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També existeixen altres mètodes per descriure tessel·lacions poligonals. Quan la tessel·lació està construïda a partir de polígons regulars, la notació més comuna és la [[configuració de vèrtex]], que és simplement una llista del nombre de costats dels polígons al voltant de cada vètex. La [[tessel·lació quadrada]] té una configuració de vèrtex de 4.4.4.4, o 4<sup>4</sup>, mentre que la d'hexàgons regulars és 6.6.6, o 6<sup>3</sup>.<ref name=Grunbaum>{{ref-llibre|cognom=Grünbaum |nom=Branko |títol=Tilings and Patterns |any=1987 |editorial=W. H. Freeman |lloc=New York |autor2=Shephard, G. C. | author2link=Geoffrey Colin Shephard |pàgina=59 |llengua=anglès}}</ref>
També existeixen altres mètodes per descriure tessel·lacions poligonals. Quan la tessel·lació està construïda a partir de polígons regulars, la notació més comuna és la [[configuració de vèrtex]], que és simplement una llista del nombre de costats dels polígons al voltant de cada vètex. La [[tessel·lació quadrada]] té una configuració de vèrtex de 4.4.4.4, o 4<sup>4</sup>, mentre que la d'hexàgons regulars és 6.6.6, o 6<sup>3</sup>.<ref name=Grunbaum>{{ref-llibre|cognom=Grünbaum |nom=Branko |títol=Tilings and Patterns |any=1987 |editorial=W. H. Freeman |lloc=New York |autor2=Shephard, G. C. | author2link=Geoffrey Colin Shephard |pàgina=59 |llengua=anglès}}</ref>

==En matemàtiques==

===Introducció a les tessel·lacions===
{{VT|Tessel·lacions euclidianes de polígons regulars|Tessel·lació regular||Llista de tessel·lacions regulars uniformes}}

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]]s about each vertex is the same.<ref name=Grunbaum/> 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 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]].<ref name=Grunbaum/>

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.<ref name=Grunbaum/>

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 single [[connected set]] or the [[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=9781855734920 |pages=172, 175}}</ref>

[[File:P5-type15-chiral_coloring.png|thumb|left|upright|The 15th convex monohedral [[pentagonal tiling]], discovered in 2015]]

A <span id="Monohedral" style="white-pace:nowrap;font-weight:bold">monohedral tiling</span> 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 | accessdate=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 A |year=1985 | volume=39 | issue=1 | pages=1–18 | doi=10.1016/0097-3165(85)90078-0 | url=http://www.sciencedirect.com/science/article/pii/0097316585900780 | accessdate=29 April 2013}}</ref><ref>{{MathWorld|title=Pentagon Tiling|urlname=PentagonTiling}}</ref>

[[File:2005-06-25 Tiles together.jpg|thumb|upright|A [[Pythagorean tiling]] ]]

An isohedral tiling is a special variation of a monohedral tiling in which all tiles belong to the same transitivity class, that is, all tiles are transforms of the same prototile under the symmetry group of the tiling.<ref name=Horne2000/> If a prototile admits a tiling, but no such tiling is isohedral, then the prototile is called anisohedral and forms [[anisohedral tiling]]s.

A [[Tiling by regular polygons|regular tessellation]] is a highly [[symmetry|symmetric]], edge-to-edge tiling made up of [[regular polygon]]s, all of the same shape. There are only three regular tessellations: those made up of [[equilateral triangle]]s, [[square (geometry)|squares]], or regular [[hexagon]]s. All three of these tilings are isogonal and monohedral.<ref name=threeregular>{{MathWorld|title=Regular Tessellations|id=RegularTessellations}}</ref>

A [[Tiling by regular polygons#Archimedean, uniform or semiregular tilings|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]]; 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 | accessdate=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 | publisher=Penguin Books | location=New York | isbn=0-14-011813-6 | pages=260–261 | contribution=two squares tessellation}}</ref> An [[edge tessellation]] is one in which each tile can be reflected over an edge to take up the position of a neighbouring tile, such as in an array of equilateral or isosceles triangles.<ref>{{cite journal |author=Kirby, Matthew; 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}}</ref>

[[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.]]

===Grups de paper pintat===
{{AP|Grup de paper pintat}}
Tilings with [[translational symmetry]] in two independent directions can be categorized by [[wallpaper group]]s, of which 17 exist.<ref name=armstrong>{{cite book | last=Armstrong | first=M.A. | title=Groups and Symmetry | 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]], [[Spain]]. Though this is disputed,<ref>{{cite journal |last=Grünbaum |first=Branko | journal=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 surprised 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 }}</ref> Of the three regular tilings two are in the ''p6m'' wallpaper group and one is in ''p4m''. Tilings in 2D with translational symmetry in just one direction can be categorized by the seven frieze groups describing the possible [[frieze pattern]]s.<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. | title=Two-Dimensional Symmetry Mutation | url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.8536 | publisher=CiteSeer | accessdate=29 May 2015 | date=1991}}</ref>

