Contents 1 History 1.1 Etymology 2 Overview 3 In mathematics 3.1 Introduction to tessellations 3.2 Wallpaper groups 3.3 Aperiodic tilings 3.4 Tessellations and colour 3.5 Tessellations with polygons 3.6 Voronoi tilings 3.7 Tessellations in higher dimensions 3.8 Tessellations in non-Euclidean geometries 4 In art 5 In manufacturing 6 In nature 7 In puzzles and recreational mathematics 8 Examples 9 Footnotes 10 References 11 Sources 12 External links

History[edit] A temple mosaic from the ancient Sumerian city of Uruk IV (3400–3100 BC), showing a tessellation pattern in coloured tiles Tessellations were used by the Sumerians (about 4000 BC) in building wall decorations formed by patterns of clay tiles.[1] Decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical 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 his Harmonices Mundi; he was possibly the first to explore and to explain the hexagonal structures of honeycomb and snowflakes.[5][6][7] Roman geometric mosaic Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries.[8][9] Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov (1964),[10] and Heinrich Heesch and Otto Kienzle (1963).[11] Etymology[edit] In Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics.[12] The word "tessella" means "small square" (from tessera, square, which in turn is from the Greek word τέσσερα for four). It corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay.

Overview[edit] A rhombitrihexagonal tiling: tiled floor of a church in Seville, Spain, using square, triangle and hexagon prototiles Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules. These rules can be varied. Common ones are that there must be no gaps between tiles, and that no corner of one tile can lie along the edge of another.[13] The tessellations created by bonded brickwork do not obey this rule. Among those that do, a regular 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 equilateral triangle, square, and regular hexagon. Any one of these three shapes can be duplicated infinitely to fill a plane 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 as pentagons, polyominoes and in fact almost any kind of geometric shape. The artist M. C. Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects.[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] Elaborate and colourful zellige tessellations of glazed tiles at the Alhambra in Spain More formally, a tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles, such that the tiles intersect only on their boundaries. These tiles may be polygons or any other shapes.[b] Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are congruent to the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane. The Conway criterion is a sufficient but not necessary set of rules for deciding if 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 if 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] The Swiss geometer Ludwig Schläfli pioneered this by defining polyschemes, which mathematicians nowadays call polytopes. These are the analogues to polygons and polyhedra in spaces with more dimensions. He further defined the Schläfli symbol notation to make it easy to describe polytopes. For example, the Schläfli symbol for an equilateral triangle is {3}, while that for a square is {4}.[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 the vertex configuration, which is simply a list of the number of sides of the polygons around a vertex. The square tiling has a vertex configuration of, or 44. The tiling of regular hexagons is noted 6.6.6, or 63.[18]

