dominoes that tile the plane if, and only if, the Turing machine does not halt. Since thehalting problemis undecidable, the problem of deciding whether a Wang domino set can tile the plane is also undecidable.[41][42][43][44][45]

RandomTruchet tiling

Truchet tilesare square tiles decorated with patterns so they do not haverotational symmetry;in 1704,Sébastien Truchetused a square tile split into two triangles of contrasting colours. These can tile the plane either periodically or randomly.[46][47]

Aneinstein tileis a single shape that forces aperiodic tiling. The first such tile, dubbed a "hat", was discovered in 2023 by David Smith, a hobbyist mathematician.[48][49]The discovery is under professional review and, upon confirmation, will be credited as solving a longstandingmathematical problem.[50]

Tessellations and colour

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Further information:Four colour theorem
At least seven colors are required if the colours of this tiling are to form a pattern by repeating this rectangle as thefundamental domain;more generally, at leastfour coloursare 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. Thefour colour theoremstates that for every tessellation of a normalEuclidean 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 left.[51]

Tessellations with polygons

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

Any triangle orquadrilateral(evennon-convex) can be used as a prototile to form a monohedral tessellation, often in more than one way. Copies of an arbitraryquadrilateralcan 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 towallpaper group p2.Asfundamental domainwe have the quadrilateral. Equivalently, we can construct aparallelogramsubtended 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.[52]

Tessellation usingTexas-shaped non-convex 12-sided polygons

If only one shape of tile is allowed, tilings exist with convexN-gons forNequal to 3, 4, 5, and 6. ForN= 5,seePentagonal tiling,forN= 6,seeHexagonal tiling,forN= 7,seeHeptagonal tilingand forN= 8,seeoctagonal tiling.

With non-convex polygons, there are far fewer limitations in the number of sides, even if only one shape is allowed.

Polyominoesare examples of tiles that are either convex of non-convex, for which various combinations, rotations, and reflections can be used to tile a plane. For results on tiling the plane withpolyominoes,seePolyomino § Uses of polyominoes.

Voronoi tilings

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AVoronoi tiling,in which the cells are always convex polygons

Voronoi or Dirichlettilings 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.)[53][54]TheVoronoi cellfor each defining point is a convex polygon. TheDelaunay triangulationis a tessellation that is thedual graphof 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.[55]Voronoi tilings with randomly placed points can be used to construct random tilings of the plane.[56]

Tessellations in higher dimensions

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Main article:Honeycomb (geometry)
Tessellating three-dimensional (3-D) space: therhombic dodecahedronis one of the solids that can be stacked tofill space exactly.

Tessellation can be extended to three dimensions. Certainpolyhedracan be stacked in a regularcrystal patternto fill (or tile) three-dimensional space, including thecube(the onlyPlatonic polyhedronto do so), therhombic dodecahedron,thetruncated octahedron,and triangular, quadrilateral, and hexagonalprisms,among others.[57]Any polyhedron that fits this criterion is known as aplesiohedron,and may possess between 4 and 38 faces.[58]Naturally occurring rhombic dodecahedra are found ascrystalsofandradite(a kind ofgarnet) andfluorite.[59][60]

Illustration of a Schmitt–Conway biprism, also called a Schmitt–Conway–Danzer tile

Tessellations in three or more dimensions are calledhoneycombs.In three dimensions there is just one regular honeycomb, which has eight cubes at each polyhedron vertex. Similarly, in three dimensions there is just one quasiregular[c]honeycomb, which has eighttetrahedraand sixoctahedraat each polyhedron vertex. However, there are many possiblesemiregular honeycombsin three dimensions.[61]Uniform honeycombs can be constructed using theWythoff construction.[62]

TheSchmitt-Conway biprismis a convex polyhedron with the property of tiling space only aperiodically.[63]

ASchwarz triangleis aspherical trianglethat can be used to tile asphere.[64]

Tessellations in non-Euclidean geometries

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Rhombitriheptagonal tilingin hyperbolic plane, seen inPoincaré disk modelprojection
The regular{3,5,3} icosahedral honeycomb,one of four regular compact honeycombs inhyperbolic 3-space

It is possible to tessellate innon-Euclideangeometries such ashyperbolic geometry.Auniform tiling in the hyperbolic plane(that may be regular, quasiregular, or semiregular) is an edge-to-edge filling of the hyperbolic plane, withregular polygonsasfaces;these arevertex-transitive(transitiveon itsvertices), and isogonal (there is anisometrymapping any vertex onto any other).[65][66]