===Tessel·lacions aperiòdiques===
{{AP|Tessel·lació aperiòdica}}
[[File:Penrose Tiling (Rhombi).svg|thumb|upright|A [[Penrose tiling]], with several symmetries but no periodic repetitions]]

[[Penrose tiling]]s, which use two different quadrilaterals, are the best known example of tiles that forcibly create non-periodic patterns. They belong to a general class of [[aperiodic 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]]s; these tilings have surprising [[self-replication|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 | url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.44.9723 | doi=10.2307/2118575 | jstor=2118575}}</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 [[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 | accessdate=29 May 2015}}</ref> A substitution rule, such as can be used to generate some 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 | accessdate=29 May 2015}}</ref> A [[Fibonacci word]] can be used to build an aperiodic tiling, and to study [[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=[[Physical Review Letters]] | pages=1761–1765 | title=Raman scattering in Fibonacci superlattices | volume=58 | issue=17 | year=1987 | doi=10.1103/physrevlett.58.1761 | bibcode=1987PhRvL..58.1761D}}</ref>

[[Image:wang tiles.svg|thumb|left|A set of 13 [[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]] 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 | author-link=Hao Wang (academic) | issue=1 | 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 | authorlink=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. | authorlink=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 }}</ref><ref>{{cite journal | last=Culik | first=Karel, II | doi=10.1016/S0012-365X(96)00118-5 | issue=1–3 | journal=[[Discrete Mathematics (journal)|Discrete Mathematics]] | mr=1417576 | pages=245–251 | title=An aperiodic set of 13 Wang tiles | volume=160 | year=1996}}</ref>

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

[[Truchet tiles]] are square tiles decorated with patterns so they do not have [[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 | authorlink=Cyril Stanley Smith | doi=10.2307/1578535 | issue=4 | journal=[[Leonardo (journal)|Leonardo]] | pages=373–385 | title=The tiling patterns of Sebastian Truchet and the topology of structural hierarchy | volume=20 | year=1987| jstor=1578535 }}</ref>

===Tessel·lacions i color===
{{VT|[[Teorema dels quatre colors]]}}

[[File:Torus with seven colours.svg|thumb|right|If the colours of this tiling are to form a pattern by repeating this rectangle as the [[fundamental domain]], at least seven colours are required; 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 which does, it is necessary to treat the colours as part of the tessellation. Here, as many as seven colours may be needed, as in the picture at right.<ref name=Hazewinkel2001>{{springer | title=Four-colour problem|id=p/f040970}}</ref>

===Tessel·lacions amb polígons===
[[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]] (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]] 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]] we have the quadrilateral. Equivalently, we can construct a [[parallelogram]] subtended by a minimal set of translation vectors, starting from a rotational centre. We can divide this by one diagonal, and take one half (a triangle) as fundamental domain. Such a triangle has the same area as the quadrilateral and can be constructed from it by cutting and pasting.<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 | authorlink=Owen Jones (architect) | year=1910 |edition=folio | origyear=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]].

For results on tiling the plane with [[polyomino]]es, see [[Polyomino tiling|Polyomino § Uses of polyominoes]].

===Tessel·lacions de Voronoi===
[[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 |authorlink= Franz Aurenhammer | 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}}</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=0-471-98635-6}}</ref> The ''Voronoi cell'' for each defining point is a convex polygon. The [[Delaunay triangulation]] is a tessellation that is the [[dual graph]] of a Voronoi tessellation. Delaunay triangulations are useful in numerical simulation, in part because among all possible triangulations of the defining points, Delaunay triangulations maximize the minimum of the angles formed by the edges.<ref name=George1998>{{cite book |author1=George, Paul Louis |author2=Borouchaki, Houman | title=Delaunay Triangulation and Meshing: Application to Finite Elements |year=1998 | publisher=Hermes | isbn=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>

[[File:HC R1.png|thumb|upright|Tessellating three-dimensional space: the [[rhombic dodecahedron]] is one of the solids that can be stacked to [[Rhombic dodecahedral honeycomb|fill space exactly]].]]

===Tessel·lacions en dimensions superiors===
{{AP|Enrajolat de l'espai}}
[[File:SCD tile.svg|thumb|upright|left|Illustration of a Schmitt-Conway biprism, also called a Schmitt–Conway–Danzer tile]]

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]] (the only [[Platonic polyhedron]] to do so), the [[rhombic dodecahedron]], the [[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]], 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 [[crystal]]s of [[andradite]] (a kind of [[garnet]]) and [[fluorite]].<ref name="Oldershaw2003">{{cite book |last=Oldershaw | first=Cally | title=Firefly Guide to Gems | url=https://books.google.com/books?id=wm_X_LzoN-cC&pg=PA107 |year=2003 |publisher=Firefly Books | isbn=978-1-55297-814-6 |page=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>

A [[Schwarz triangle]] is a [[spherical triangle]] that can be used to tile a [[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}}</ref>