In mathematics[edit] Introduction to tessellations[edit] Further information: Euclidean tilings of regular polygons, Uniform tiling, and List of convex uniform tilings 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.[18] The fundamental 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 of four squares at every vertex.[18] 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.[18] 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.[23] 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.[1] 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, 3π/5, is not a divisor of 2π.[24][25][26] 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.[23] 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.[27] 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).[28] 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).[29] 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.[30] 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.[31] This tessellated, monohedral street pavement uses curved shapes instead of polygons. It belongs to wallpaper group p3. Wallpaper groups[edit] Main article: Wallpaper group Tilings with translational symmetry in two independent directions can be categorized by wallpaper groups, of which 17 exist.[32] It has been claimed that all seventeen of these groups are represented in the Alhambra palace in Granada, Spain. Though this is disputed,[33] the variety and sophistication of the Alhambra tilings have surprised modern researchers.[34] 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.[35] Orbifold notation can be used to describe wallpaper groups of the Euclidean plane.[36] Aperiodic tilings[edit] Main articles: Aperiodic tiling and List of aperiodic sets of tiles 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.[37] Pinwheel tilings are non-periodic, using a rep-tile construction; the tiles appear in infinitely many orientations.[38] 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.[39] A substitution rule, such as can be used to generate some Penrose patterns using assemblies of tiles called rhombs, illustrates scaling symmetry.[40] A Fibonacci word can be used to build an aperiodic tiling, and to study quasicrystals, which are structures with aperiodic order.[41] 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.[42][43][44][45][46] 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.[47][48] Tessellations and colour[edit] Further information: four colour theorem 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.[49] Tessellations with polygons[edit] 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.[50] 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 and for N = 6, see Hexagonal tiling. For results on tiling the plane with polyominoes, see Polyomino § Uses of polyominoes. Voronoi tilings[edit] 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.)[51][52] 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.[53] Voronoi tilings with randomly placed points can be used to construct random tilings of the plane.[54] Tessellating three-dimensional space: the rhombic dodecahedron is one of the solids that can be stacked to fill space exactly. Tessellations in higher dimensions[edit] Main article: Honeycomb (geometry) 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.[55] Any polyhedron that fits this criterion is known as a plesiohedron, and may possess between 4 and 38 faces.[56] Naturally occurring rhombic dodecahedra are found as crystals of andradite (a kind of garnet) and fluorite.[57][58] A Schwarz triangle is a spherical triangle that can be used to tile a sphere.[59] 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.[60] Uniform polyhedra can be constructed using the Wythoff construction.[61] The Schmitt-Conway biprism is a convex polyhedron with the property of tiling space only aperiodically.[62] Tessellations in non-Euclidean geometries[edit] 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).[63][64] 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.[65]

In art[edit] Further information: Mathematics and art A quilt showing a regular tessellation pattern. Roman mosaic floor panel of stone, tile and glass, from a villa near Antioch in Roman Syria. 2nd century AD 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 the Moorish wall tilings of Islamic architecture, using Girih and Zellige tiles in buildings such as the Alhambra[66] and La Mezquita.[67] Tessellations frequently appeared in the graphic art of M. C. Escher; he was inspired by the Moorish use of symmetry in places such as the Alhambra when he visited Spain in 1936.[68] Escher made four "Circle Limit" drawings of tilings that use hyperbolic geometry.[69][70] For his woodcut "Circle Limit IV" (1960), Escher prepared a pencil and ink study showing the required geometry.[71] 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."[72] Tessellated designs often appear on textiles, whether woven, stitched in or printed. Tessellation patterns have been used to design interlocking motifs of patch shapes in quilts.[73][74] Tessellations are also a main genre in origami (paper folding), where pleats are used to connect molecules such as twist folds together in a repeating fashion.[75]

In manufacturing[edit] Tessellation is used in manufacturing industry to reduce the wastage of material (yield losses) such as sheet metal when cutting out shapes for objects like car doors or drinks cans.[76] Tessellation is apparent in the mudcrack-like cracking of thin films[77][78] – with a degree of self-organisation being observed using micro and nanotechnologies.[79]

In nature[edit] Tessellate pattern in a Colchicum flower Main article: Patterns in nature The honeycomb provides a well-known example of tessellation in nature with its hexagonal cells.[80] In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including the fritillary[81] and some species of Colchicum are characteristically tessellate.[82] Many patterns in nature are formed by cracks in sheets of materials. These patterns can be described by Gilbert tessellations,[83] also known as random crack networks.[84] The Gilbert tessellation is a mathematical model for the formation of mudcracks, needle-like crystals, and similar structures. The model, named after Edgar Gilbert, allows cracks to form starting from randomly scattered over the plane; each crack propagates in two opposite directions along a line through the initiation point, its slope chosen at random, creating a tessellation of irregular convex polygons.[85] Basaltic lava flows often display columnar jointing as a result of contraction forces causing cracks as the lava cools. The extensive crack networks that develop often produce hexagonal columns of lava. One example of such an array of columns is the Giant's Causeway in Northern Ireland.[86] Tessellated pavement, a characteristic example of which is found at Eaglehawk Neck on the Tasman Peninsula of Tasmania, is a rare sedimentary rock formation where the rock has fractured into rectangular blocks.[87] Other natural patterns occur in foams; these are packed according to Plateau's laws, which require minimal 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, the bitruncated cubic honeycomb with very slightly curved faces. In 1993, Denis Weaire and Robert Phelan proposed the Weaire–Phelan structure, which uses less surface area to separate cells of equal volume than Kelvin's foam.[88]

