“Tessellation”的意思、由来-开放百科全书

A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.

A periodic tiling has a repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is also called a tessellation of space.

A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor or wall coverings. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs.

History

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.^[4]{{sfn|Gullberg|1997|p=395}}{{sfn|Stewart|2001|p=13}}

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.^[5]^[6] Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Aleksei Shubnikov and Nikolai Belov (1964),^[7] and Heinrich Heesch and Otto Kienzle (1963).^[8]

Etymology

In Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics.^[9] 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

Tessellation in two dimensions, also called planar tiling, is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules. These rules can be varied. Common ones are that there must be no gaps between tiles, and that no corner of one tile can lie along the edge of another.^[10] The tessellations created by bonded brickwork do not obey this rule. Among those that do, a regular tessellation has both identical{{efn|The mathematical term for identical shapes is "congruent" – in mathematics, "identical" means they are the same tile.}} regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile.^[11] 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.{{sfn|Gullberg|1997|p=395}}

Many other types of tessellation are possible under different constraints. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner.^[12] Irregular tessellations can also be made from other shapes such as pentagons, polyominoes and in fact almost any kind of geometric shape. The artist M. C. Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects.{{sfn|Escher|1974|pp=11–12, 15–16}} If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, and these can be used to decorate physical surfaces such as church floors.^[13]

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.{{efn|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.^[15]}} 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.^[14] 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.^[15]

Mathematically, tessellations can be extended to spaces other than the Euclidean plane.{{sfn|Gullberg|1997|p=395}} 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}.^[15] 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}.^[16]

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 4.4.4.4, or 4⁴. The tiling of regular hexagons is noted 6.6.6, or 6³.^[17]

In mathematics

Introduction to tessellations

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.^[17] The fundamental region is a shape such as a rectangle that is repeated to form the tessellation.^[18] For example, a regular tessellation of the plane with squares has a meeting of four squares at every vertex.^[17]

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.^[17]

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.^[19]

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, {{sfrac|3{{pi}}|5}}, is not a divisor of 2{{pi}}.^[20]^[21]^[22]

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.^[19] 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.^[23]

A semi-regular (or Archimedean) tessellation uses more than one type of regular polygon in an isogonal arrangement. There are eight semi-regular tilings (or nine if the mirror-image pair of tilings counts as two).{{sfn|Stewart|2001|p=75}} These can be described by their vertex configuration; 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).^[24] 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.^[25] 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.^[26]

Wallpaper groups

Tilings with translational symmetry in two independent directions can be categorized by wallpaper groups, of which 17 exist.^[27] It has been claimed that all seventeen of these groups are represented in the Alhambra palace in Granada, Spain. Though this is disputed,^[28] the variety and sophistication of the Alhambra tilings have surprised modern researchers.^[29] 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.^[30] Orbifold notation can be used to describe wallpaper groups of the Euclidean plane.^[31]

Aperiodic tilings

Tessellations and colour

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.^[43]

Tessellations with 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.^[44]

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 polyominoes, see Polyomino § Uses of polyominoes.

Voronoi tilings

Tessellations in higher dimensions

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.^[49] Any polyhedron that fits this criterion is known as a plesiohedron, and may possess between 4 and 38 faces.^[50] Naturally occurring rhombic dodecahedra are found as crystals of andradite (a kind of garnet) and fluorite.^[51]^[52]

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{{efn|In this context, quasiregular means that the cells are regular (solids), and the vertex figures are semiregular.}} honeycomb, which has eight tetrahedra and six octahedra at each polyhedron vertex. However, there are many possible semiregular honeycombs in three dimensions.^[53] Uniform polyhedra can be constructed using the Wythoff construction.^[54]

The Schmitt-Conway biprism is a convex polyhedron with the property of tiling space only aperiodically.^[55]

A Schwarz triangle is a spherical triangle that can be used to tile a sphere.^[57]

Tessellations in non-Euclidean geometries

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).^[58]^[59]

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.^[60]

In art

In architecture, tessellations have been used to create decorative motifs since ancient times. Mosaic tilings often had geometric patterns.^[61] 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^[62] and La Mezquita.^[63]