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://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA3 |date=1995 |publisher=John Wiley & Sons |isbn=978-0-471-01003-6 |page=3 and passim}}</ref> Uniform polyhedra can be constructed using the [[Wythoff construction]].<ref>{{mathworld | urlname=WythoffConstruction | title=Wythoff construction}}</ref>

The 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 |url=https://books.google.com/books?id=LdQ8AAAAIAAJ&pg=PA209 |date=26 September 1996 |publisher=CUP Archive |isbn=978-0-521-57541-6 |page=209}}</ref><!--<ref name="Kennard2015">{{cite book |last=Kennard |first=Fredrick |title=Unsolved Problems in Mathematics |url=https://books.google.com/books?id=OaNsCQAAQBAJ&pg=PA42 |date=21 February 2015|publisher=Lulu.com |isbn=978-1-312-93811-3 |page=42}}</ref> self-published source, possibly usable-->

===Tessel·lacions en geometries no euclidianes===
[[File:Uniform tiling 73-t02.png|thumb|left|upright|[[Rhombitriheptagonal tiling]] in hyperbolic plane, seen in [[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]]. A [[Uniform tilings in hyperbolic plane|uniform tiling in the hyperbolic plane]] (which may be regular, quasiregular or semiregular) is an edge-to-edge filling of the hyperbolic plane, with [[regular polygon]]s as [[Face (geometry)|faces]]; these are [[vertex-transitive]] ([[Transitive group action|transitive]] on its [[vertex (geometry)|vertices]]), and isogonal (there is an [[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 | accessdate=27 May 2015}}</ref>

A [[Uniform honeycombs in hyperbolic space|uniform honeycomb in hyperbolic space]] is a uniform tessellation of [[uniform polyhedron|uniform polyhedral]] [[Cell (geometry)|cells]]. In 3-dimensional hyperbolic space there are nine [[Coxeter group]] families of compact [[convex uniform honeycomb]]s, generated as [[Wythoff construction]]s, and represented by [[permutation]]s of [[Coxeter-Dynkin diagram#Application with uniform polytopes|rings]] of the [[Coxeter diagram]]s for each family.<ref>{{cite book | author=Coxeter, H.S.M. | work=The Beauty of Geometry: Twelve Essays | publisher=Dover Publications | date=1999 | isbn=0-486-40919-8 | title=Chapter 10: Regular honeycombs in hyperbolic space | pages=212–213}}</ref>



== Notes ==
== Notes ==

Revisió del 18:16, 3 jul 2018

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Peces de terracota d'un zellige de Marraquèix que formen diferents tipus de tessel·lacions.

Els termes tessel·lació[1] i tessel·lat[2] fan referència a una regularitat o patró de figures que recobreixen o pavimenten completament una superfície plana de manera que es no queden espais buits ni se superposen les figures (o tessel·les). En matemàtiques, les tessel·lacions es poden generalitzar a dimensions superiors i a una diversitat de geometries.

Una tessel·lació periòdica té un patró que es repeteix. Alguns tipus especials inclouen les tessel·lacions regulars amb tessel·les totes de la mateixa forma de polígons regulars, i tessel·lacions semiregulars amb més d'una forma i amb cada cantonada arranjada idènticament. Els patrons formats per tessel·lacions periòdiques es poden categoritzar en 17 grups de paper pintat. Una tessel·lació sense patró repetitiu s'anomena «no periòdica». Una tessel·lació aperiòdica utilitza un petit conjunt de formes que no poden formar un patró de repetició. En la geometria de dimensions superiors, un enrajolat també s'anomena «tessel·lació de l'espai».

Una tessel·lació real és aquella feta de peces quadrades o hexagonals de ceràmica cimentada. Aquestes tessel·lacions o mosaics poden ser patrons decoratius, o bé poden tenir funcions com ara dotar de durabilitat i resistència a l'aigua un paviment, terra o paret. Històricament, les tessel·lacions foren utilitzades en l'Antiga Roma i en l'art islàmic com, per exemple, en els mosaics decoratius geomètrics de l'Alhambra. En el segle XX, l'obra de M. C. Escher sovint féu ús de les tessel·lacions, tant en la geometria euclidiana ordinària com en la geometria hiperbòlica, per assolir un efecte artístic. Les tessel·lacions formen una classe de patrons a la naturalesa com, per exemple, en les matrius de cel·les hexagonals d'un rusc.

Història

Un mosaic de l'antiga ciutat sumèria d'Uruk IV (3400-3100 aC) que mostra un patró de tessel·lació amb peces acolorides.