In puzzles and recreational mathematics[edit] Traditional tangram dissection puzzle Main articles: Tiling puzzle and recreational mathematics Tessellations have given rise to many types of tiling puzzle, from traditional jigsaw puzzles (with irregular pieces of wood or cardboard)[89] and the tangram[90] to more modern puzzles which often have a mathematical basis. For example, polyiamonds and polyominoes are figures of regular triangles and squares, often used in tiling puzzles.[91][92] Authors such as Henry Dudeney and Martin Gardner have made many uses of tessellation in recreational mathematics. For example, Dudeney invented the hinged dissection,[93] while Gardner wrote about the rep-tile, a shape that can be dissected into smaller copies of the same shape.[94][95] Inspired by Gardner's articles in Scientific American, the amateur mathematician Marjorie Rice found four new tessellations with pentagons.[96][97] Squaring the square is the problem of tiling an integral square (one whose sides have integer length) using only other integral squares.[98][99] 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.[100]

Examples[edit] Triangular tiling, one of the three regular tilings of the plane. Snub hexagonal tiling, a semiregular tiling of the plane Floret pentagonal tiling, dual to a semiregular tiling and one of 15 monohedral pentagon tilings. A honeycomb is a natural tessellated structure. The Voderberg tiling, a spiral, monohedral tiling made of enneagons. Alternated octagonal or tritetragonal tiling is a uniform tiling of the hyperbolic plane.