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.{{sfn|Escher|1974|pp=5, 17}} Escher made four "Circle Limit" drawings of tilings that use hyperbolic geometry.^[64]^[65] For his woodcut "Circle Limit IV" (1960), Escher prepared a pencil and ink study showing the required geometry.{{sfn|Escher|1974|pp=142–143}} Escher explained that "No single component of all the series, which from infinitely far away rise like rockets perpendicularly from the limit and are at last lost in it, ever reaches the boundary line."{{sfn|Escher|1974|p=16}}

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.^[66]^[67]

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.^[68]

In manufacturing

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.^[69]

Tessellation is apparent in the mudcrack-like cracking of thin films^[70]^[71] – with a degree of self-organisation being observed using micro and nanotechnologies.^[72]

In nature

The honeycomb is a well-known example of tessellation in nature with its hexagonal cells.^[73]

In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including the fritillary^[74] and some species of Colchicum are characteristically tessellate.^[75]

Many patterns in nature are formed by cracks in sheets of materials. These patterns can be described by Gilbert tessellations,^[76] also known as random crack networks.^[77] 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.^[78] 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.^[79] 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.^[80]

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.^[81]

In puzzles and recreational mathematics

Tessellations have given rise to many types of tiling puzzle, from traditional jigsaw puzzles (with irregular pieces of wood or cardboard)^[82] and the tangram^[83] 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.^[84]^[85] Authors such as Henry Dudeney and Martin Gardner have made many uses of tessellation in recreational mathematics. For example, Dudeney invented the hinged dissection,^[86] while Gardner wrote about the rep-tile, a shape that can be dissected into smaller copies of the same shape.^[87]^[88] Inspired by Gardner's articles in Scientific American, the amateur mathematician Marjorie Rice found four new tessellations with pentagons.^[89]^[90] Squaring the square is the problem of tiling an integral square (one whose sides have integer length) using only other integral squares.^[91]^[92] 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.^[93]