Les tessel·lacions foren utilitzades pels sumeris (ca. 4000 aC) per crear decoracions de parets formades per patrons de peces d'argila.[3] En l'antiguitat clàssica també es realitzaren tessel·lats decoratius (mosaics) fets de petits blocs petits anomenats tesserae,[4] els quals de vegades mostren patrons geomètrics.[5][6]

El 1619 Johannes Kepler dugué a terme el primer estudi conegut de les tessel·lacions. Escrigué sobre les tessel·lacions regulars i semiregulars a la seva obra Harmonices Mundi; possiblement fou el primer en explorar i explicar les estructures hexagonals dels ruscs i les volves de neu.[7][8][9]

Mosaic geomètric de l'antiga Roma

Uns dos-cents anys més tard, el 1891, el cristal·lògraf rus Ievgraf Fiódorov provà que tota tessel·lació periòdica en el pla mostra algun dels disset grups diferents d'isometries.[10][11] L'obra de Fyodorov marcà l'inici no oficial de l'estudi matemàtic de les tessel·lacions. Altres contribuïdors prominents inclouen Xúbnikov i Belov (1964),[12] i Heinrich Heesch i Otto Kienzle (1963).[13]

Etimologia

Article principal: Tessel·la

En llatí, tessella és una petita peça cúbica d'argila, roca o vidre utilitzada per fer mosaics.[14] El mot tessella significa 'petit quadrat (de tessera, 'quadrat', que al seu torn prové del grec τέσσερα, 'quatre').

Descripció general

Una tessel·lació rombitrihexagonal: paviment tessel·lat d'una església de Sevilla que utilitza prototessel·les quadrades, triangulars i hexagonals.

El tessel·lat bidimensional és un tema en geometria que tracta de com les formes, conegudes com a tessel·les, poden ser arranjades per emplenar un pla sense deixar cap espai buit seguint una sèrie de regles, les quals poden canviar. Unes regles comunes, per exemple, són que no hi pot haver espais entre les tessel·les i que cap vèrtex d'una d'elles pot sobreposar-se a l'aresta d'una altra.[15] Les tessel·lacions creades en forma d'aparell no obeeixen aquesta regla; d'entre les que sí que l'obeeixen , una tessel·lació regular té tessel·les regulars idèntiques[a] i vèrtexs regulars idèntics, amb el mateix angle entre arestes adjacents de cada tessel·la.[16] Només hi ha tres formes que poden formar tals tessel·lacions: el triangle equilàter, el quadrat, i l'hexàgon regular. Qualsevol d'aquestes tres formes pot ser replicada infinitament per emplenar un pla sense espais buits.[8]

Molts altres tipus de tessel·lacions són possibles sota restriccions diferents. Per exemple, hi ha vuit tipus de tessel·lacions semiregulars, construïdes amb més d'un tipus de polígon regular però amb el mateix arranjament de polígons a cada cantonada.[17] Les tessel·lacions irregulars poden també ser construïdes a partir d'altres formes com pentàgon (geometria)s, poliòminos i, de fet, gairebé qualsevol altra forma geomètrica. L'artista M. C. Escher és famós per les seves tessel·lacions amb peces irregulars amb forma d'animals i altres objectes naturals.[18] Si s'escullen colors amb forts contrastos per les tessel·les de diferent forma, s'obtenen patrons que es poden utilitzar per decorar superíficies físiques tals com el terra d'esglésies.[19]

Tessel·lacions d'un zellige de l'Alhambra

Més formalment, una tessel·lació és un recobriment del pla euclidià d'un nombre comptable de conjunts tancats, anomenats tessel·les, tals que les tessel·les s'intersequen solament en les seves fronteres. Aquestes tessel·les poden ser polígons o qualsevol altra forma.[b] Moltes tessel·lacions es formen a partir d'un nombre finit de prototessel·les en les quals totes les tessel·les de la tessel·lació són congruents amb les prototessel·les donades. Si una forma geomètrica es pot utilitzar com a prototessel·la per rear una tessel·lació, aquesta forma es diu que tessel·la el pla. El criteri de Conway és un conjunt de regles necessàries però no suficients per decidir si una forma donada tessel·la el pla periòdicament sense reflexions: algunes tessel·les fallen el criteri però tot i això tessel·len el pla.[21] No s'ha trobat cap regla general per determinar si una forma donada pot tessel·lar el pla o no, la qual cosa significa que resten molts problemes sense resoldre al voltant de les tessel·lacions.[20]

Matemàticament, les tessel·lacions es poden estendre a espais que no siguin el pla euclidià.[8] El geòmetra suís Ludwig Schläfli fou el pioner en aquest tema i definí els poliesquemes, que els matemàtics actualment anomenen polítops; aquests són els anàlegs dels polígons i políedres en espais de més dimensions. També definí la notació del símbol de Schläfli per facilitar la descripció dels polítops. Per exemple, el símbol de Schläfli del triangle equilàter és {3}, i el del quadrat és {4}.[22] La notació de Schläfli permet descriure les tessel·lacions de manera compacta; per exemple, una tessel·lació d'hexàgons regulars té tres polígons de sis costats a cada vèrtex, per la qual cosa el seu símbol de Schläfli és {6,3}.[23]