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

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"Über Wirkungsbereichsteilungen von kubischer Symmetrie". Zeitschrift für Kristallographie, Kristallgeometrie, Kristallphysik, Kristallchemie. 154 (3–4): 199–215. doi:10.1524/zkri.1981.154.3-4.199. MR 0598811. . ^ Oldershaw, Cally (2003). Firefly Guide to Gems. Firefly Books. p. 107. ISBN 978-1-55297-814-6.  ^ Kirkaldy, J. F. (1968). Minerals and Rocks in Colour (2nd ed.). Blandford. pp. 138–139.  ^ Schwarz, H. A. (1873). "Ueber diejenigen Fälle in welchen die Gaussichen hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt". Journal für die reine und angewandte Mathematik. 75: 292–335. doi:10.1515/crll.1873.75.292. ISSN 0075-4102.  ^ Coxeter, Harold Scott Macdonald; Sherk, F. Arthur; Canadian Mathematical Society (1995). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. John Wiley & Sons. p. 3 and passim. ISBN 978-0-471-01003-6.  ^ Weisstein, Eric W. "Wythoff construction". MathWorld.  ^ Senechal, Marjorie (26 September 1996). Quasicrystals and Geometry. CUP Archive. p. 209. ISBN 978-0-521-57541-6.  ^ Margenstern, Maurice (4 January 2011). "Coordinates for a new triangular tiling of the hyperbolic plane". arXiv:1101.0530 .  ^ Zadnik, Gašper. "Tiling the Hyperbolic Plane with Regular Polygons". Wolfram. Retrieved 27 May 2015.  ^ Coxeter, H.S.M. (1999). Chapter 10: Regular honeycombs in hyperbolic space. The Beauty of Geometry: Twelve Essays. Dover Publications. pp. 212–213. ISBN 0-486-40919-8.  ^ "Mathematics in Art and Architecture". National University of Singapore. Retrieved 17 May 2015.  ^ Whittaker, Andrew (2008). Speak the Culture: Spain. Thorogood Publishing. p. 153. ISBN 978-1-85418-605-8.  ^ Escher 1974, pp. 5, 17. ^ Gersten, S. M. "Introduction to Hyperbolic and Automatic Groups" (PDF). University of Utah. Retrieved 27 May 2015. 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  ^ Leys, Jos (2015). "Hyperbolic Escher". Retrieved 27 May 2015.  ^ Escher 1974, pp. 142–143. ^ Escher 1974, p. 16. ^ Porter, Christine (2006). Tessellation Quilts: Sensational Designs From Interlocking Patterns. F+W Media. pp. 4–8. ISBN 9780715319413.  ^ Beyer, Jinny (1999). Designing tessellations: the secrets of interlocking patterns. Contemporary Book. pp. Ch. 7. ISBN 9780809228669.  ^ Gjerde, Eric (2008). Origami Tessellations. Taylor and Francis. ISBN 978-1-568-81451-3.  ^ "Reducing yield losses: using less metal to make the same thing". UIT Cambridge. Retrieved 29 May 2015.  ^ Thouless, M. D. (1990). "Crack Spacing in Brittle Films on Elastic Substrates". J. Am. Chem. Soc. 73: 2144. doi:10.1111/j.1151-2916.1990.tb05290.x.  ^ Z. C. Xia, J. W. Hutchinson "Crack patterns in thin films" J. Mech. Phys. Solids 48, 1107 (2000). doi:10.1016/S0022-5096(99)00081-2 ^ Seghir, R.; Arscott, S. (2015). "Controlled mud-crack patterning and self-organized cracking of polydimethylsiloxane elastomer surfaces". Sci. Rep. 5: 14787. doi:10.1038/srep14787.  ^ Ball, Philip. "How honeycombs can build themselves". Nature. Retrieved 7 November 2014.  ^ Shorter Oxford English dictionary (6th ed.). United Kingdom: Oxford University Press. 2007. p. 3804. ISBN 0199206872.  ^ Purdy, Kathy (2007). "Colchicums: autumn's best-kept secret". American Gardener (September/October): 18–22.  ^ Schreiber, Tomasz; Soja, Natalia (2010). "Limit theory for planar Gilbert tessellations". arXiv:1005.0023 .  ^ Gray, N. H.; Anderson, J. B.; Devine, J. D.; Kwasnik, J. M. (1976). "Topological properties of random crack networks". Mathematical Geology. 8 (6): 617–626. doi:10.1007/BF01031092.  ^ Gilbert, E. N. (1967). "Random plane networks and needle-shaped crystals". In Noble, B. Applications of Undergraduate Mathematics in Engineering. New York: Macmillan.  ^ Weaire, D.; Rivier, N. (1984). "Soap, cells and statistics: Random patterns in two dimensions". Contemporary Physics. 25 (1): 59–99. doi:10.1080/00107518408210979.  ^ Branagan, D.F. (1983). Young, R.W.; Nanson, G.C., eds. Tesselated pavements. Aspects of Australian sandstone landscapes. Special Publication No. 1, Australian and New Zealand Geomorphology. University of Wollongong. pp. 11–20. ISBN 0-864-18001-2.  ^ Ball, Philip (2009). Shapes. Oxford University Press. pp. 73–76. ISBN 978-0-199-60486-9.  ^ McAdam, Daniel. "History of Jigsaw Puzzles". American Jigsaw Puzzle Society. Archived from the original on 11 February 2014. Retrieved 28 May 2015.  ^ Slocum, Jerry (2001). The Tao of Tangram. Barnes & Noble. p. 9. ISBN 978-1-4351-0156-2.  ^ Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton University Press. ISBN 0-691-02444-8.  ^ Martin, George E. (1991). Polyominoes: A guide to puzzles and problems in tiling. Mathematical Association of America.  ^ Frederickson, Greg N. (2002). Hinged Dissections: Swinging and Twisting. Cambridge University Press. ISBN 0521811929.  ^ Gardner, Martin (May 1963). "On 'Rep-tiles,' Polygons that can make larger and smaller copies of themselves". Scientific American. Vol. 208 no. May. pp. 154–164.  ^ Gardner, Martin (14 December 2006). Aha! A Two Volume Collection: Aha! Gotcha Aha! Insight. MAA. p. 48. ISBN 978-0-88385-551-5.  ^ Suri, Mani (12 October 2015). "The Importance of Recreational Math". New York Times.  ^ Schattschneider, Doris (1978). "Tiling the Plane with Congruent Pentagons" (PDF). Mathematics Magazine. MAA. 51 (1): 29–44. doi:10.2307/2689644.  ^ Tutte, W. T. "Squaring the Square". Retrieved 29 May 2015.  ^ Gardner, Martin; Tutte, William T. (November 1958). "Mathematical Games". Scientific American.  ^ Henle, Frederick V.; Henle, James M. (2008). "Squaring the plane" (PDF). American Mathematical Monthly. 115: 3–12. JSTOR 27642387. 

