Atessellation ortiling is the covering of asurface, often aplane, using one or moregeometric shapes, calledtiles, with no overlaps and no gaps. Inmathematics, tessellation can be generalized tohigher dimensions and a variety of geometries.
Aperiodic tiling has a repeating pattern. Some special kinds includeregular tilings withregular polygonal tiles all of the same shape, andsemiregular 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 17wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". Anaperiodic tiling uses 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 ascementedceramic squares 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 Rome and inIslamic art such as in theMoroccan architecture anddecorative geometric tiling of theAlhambra palace. In the twentieth century, the work ofM. C. Escher often made use of tessellations, both in ordinaryEuclidean geometry and 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 cells found inhoneycombs.
Tessellations were used by theSumerians (about 4000 BC) in building wall decorations formed by patterns of clay tiles.[1]
Decorativemosaic tilings made of small squared blocks calledtesserae were widely employed inclassical antiquity,[2] sometimes displaying geometric patterns.[3][4]
In 1619,Johannes Kepler made 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]
Some two hundred years later in 1891, the Russian crystallographerYevgraf Fyodorov proved 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 Shubnikov andNikolai Belov in their bookColored Symmetry (1964),[10] andHeinrich Heesch and Otto Kienzle (1963).[11]
In Latin,tessella is a small cubical piece ofclay,stone, orglass used 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 ofglazed clay.
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 brickwork do not obey this rule. Among those that do, aregular tessellation has both identical[a]regular tiles and 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,square and the regularhexagon. Any one of these three shapes can be duplicated infinitely to fill aplane with 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,polyominoes and in fact almost any kind of geometric shape. The artistM. C. Escher is 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]
More formally, a tessellation or tiling is acover of the Euclidean plane by acountable number 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 ofprototiles in which all tiles in the tessellation arecongruent to the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said totessellate or totile the plane. TheConway 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.[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 Schläfli pioneered this by definingpolyschemes, which mathematicians nowadays callpolytopes. These are the analogues to polygons andpolyhedra in spaces with more dimensions. He further defined theSchläfli 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}.[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]
Mathematicians use some technical terms when discussing tilings. Anedge is the intersection between two bordering tiles; it is often a straight line. Avertex is the point of intersection of three or more bordering tiles. Using these terms, anisogonal orvertex-transitive tiling is a tiling where every vertex point is identical; that is, the arrangement ofpolygons about each vertex is the same.[18] Thefundamental region is 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 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.[18]
Anormal tiling is a tessellation for which every tile istopologically equivalent to adisk, the intersection of any two tiles is aconnected set or 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]
Amonohedral tiling is 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 tiling has a unit tile that is a nonconvexenneagon.[1] TheHirschhorn tiling, published by Michael D. Hirschhorn and D. C. Hunt in 1985, is apentagon tiling using irregular pentagons: regular pentagons cannot tile the Euclidean plane as theinternal angle of 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 thesymmetry group 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 tessellation is 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]
Asemi-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).[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 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.[30]
Tilings withtranslational symmetry in 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 theAlhambra palace 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 thep6m wallpaper 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 notation can be used to describe wallpaper groups of the Euclidean plane.[35]
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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 process ofsubstitution tiling is a method of generating aperiodic tilings. One class that can be generated in this way is therep-tiles; these tilings have unexpectedself-replicating properties.[36]Pinwheel tilings are 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 word can be used to build an aperiodic tiling, and to studyquasicrystals, which are structures with aperiodic order.[40]
Wang tiles are 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 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 thehalting problem is undecidable, the problem of deciding whether a Wang domino set can tile the plane is also undecidable.[41][42][43][44][45]
Truchet tiles are square tiles decorated with patterns so they do not haverotational 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.[46][47]
Aneinstein 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.[48][49] The discovery is under professional review and, upon confirmation, will be credited as solving a longstandingmathematical problem.[50]
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 theorem states 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]
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 arbitraryquadrilateral 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 towallpaper group p2. Asfundamental domain we have the quadrilateral. Equivalently, we can construct aparallelogram 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.[52]
If only one shape of tile is allowed, tilings exist with convexN-gons forN equal to 3, 4, 5, and 6. ForN = 5, seePentagonal tiling, forN = 6, seeHexagonal tiling, forN = 7, seeHeptagonal tiling and 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.
Polyominoes are 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 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.)[53][54] TheVoronoi cell for each defining point is a convex polygon. TheDelaunay triangulation is a tessellation that is thedual 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.[55] Voronoi tilings with randomly placed points can be used to construct random tilings of the plane.[56]
Tessellation can be extended to three dimensions. Certainpolyhedra can be stacked in a regularcrystal pattern to fill (or tile) three-dimensional space, including thecube (the onlyPlatonic polyhedron to 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 ascrystals ofandradite (a kind ofgarnet) andfluorite.[59][60]
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 eighttetrahedra and sixoctahedra at each polyhedron vertex. However, there are many possiblesemiregular honeycombs in three dimensions.[61] Uniform honeycombs can be constructed using theWythoff construction.[62]
TheSchmitt-Conway biprism is a convex polyhedron with the property of tiling space only aperiodically.[63]
ASchwarz triangle is aspherical triangle that can be used to tile asphere.[64]
It is possible to tessellate innon-Euclidean geometries 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 polygons asfaces; these arevertex-transitive (transitive on itsvertices), and isogonal (there is anisometry mapping any vertex onto any other).[65][66]
Auniform honeycomb in hyperbolic space is a uniform tessellation ofuniform polyhedralcells. In three-dimensional (3-D) hyperbolic space there are nineCoxeter group families of compactconvex uniform honeycombs, generated asWythoff constructions, and represented bypermutations ofrings of theCoxeter diagrams for each family.[67]
In architecture, tessellations have been used to create decorative motifs since ancient times.Mosaic tilings often had geometric patterns.[4] Later civilisations also used larger tiles, either plain or individually decorated. Some of the most decorative were theMoorish wall tilings ofIslamic architecture, usingGirih andZellige tiles 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 visitedSpain in 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]
Tessellated designs often appear on textiles, whether woven, stitched in, or printed. Tessellation patterns have been used to design interlockingmotifs of 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]
Tessellation is used inmanufacturing industry to reduce the wastage of material (yield losses) such assheet metal when cutting out shapes for objects such ascar doors ordrink cans.[78]
Tessellation is apparent in themudcrack-likecracking ofthin films[79][80] – with a degree ofself-organisation being observed usingmicro andnanotechnologies.[81]
Thehoneycomb is 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 nature are 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 flows often displaycolumnar jointing as a result ofcontraction 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 theGiant's Causeway in Northern Ireland.[88]Tessellated pavement, a characteristic example of which is found atEaglehawk Neck on theTasman Peninsula ofTasmania, is a rare sedimentary rock formation where the rock has fractured into rectangular blocks.[89]
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 Kelvin proposed a packing using only one solid, thebitruncated cubic honeycomb with 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]
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,polyiamonds andpolyominoes are figures of regular triangles and squares, often used in tiling puzzles.[93][94] Authors such asHenry Dudeney andMartin Gardner have 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 bedissected into smaller copies of the same shape.[96][97] Inspired by Gardner's articles inScientific American, the amateur mathematicianMarjorie Rice found four new tessellations with pentagons.[98][99]Squaring the square is 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]
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
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