També existeixen altres mètodes per descriure tessel·lacions poligonals. Quan la tessel·lació està construïda a partir de polígons regulars, la notació més comuna és la configuració de vèrtex, que és simplement una llista del nombre de costats dels polígons al voltant de cada vètex. La tessel·lació quadrada té una configuració de vèrtex de 4.4.4.4, o 44, mentre que la d'hexàgons regulars és 6.6.6, o 63.[20]

En matemàtiques

Introducció a les tessel·lacions

Vegeu també: Tessel·lacions euclidianes de polígons regulars, Tessel·lació regular, i Llista de tessel·lacions regulars uniformes

Mathematicians use some technical terms when discussing tilings. An edge is the intersection between two bordering tiles; it is often a straight line. A vertex is the point of intersection of three or more bordering tiles. Using these terms, an isogonal or vertex-transitive tiling is a tiling where every vertex point is identical; that is, the arrangement of polygons about each vertex is the same.[20] The fundamental region is a shape such as a rectangle that is repeated to form the tessellation.[24] For example, a regular tessellation of the plane with squares has a meeting of four squares at every vertex.[20]

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.[20]

A normal tiling is a tessellation for which every tile is topologically equivalent to a disk, the intersection of any two tiles is a single connected set or the 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.[25]

The 15th convex monohedral pentagonal tiling, discovered in 2015

A monohedral tiling is a tessellation in which all tiles are congruent; it has only one prototile. A particularly interesting type of monohedral tessellation is the spiral monohedral tiling. The first spiral monohedral tiling was discovered by Heinz Voderberg in 1936; the Voderberg tiling has a unit tile that is a nonconvex enneagon.[3] 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, 3Plantilla:Pi/5, is not a divisor of 2Plantilla:Pi.[26][27][28]

A Pythagorean tiling

An isohedral tiling is a special variation of a monohedral tiling in which all tiles belong to the same transitivity class, that is, all tiles are transforms of the same prototile under the symmetry group of the tiling.[25] If a prototile admits a tiling, but no such tiling is isohedral, then the prototile is called anisohedral and forms anisohedral tilings.

A regular tessellation is a highly symmetric, edge-to-edge tiling made up of regular polygons, all of the same shape. There are only three regular tessellations: those made up of equilateral triangles, squares, or regular hexagons. All three of these tilings are isogonal and monohedral.[29]

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).[30] These can be described by their vertex configuration; for example, a semi-regular tiling using squares and regular octagons has the vertex configuration 4.82 (each vertex has one square and two octagons).[31] Many non-edge-to-edge tilings of the Euclidean plane are possible, including the family of Pythagorean tilings, tessellations that use two (parameterised) sizes of square, each square touching four squares of the other size.[32] An edge tessellation is one in which each tile can be reflected over an edge to take up the position of a neighbouring tile, such as in an array of equilateral or isosceles triangles.[33]

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

Grups de paper pintat

Article principal: Grup de paper pintat

Tilings with translational symmetry in two independent directions can be categorized by wallpaper groups, of which 17 exist.[34] It has been claimed that all seventeen of these groups are represented in the Alhambra palace in Granada, Spain. Though this is disputed,[35] the variety and sophistication of the Alhambra tilings have surprised modern researchers.[36] Of the three regular tilings two are in the p6m wallpaper group and one is in p4m. Tilings in 2D with translational symmetry in just one direction can be categorized by the seven frieze groups describing the possible frieze patterns.[37] Orbifold notation can be used to describe wallpaper groups of the Euclidean plane.[38]

Tessel·lacions aperiòdiques

Article principal: Tessel·lació aperiòdica
A Penrose tiling, with several symmetries but no periodic repetitions

Penrose tilings, which use two different quadrilaterals, are the best known example of tiles that forcibly create non-periodic patterns. They belong to a general class of aperiodic tilings, which use tiles that cannot tessellate periodically. The recursive process of substitution tiling is a method of generating aperiodic tilings. One class that can be generated in this way is the rep-tiles; these tilings have surprising self-replicating properties.[39] Pinwheel tilings are non-periodic, using a rep-tile construction; the tiles appear in infinitely many orientations.[40] It might be thought that a non-periodic pattern would be entirely without symmetry, but this is not so. Aperiodic tilings, while lacking in translational symmetry, 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.[41] A substitution rule, such as can be used to generate some Penrose patterns using assemblies of tiles called rhombs, illustrates scaling symmetry.[42] A Fibonacci word can be used to build an aperiodic tiling, and to study quasicrystals, which are structures with aperiodic order.[43]

A set of 13 Wang tiles that tile the plane only aperiodically

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 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 is undecidable, the problem of deciding whether a Wang domino set can tile the plane is also undecidable.[44][45][46][47][48]

Random Truchet tiling

Truchet tiles are square tiles decorated with patterns so they do not have 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.[49][50]

Tessel·lacions i color

Vegeu també: [[:Teorema dels quatre colors]]
If the colours of this tiling are to form a pattern by repeating this rectangle as the fundamental domain, at least seven colours are required; more generally, at least 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 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 which does, it is necessary to treat the colours as part of the tessellation. Here, as many as seven colours may be needed, as in the picture at right.[51]