Sources[edit] Coxeter, H.S.M. (1973). Regular Polytopes, Section IV : Tessellations and Honeycombs. Dover. ISBN 0-486-61480-8.  Escher, M. C. (1974). J. L. Locher, ed. The World of M. C. Escher (New Concise NAL ed.). Abrams. ISBN 0-451-79961-5.  Gardner, Martin (1989). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.  Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. ISBN 0-7167-1193-1.  Gullberg, Jan (1997). Mathematics From the Birth of Numbers. Norton. ISBN 0-393-04002-X.  Magnus, Wilhelm (1974). Noneuclidean Tesselations and Their Groups. Academic Press. ISBN 978-0-12-465450-1.  Stewart, Ian (2001). What Shape is a Snowflake?. Weidenfeld and Nicolson. ISBN 0-297-60723-5. 

External links[edit] Wikimedia Commons has media related to Tessellation. Wolfram MathWorld: Tessellation (good bibliography, drawings of regular, semiregular and demiregular tessellations) Tilings Encyclopedia (extensive information on substitution tilings, including drawings, people, and references) (how-to guides, Escher tessellation gallery, galleries of tessellations by other artists, lesson plans, history) Eppstein, David. "The Geometry Junkyard: Hyperbolic Tiling".  (list of web resources including articles and galleries) v t e Mathematics and art Concepts Algorithm Catenary Fractal Golden ratio Plastic number Hyperboloid structure Minimal surface Paraboloid Perspective Camera lucida Camera obscura Projective geometry Proportion Architecture Human Symmetry Tessellation Wallpaper group Forms Algorithmic art Anamorphic art Computer art 4D art Fractal art Islamic geometric patterns Girih Jali Muqarnas Zellige Knotting Architecture Geodesic dome Islamic Mughal Pyramid Vastu shastra Music Origami Textiles String art Sculpture Tiling Artworks List of works designed with the golden ratio Continuum Octacube Pi Pi in the Sky Buildings Hagia Sophia Pantheon Parthenon Pyramid of Khufu Sagrada Família St Mary's Cathedral Sydney Opera House Taj Mahal Artists Renaissance Paolo Uccello Piero della Francesca Albrecht Dürer Leonardo da Vinci Vitruvian Man Parmigianino Self-portrait in a Convex Mirror 19th–20th Century William Blake The Ancient of Days Newton Jean Metzinger Danseuse au café L'Oiseau bleu Man Ray René Magritte La condition humaine Salvador Dalí Crucifixion The Swallow's Tail Giorgio de Chirico M. C. Escher Circle Limit III Print Gallery Relativity Reptiles Waterfall Contemporary Martin and Erik Demaine Scott Draves Jan Dibbets John Ernest Helaman Ferguson Peter Forakis Bathsheba Grossman George W. Hart Desmond Paul Henry John A. Hiigli Anthony Hill Charles Jencks Garden of Cosmic Speculation Robert Longhurst István Orosz Hinke Osinga Hamid Naderi Yeganeh A Bird in Flight Boat Tony Robbin Oliver Sin Hiroshi Sugimoto Daina Taimina Roman Verostko Theorists Ancient Polykleitos Canon Vitruvius De architectura Renaissance Luca Pacioli De divina proportione Piero della Francesca De prospectiva pingendi Leon Battista Alberti De pictura De re aedificatoria Sebastiano Serlio Regole generali d'architettura