Examples

Footnotes

References

1. ^¹{{cite book |last=Pickover |first=Clifford A. |title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics |year=2009 |publisher=Sterling |isbn=9781402757969 |page=372}}
2. ^{{cite book |last1=Dunbabin | first1=Katherine M. D. | title=Mosaics of the Greek and Roman world | date=2006 | publisher=Cambridge University Press | page=280}}
3. ^{{cite web | title=The Brantingham Geometric Mosaics | url=http://www.hullcc.gov.uk/museumcollections/collections/storydetail.php?irn=410 | publisher=Hull City Council | accessdate=26 May 2015 | date=2008}}
4. ^{{cite book | authorlink=Johannes Kepler | author=Kepler, Johannes | date=1619 | titlelink=Harmonices Mundi | title=Harmonices Mundi |trans-title=Harmony of the Worlds }}
5. ^{{cite web | url=http://public.lanl.gov/djidjev/papers/coverage_chapter.pdf | title=Dynamic Coverage Problems in Sensor Networks | publisher=Los Alamos National Laboratory | year=2012 | accessdate=6 April 2013 |author1=Djidjev, Hristo |author2=Potkonjak, Miodrag |page=2}}
6. ^{{cite journal | author=Fyodorov, Y. | authorlink=Yevgraf Fyodorov | date=1891 | title=Simmetrija na ploskosti [Symmetry in the plane] | journal=Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva [Proceedings of the Imperial St. Petersburg Mineralogical Society] |series=2 | volume=28 | pages=245–291| language=Russian}}
7. ^{{cite book | last1=Shubnikov | first1=Alekseĭ Vasilʹevich | last2=Belov | first2=Nikolaĭ Vasilʹevich | title=Colored Symmetry | url=https://books.google.com/books?id=QKk9AAAAIAAJ | year=1964 | publisher=Macmillan}}
8. ^{{cite book | last1=Heesch | first1=H. | last2=Kienzle | first2=O. | title=Flächenschluss: System der Formen lückenlos aneinanderschliessender Flächteile | date=1963 | publisher=Springer | language=German}}
9. ^{{cite web | url=http://m-w.com/dictionary/tessellate | title=Tessellate | publisher=Merriam-Webster Online | accessdate=26 May 2015}}
10. ^{{cite book |author1=Conway, R. |author2=Burgiel, H. |author3=Goodman-Strauss, G. | title=The Symmetries of Things | publisher=Peters | date=2008}}
11. ^{{cite book | author=Coxeter, H. S. M. |authorlink=H. S. M. Coxeter | title=Regular Polytopes | edition=3rd | publisher=Dover | date=1973}}
12. ^{{cite book | author=Cundy and Rollett | title=Mathematical Models | edition=2nd | publisher=Oxford | date=1961 | pages=61–62}}
13. ^{{cite web | url=http://www.basilicasanmarco.it/basilica/architettura/il-pavimento-tessulare/?lang=en | title=Basilica di San Marco | publisher=Basilica di San Marco | work=Section: Tessellated floor | accessdate=26 April 2013}}
14. ^{{cite magazine | title=Will It Tile? Try the Conway Criterion! | author=Schattschneider, Doris |authorlink=Doris Schattschneider |magazine=Mathematics Magazine | volume=53 | issue=4 | date=September 1980 |pages=224–233 | doi=10.2307/2689617}}
15. ^{{cite book | url=https://books.google.com/books?id=iWvXsVInpgMC&lpg=PP1 | title=Regular Polytopes | publisher=Methuen | author=Coxeter, H. S. M. |authorlink=H. S. M. Coxeter | year=1948 |pages=14, 69, 149| isbn=9780486614809 }}
16. ^{{MathWorld|title=Tessellation|id=Tessellation}}
17. ^¹²³⁴⁵{{cite book | last=Grünbaum |first=Branko | title=Tilings and Patterns | year=1987 | publisher=W. H. Freeman | location=New York | author2=Shephard, G. C. | author2link=Geoffrey Colin Shephard | page=59}}
18. ^{{cite book |last1=Emmer |first1=Michele |last2=Schattschneider |first2=Doris |title=M.C. Escher's Legacy: A Centennial Celebration |url=https://books.google.com/books?id=5DDyBwAAQBAJ&pg=PA325 |date=8 May 2007 |publisher=Springer |location=Berlin Heidelberg |isbn=978-3-540-28849-7 |page=325}}
19. ^¹{{cite book |last=Horne | first=Clare E. | title=Geometric Symmetry in Patterns and Tilings | year=2000 |publisher=Woodhead Publishing |isbn=9781855734920 |pages=172, 175}}
20. ^{{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}}
21. ^{{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}}
22. ^{{MathWorld|title=Pentagon Tiling|urlname=PentagonTiling}}
23. ^{{MathWorld|title=Regular Tessellations|id=RegularTessellations}}
24. ^{{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)}}
25. ^{{cite book | last=Wells | first=David | year=1991 | title=The Penguin Dictionary of Curious and Interesting Geometry | publisher=Penguin Books | location=New York | isbn=978-0-14-011813-1 | pages=260–261 | contribution=two squares tessellation}}
26. ^{{cite journal |author=Kirby, Matthew |author2=Umble, Ronald |title=Edge Tessellations and Stamp Folding Puzzles |journal=Mathematics Magazine |volume=84 |issue=4 |date=2011 |pages=283–89 |doi=10.4169/math.mag.84.4.283}}
27. ^{{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}}
28. ^{{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}}
29. ^{{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 }}
30. ^{{MathWorld|title=Frieze Group|id=FriezeGroup}}
31. ^{{cite journal | last1=Huson | first1=Daniel H. | title=Two-Dimensional Symmetry Mutation | publisher=CiteSeer | date=1991| citeseerx=10.1.1.30.8536 }}