Auniform honeycomb in hyperbolic spaceis a uniform tessellation ofuniform polyhedralcells.In three-dimensional (3-D) hyperbolic space there are nineCoxeter groupfamilies of compactconvex uniform honeycombs,generated asWythoff constructions,and represented bypermutationsofringsof theCoxeter diagramsfor each family.[67]

In art

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Further information:Mathematics and art
Romanmosaicfloor panel of stone, tile, and glass, from a villa nearAntiochin Roman Syria. second century AD

In architecture, tessellations have been used to create decorative motifs since ancient times.Mosaictilings often had geometric patterns.[4]Later civilisations also used larger tiles, either plain or individually decorated. Some of the most decorative were theMoorishwall tilings ofIslamic architecture,usingGirihandZelligetiles in buildings such as theAlhambra[68]andLa Mezquita.[69]

Tessellations frequently appeared in the graphic art ofM. C. Escher;he was inspired by the Moorish use of symmetry in places such as the Alhambra when he visitedSpainin 1936.[70]Escher made four "Circle Limit"drawings of tilings that use hyperbolic geometry.[71][72]For hiswoodcut"Circle Limit IV" (1960), Escher prepared a pencil and ink study showing the required geometry.[73]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."[74]

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 interlockingmotifsof patch shapes inquilts.[75][76]

Tessellations are also a main genre inorigami(paper folding), where pleats are used to connect molecules, such as twist folds, together in a repeating fashion.[77]

In manufacturing

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Tessellation is used inmanufacturing industryto reduce the wastage of material (yield losses) such assheet metalwhen cutting out shapes for objects such ascar doorsordrink cans.[78]

Tessellation is apparent in themudcrack-likecrackingofthin films[79][80]– with a degree ofself-organisationbeing observed usingmicroandnanotechnologies.[81]

In nature

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Main article:Patterns in nature § Tessellations
Ahoneycombis a natural tessellated structure.

Thehoneycombis a well-known example of tessellation in nature with its hexagonal cells.[82]

In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including thefritillary,[83]and some species ofColchicum,are characteristically tessellate.[84]

Manypatterns in natureare formed by cracks in sheets of materials. These patterns can be described byGilbert tessellations,[85]also known as random crack networks.[86]The Gilbert tessellation is a mathematical model for the formation ofmudcracks,needle-likecrystals,and similar structures. The model, named afterEdgar 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.[87]Basalticlava flowsoften displaycolumnar jointingas a result ofcontractionforces 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 theGiant's Causewayin Northern Ireland.[88]Tessellated pavement,a characteristic example of which is found atEaglehawk Neckon theTasman PeninsulaofTasmania,is a rare sedimentary rock formation where the rock has fractured into rectangular blocks.[89]

Tessellate pattern in aColchicumflower

Other natural patterns occur infoams;these are packed according toPlateau's laws,which requireminimal surfaces.Such foams present a problem in how to pack cells as tightly as possible: in 1887,Lord Kelvinproposed a packing using only one solid, thebitruncated cubic honeycombwith very slightly curved faces. In 1993, Denis Weaire and Robert Phelan proposed theWeaire–Phelan structure,which uses less surface area to separate cells of equal volume than Kelvin's foam.[90]

In puzzles and recreational mathematics

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Traditionaltangramdissection puzzle
Main articles:Tiling puzzleandrecreational mathematics

Tessellations have given rise to many types oftiling puzzle,from traditionaljigsaw puzzles(with irregular pieces of wood or cardboard)[91]and thetangram,[92]to more modern puzzles that often have a mathematical basis. For example,polyiamondsandpolyominoesare figures of regular triangles and squares, often used in tiling puzzles.[93][94]Authors such asHenry DudeneyandMartin Gardnerhave made many uses of tessellation inrecreational mathematics.For example, Dudeney invented thehinged dissection,[95]while Gardner wrote about the "rep-tile",a shape that can bedissectedinto smaller copies of the same shape.[96][97]Inspired by Gardner's articles inScientific American,the amateur mathematicianMarjorie Ricefound four new tessellations with pentagons.[98][99]Squaring the squareis the problem of tiling an integral square (one whose sides have integer length) using only other integral squares.[100][101]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.[102]

Examples

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See also

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Explanatory footnotes

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  1. ^The mathematical term for identical shapes is "congruent" – in mathematics, "identical" means they are the same tile.
  2. ^The tiles are usually required to behomeomorphic(topologically equivalent) to aclosed disk,which means bizarre shapes with holes, dangling line segments, or infinite areas are excluded.[18]
  3. ^In this context, quasiregular means that the cells are regular (solids), and the vertex figures are semiregular.

References

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