Andrea Palladio I quattro libri dell'architettura Albrecht Dürer Vier Bücher von Menschlicher Proportion Romantic Frederik Macody Lund "Ad Quadratum" Jay Hambidge "The Greek Vase" Samuel Colman "Nature's Harmonic Unity" Modern Owen Jones The Grammar of Ornament Ernest Hanbury Hankin The Drawing of Geometric Patterns in Saracenic Art G. H. Hardy A Mathematician's Apology George David Birkhoff Aesthetic Measure Douglas Hofstadter Gödel, Escher, Bach Nikos Salingaros The 'Life' of a Carpet Publications Journal of Mathematics and the Arts Organizations Ars Mathematica The Bridges Organization European Society for Mathematics and the Arts Goudreau Museum of Mathematics in Art and Science Institute For Figuring Museum of Mathematics Related topics Droste effect Mathematical beauty Patterns in nature Sacred geometry Category v t e Tessellation Periodic Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling & honeycomb Coloring Convex Kisrhombille Wallpaper group Wythoff Aperiodic Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pentagonal Pinwheel Quaquaversal Rep-tile & Self-tiling Sphinx Truchet Other Anisohedral & Isohedral Architectonic & catoptric Circle Limit III Computer graphics Honeycomb Isotoxal List Packing Problems Domino Wang Heesch's Squaring Prototile Conway criterion Girih Regular Division of the Plane Regular grid Substitution Voronoi Voderberg By vertex type Spherical 2n 33.n V33.n 42.n V42.n Regular 2∞ 36 44 63 Semi- regular V32.4.3.4 33.42 33.∞ 34.6 V34.6 (3.6)2 3.122 42.∞ 4.6.12 4.82 Hyper- bolic 32.4.3.∞ 32.∞.3.∞ 34.7 34.8 34.∞ 35.4 37 38 3∞ (3.4)3 (3.4)4 3.4.∞.4 (3.7)2 (3.8)2 3.142 3.162 (3.∞)2 3.∞2 42.5.4 42.6.4 42.7.4 42.8.4 42.∞.4 45 46 47 48 4∞ (4.5)2 (4.6)2 4.6.12 4.6.14 V4.6.14 4.6.16 V4.6.16 4.6.∞ (4.7)2 (4.8)2 4.8.10 V4.8.10 4.8.12 4.8.14 4.8.16 4.8.∞ 4.102 4.10.12 4.122 4.12.16 4.142 4.162 4.∞2 (4.∞)2 54 55 56 5∞ (5.6)2 5.82 5.102 5.122 (5.∞)2 64 65 66 68 (6.8)2 6.82 6.102 6.122 6.162 73 74 77 7.62 7.82 7.142 83 84 86 88 812 8.62 8.162 ∞3 ∞4 ∞5 ∞∞ ∞.62 ∞.82 v t e Patterns in nature Patterns Crack Dune Foam Meander Phyllotaxis Soap bubble Symmetry in crystals Quasicrystals in flowers in biology Tessellation Vortex street Wave Widmanstätten pattern Causes Pattern formation Biology Natural selection Camouflage Mimicry Sexual selection Mathematics Chaos theory Fractal Logarithmic spiral Physics Crystal Fluid dynamics Plateau's laws Self-organization People Plato Pythagoras Empedocles Leonardo Fibonacci Liber Abaci Adolf Zeising Ernst Haeckel Joseph Plateau Wilson Bentley D'Arcy Wentworth Thompson On Growth and Form Alan Turing The Chemical Basis of Morphogenesis Aristid Lindenmayer Benoît Mandelbrot How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension Related Pattern recognition Emergence Mathematics and art Retrieved from "" Categories: SymmetryMosaicTessellationHidden categories: CS1 Russian-language sources (ru)CS1 German-language sources (de)CS1 maint: Multiple names: authors listGood articles

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