Tessel·lacions amb polígons

A 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 (even non-convex) can be used as a prototile to form a monohedral tessellation, often in more than one way. Copies of an arbitrary quadrilateral can form a tessellation with translational symmetry and 2-fold rotational symmetry with centres at the midpoints of all sides. For an asymmetric quadrilateral this tiling belongs to wallpaper group p2. As fundamental domain we have the quadrilateral. Equivalently, we can construct a parallelogram subtended by a minimal set of translation vectors, starting from a rotational centre. We can divide this by one diagonal, and take one half (a triangle) as fundamental domain. Such a triangle has the same area as the quadrilateral and can be constructed from it by cutting and pasting.[52]

If only one shape of tile is allowed, tilings exists with convex N-gons for N equal to 3, 4, 5 and 6. For N = 5, see Pentagonal tiling, for N = 6, see Hexagonal tiling,for N = 7, see Heptagonal tiling and for N = 8, see octagonal tiling.

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

Tessel·lacions de Voronoi

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] The Voronoi cell for each defining point is a convex polygon. The Delaunay triangulation is a tessellation that is the dual graph of a Voronoi tessellation. Delaunay triangulations are useful in numerical simulation, in part because among all possible triangulations of the defining points, Delaunay triangulations maximize the minimum of the angles formed by the edges.[55] Voronoi tilings with randomly placed points can be used to construct random tilings of the plane.[56]

Tessellating three-dimensional space: the rhombic dodecahedron is one of the solids that can be stacked to fill space exactly.

Tessel·lacions en dimensions superiors

Article principal: Enrajolat de l'espai
Illustration of a Schmitt-Conway biprism, also called a Schmitt–Conway–Danzer tile

Tessellation can be extended to three dimensions. Certain polyhedra can be stacked in a regular crystal pattern to fill (or tile) three-dimensional space, including the cube (the only Platonic polyhedron to do so), the rhombic dodecahedron, the truncated octahedron, and triangular, quadrilateral, and hexagonal prisms, among others.[57] Any polyhedron that fits this criterion is known as a plesiohedron, and may possess between 4 and 38 faces.[58] Naturally occurring rhombic dodecahedra are found as crystals of andradite (a kind of garnet) and fluorite.[59][60]

A Schwarz triangle is a spherical triangle that can be used to tile a sphere.[61]

Tessellations in three or more dimensions are called honeycombs. In three dimensions there is just one regular honeycomb, which has eight cubes at each polyhedron vertex. Similarly, in three dimensions there is just one quasiregular[c] honeycomb, which has eight tetrahedra and six octahedra at each polyhedron vertex. However, there are many possible semiregular honeycombs in three dimensions.[62] Uniform polyhedra can be constructed using the Wythoff construction.[63]

The Schmitt-Conway biprism is a convex polyhedron with the property of tiling space only aperiodically.[64]

Tessel·lacions en geometries no euclidianes

Rhombitriheptagonal tiling in hyperbolic plane, seen in Poincaré disk model projection
The regular {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. A uniform tiling in the hyperbolic plane (which may be regular, quasiregular or semiregular) is an edge-to-edge filling of the hyperbolic plane, with regular polygons as faces; these are vertex-transitive (transitive on its vertices), and isogonal (there is an isometry mapping any vertex onto any other).[65][66]

A uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.[67]


Notes

  1. ↑ El terme matemàtic per formes idèntiques és "congruent"; en matemàtiques, "idèntic" significa que són la mateixa tessel·la.
  2. ↑ Normalment es requereix que les tessel·les siguin homeomorfes (topològicament equivalents) a un disc tancat, la qual cosa significa que s'exclouen formes estranyes com, per exemple, amb forats o àrees infinites.[20]
  3. ↑ In this context, quasiregular means that the cells are regular (solids), and the vertex figures are semiregular.

Referències

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  2. ↑ «tessel·lat». Gran Diccionari de la Llengua Catalana. Barcelona: Grup Enciclopèdia Catalana.
  3. ↑ 3,0 3,1 Pickover, Clifford A. The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics (en anglès). Sterling, 2009, p. 372. ISBN 9781402757969. 
  4. ↑ Dunbabin, Katherine M. D.. Mosaics of the Greek and Roman world (en anglès). Cambridge University Press, 2006, p. 280. 
  5. ↑ «The Brantingham Geometric Mosaics» (en anglès). Hull City Council, 2008.
  6. ↑ Error de citació: Etiqueta <ref> no vàlida; no s'ha proporcionat text per les refs nomenades Field1988
  7. ↑ Kepler, Johannes. Harmonices Mundi (en anglès), 1619. 
  8. ↑ 8,0 8,1 8,2 Gullberg, 1997, p. 395.
  9. ↑ Stewart, 2001, p. 13.
  10. ↑ ; Potkonjak, Miodrag«Dynamic Coverage Problems in Sensor Networks» (en anglès) p. 2. Los Alamos National Laboratory, 2012.
  11. ↑ Fyodorov, Y. «Simmetrija na ploskosti [Simetria en el pla]» (en rus). Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva [Proceedings of the Imperial St. Petersburg Mineralogical Society], series 2, 28, 1891, pàg. 245–291.
  12. ↑ Shubnikov, Alekseĭ Vasilʹevich; Belov, Nikolaĭ Vasilʹevich. Colored Symmetry (en anglès). Macmillan, 1964. 
  13. ↑ Heesch, H.; Kienzle, O. Flächenschluss: System der Formen lückenlos aneinanderschliessender Flächteile (en alemany). Springer, 1963. 
  14. ↑ «Tessellate» (en anglès). Merriam-Webster Online.
  15. ↑ Conway, R.; Burgiel, H.; Goodman-Strauss, G. The Symmetries of Things. Peters, 2008. 
  16. ↑ Coxeter, H. S. M.. Regular Polytopes. 3rd. Dover, 1973. 
  17. ↑ Cundy and Rollett. Mathematical Models. 2a. Oxford, 1961, p. 61–62. 
  18. ↑ Escher, 1974, p. 11–12, 15–16.
  19. ↑ «Basilica di San Marco». Secció: Tessellated floor. Basilica di San Marco.
  20. ↑ 20,0 20,1 20,2 20,3 20,4 20,5 Grünbaum, Branko; Shephard, G. C.. Tilings and Patterns (en anglès). New York: W. H. Freeman, 1987, p. 59. 
  21. ↑ Schattschneider, Doris (setembre 1980). «Will It Tile? Try the Conway Criterion!». Mathematics Magazine 53 (4): 224–233. doi:10.2307/2689617.  Paràmetre desconegut |llengua= ignorat (ajuda);
  22. ↑ Coxeter, H. S. M.. Regular Polytopes (en anglès). Methuen, 1948, p. 14, 69, 149. ISBN 9780486614809. 
  23. ↑ Weisstein, Eric W., «Tessellation» a MathWorld (en anglès).
  24. ↑ Emmer, Michele; Schattschneider, Doris. M.C. Escher's Legacy: A Centennial Celebration. Springer Berlin Heidelberg, 8 May 2007, p. 325. ISBN 978-3-540-28849-7. 
  25. ↑ 25,0 25,1 Horne, Clare E. Geometric Symmetry in Patterns and Tilings. Woodhead Publishing, 2000, p. 172, 175. ISBN 9781855734920. 
  26. ↑ Dutch, Steven. «Some Special Radial and Spiral Tilings». University of Wisconsin, 29-07-1999. [Consulta: 6 abril 2013].
  27. ↑ Hirschhorn, M. D. «Equilateral convex pentagons which tile the plane». Journal of Combinatorial Theory, Series A, vol. 39, 1, 1985, pàg. 1–18. DOI: 10.1016/0097-3165(85)90078-0 [Consulta: 29 abril 2013].
  28. ↑ Weisstein, Eric W., «Pentagon Tiling» a MathWorld (en anglès).
  29. ↑ Weisstein, Eric W., «Regular Tessellations» a MathWorld (en anglès).
  30. ↑ Stewart, 2001, p. 75.
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  33. ↑ Kirby, Matthew; Umble, Ronald «Edge Tessellations and Stamp Folding Puzzles». Mathematics Magazine, vol. 84, 4, 2011, pàg. 283–89. DOI: 10.4169/math.mag.84.4.283.
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  35. ↑ Grünbaum, Branko «What symmetry groups are present in the Alhambra?». Notices of the American Mathematical Society, vol. 53, 6, June–July 2006, pàg. 670–673.
  36. ↑ Lu, Peter J. «Decagonal and quasi-crystalline tilings in medieval Islamic architecture». Science, vol. 315, 5815, 23-02-2007, pàg. 1106–10. Bibcode: 2007Sci...315.1106L. DOI: 10.1126/science.1135491. PMID: 17322056.
  37. ↑ Weisstein, Eric W., «Frieze Group» a MathWorld (en anglès).
  38. ↑ Huson, Daniel H. «Two-Dimensional Symmetry Mutation». CiteSeer, 1991. [Consulta: 29 maig 2015].
  39. ↑ Gardner, 1989, p. 1-18.
  40. ↑ Radin, C. «The Pinwheel Tilings of the Plane». Annals of Mathematics, vol. 139, 3, May 1994, pàg. 661–702. DOI: 10.2307/2118575. JSTOR: 2118575.
  41. ↑ Austin, David. «Penrose Tiles Talk Across Miles». American Mathematical Society. [Consulta: 29 maig 2015].
  42. ↑ Harriss, E. O. «Aperiodic Tiling». University of London and EPSRC. [Consulta: 29 maig 2015].
  43. ↑ Dharma-wardana, M. W. C.; MacDonald, A. H.; Lockwood, D. J.; Baribeau, J.-M.; Houghton, D. C. «Raman scattering in Fibonacci superlattices». Physical Review Letters, vol. 58, 17, 1987, pàg. 1761–1765. Bibcode: 1987PhRvL..58.1761D. DOI: 10.1103/physrevlett.58.1761.
  44. ↑ Wang, Hao «Proving theorems by pattern recognition—II». Bell System Technical Journal, vol. 40, 1, 1961, pàg. 1–41. DOI: 10.1002/j.1538-7305.1961.tb03975.x.
  45. ↑ Wang, Hao (November 1965). «Games, logic and computers». Scientific American: 98–106. 
  46. ↑ Berger, Robert «The undecidability of the domino problem». Memoirs of the American Mathematical Society, vol. 66, 66, 1966, pàg. 72. DOI: 10.1090/memo/0066.
  47. ↑ Robinson, Raphael M. «Undecidability and nonperiodicity for tilings of the plane». Inventiones Mathematicae, vol. 12, 3, 1971, pàg. 177–209. Bibcode: 1971InMat..12..177R. DOI: 10.1007/bf01418780.
  48. ↑ Culik, Karel, II «An aperiodic set of 13 Wang tiles». Discrete Mathematics, vol. 160, 1–3, 1996, pàg. 245–251. DOI: 10.1016/S0012-365X(96)00118-5.
  49. ↑ Browne, Cameron «Truchet curves and surfaces». Computers & Graphics, vol. 32, 2, 2008, pàg. 268–281. DOI: 10.1016/j.cag.2007.10.001.
  50. ↑ Smith, Cyril Stanley «The tiling patterns of Sebastian Truchet and the topology of structural hierarchy». Leonardo, vol. 20, 4, 1987, pàg. 373–385. DOI: 10.2307/1578535. JSTOR: 1578535.
  51. ↑ Michiel Hazewinkel (ed.). Four-colour problem. Encyclopedia of Mathematics (en anglès). Springer, 2001. ISBN 978-1-55608-010-4. 
  52. ↑ Jones, Owen [1856]. The Grammar of Ornament. folio. Bernard Quaritch, 1910. 
  53. ↑ Aurenhammer, Franz «Voronoi Diagrams – A Survey of a Fundamental Geometric Data Structure». ACM Computing Surveys, vol. 23, 3, 1991, pàg. 345–405. DOI: 10.1145/116873.116880.
  54. ↑ Okabe, Atsuyuki. Spatial Tessellations – Concepts and Applications of Voronoi Diagrams. 2nd. John Wiley, 2000. ISBN 0-471-98635-6. 
  55. ↑ George, Paul Louis. Delaunay Triangulation and Meshing: Application to Finite Elements. Hermes, 1998, p. 34–35. ISBN 2-86601-692-0. 
  56. ↑ Moller, Jesper. Lectures on Random Voronoi Tessellations. Springer, 1994. ISBN 978-1-4612-2652-9. 
  57. ↑ Grünbaum, Branko «Uniform tilings of 3-space». Geombinatorics, vol. 4, 2, 1994, pàg. 49–56.
  58. ↑ Engel, Peter «Über Wirkungsbereichsteilungen von kubischer Symmetrie». Zeitschrift für Kristallographie, Kristallgeometrie, Kristallphysik, Kristallchemie, vol. 154, 3–4, 1981, pàg. 199–215. Bibcode: 1981ZK....154..199E. DOI: 10.1524/zkri.1981.154.3-4.199..
  59. ↑ Oldershaw, Cally. Firefly Guide to Gems. Firefly Books, 2003, p. 107. ISBN 978-1-55297-814-6. 
  60. ↑ Kirkaldy, J. F.. Minerals and Rocks in Colour. 2nd. Blandford, 1968, p. 138–139. 
  61. ↑ Schwarz, H. A. «Ueber diejenigen Fälle in welchen die Gaussichen hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt». Journal für die reine und angewandte Mathematik, vol. 1873, 75, 1873, pàg. 292–335. DOI: 10.1515/crll.1873.75.292. ISSN: 0075-4102.
  62. ↑ Coxeter, Harold Scott Macdonald; Sherk, F. Arthur. Kaleidoscopes: Selected Writings of H.S.M. Coxeter. John Wiley & Sons, 1995, p. 3 and passim. ISBN 978-0-471-01003-6. 
  63. ↑ Weisstein, Eric W., «Wythoff construction» a MathWorld (en anglès).
  64. ↑ Senechal, Marjorie. Quasicrystals and Geometry. CUP Archive, 26 September 1996, p. 209. ISBN 978-0-521-57541-6. 
  65. ↑ Coordinates for a new triangular tiling of the hyperbolic plane. 
  66. ↑ Zadnik, Gašper. «Tiling the Hyperbolic Plane with Regular Polygons». Wolfram. [Consulta: 27 maig 2015].
  67. ↑ Coxeter, H.S.M.. Chapter 10: Regular honeycombs in hyperbolic space. Dover Publications, 1999, p. 212–213. ISBN 0-486-40919-8. 
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