32. ^{{cite journal | author=Radin, C. | title=The Pinwheel Tilings of the Plane | journal=Annals of Mathematics | volume=139 | issue=3 | pages=661–702 | date=May 1994 | doi=10.2307/2118575 | jstor=2118575| citeseerx=10.1.1.44.9723 }}
33. ^{{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}}
34. ^{{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}}
35. ^{{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}}
36. ^{{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}}
37. ^{{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}}
38. ^{{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}}
39. ^{{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 }}
40. ^{{cite journal | last=Culik | first=Karel, II | doi=10.1016/S0012-365X(96)00118-5 | issue=1–3 | journal=Discrete Mathematics | mr=1417576 | pages=245–251 | title=An aperiodic set of 13 Wang tiles | volume=160 | year=1996}}
41. ^{{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}}
42. ^{{cite journal | last=Smith | first=Cyril Stanley | authorlink=Cyril Stanley Smith | doi=10.2307/1578535 | issue=4 | journal=Leonardo | pages=373–385 | title=The tiling patterns of Sebastian Truchet and the topology of structural hierarchy | volume=20 | year=1987| jstor=1578535 }}
43. ^{{springer | title=Four-colour problem|id=p/f040970}}
44. ^{{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}}
45. ^{{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}}
46. ^{{cite book |author1=Okabe, Atsuyuki |author2=Boots, Barry |author3=Sugihara, Kokichi |author4=Chiu, Sung Nok | date=2000 | title=Spatial Tessellations – Concepts and Applications of Voronoi Diagrams | edition=2nd | publisher=John Wiley | isbn=978-0-471-98635-5}}
47. ^{{cite book |author1=George, Paul Louis |author2=Borouchaki, Houman | title=Delaunay Triangulation and Meshing: Application to Finite Elements |year=1998 | publisher=Hermes | isbn=978-2-86601-692-0 | pages=34–35}}
48. ^{{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}}
49. ^{{cite journal | author=Grünbaum, Branko | date=1994 | title=Uniform tilings of 3-space | journal= Geombinatorics | volume=4 | issue=2 | pages=49–56}}
50. ^{{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 }}.
51. ^{{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}}
52. ^{{cite book | last1=Kirkaldy | first1=J. F. | title=Minerals and Rocks in Colour | date=1968 | publisher=Blandford | pages=138–139 | edition=2nd}}
53. ^{{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}}
54. ^{{mathworld | urlname=WythoffConstruction | title=Wythoff construction}}
55. ^{{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}}
56. ^{{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}}
57. ^{{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}}
58. ^{{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 }}
59. ^{{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}}
60. ^{{cite book | author=Coxeter, H.S.M. | work=The Beauty of Geometry: Twelve Essays | publisher=Dover Publications | date=1999 | isbn=978-0-486-40919-1 | title=Chapter 10: Regular honeycombs in hyperbolic space | pages=212–213}}
61. ^¹{{cite book | title=Geometric Patterns from Roman Mosaics | publisher=Tarquin | author=Field, Robert | year=1988 | isbn=978-0-906-21263-9}}
62. ^{{cite web | url=http://www.math.nus.edu.sg/aslaksen/teaching/math-art-arch.shtml#Symmetry | title=Mathematics in Art and Architecture | publisher=National University of Singapore |accessdate=17 May 2015}}
63. ^{{cite book | last=Whittaker | first=Andrew |title=Speak the Culture: Spain | url=https://books.google.com/books?id=2rZS8SWMroUC&pg=PT153 | year=2008 | publisher=Thorogood Publishing | isbn=978-1-85418-605-8 | page=153}}
64. ^{{cite web | last1=Gersten |first1=S. M. | title=Introduction to Hyperbolic and Automatic Groups |url=http://www.math.utah.edu/~sg/Papers/banff.pdf | publisher=University of Utah | accessdate=27 May 2015 | quote=Figure 1 is part of a tiling of the Euclidean plane, which we imagine as continued in all directions, and Figure 2 [Circle Limit IV] is a beautiful tesselation of the Poincaré unit disc model of the hyperbolic plane by white tiles representing angels and black tiles representing devils. An important feature of the second is that all white tiles are mutually congruent as are all black tiles; of course this is not true for the Euclidean metric, but holds for the Poincaré metric}}
65. ^{{cite web | last1=Leys | first1=Jos | title=Hyperbolic Escher | url=http://www.josleys.com/show_gallery.php?galid=325 | accessdate=27 May 2015 | date=2015}}
66. ^{{cite book | last=Porter | first=Christine | title=Tessellation Quilts: Sensational Designs From Interlocking Patterns | year=2006 | publisher=F+W Media | isbn=9780715319413 | pages=4–8}}
67. ^{{cite book |last=Beyer |first=Jinny | title=Designing tessellations: the secrets of interlocking patterns |year=1999 |publisher=Contemporary Book |isbn=9780809228669 |pages=Ch. 7}}
68. ^{{Cite book |title=Origami Tessellations |last=Gjerde |first=Eric |publisher=Taylor and Francis |year=2008|isbn=978-1-568-81451-3}}
69. ^{{cite web | title=Reducing yield losses: using less metal to make the same thing | url=http://www.withbotheyesopen.com/pdftransponder.php?c=13|publisher=UIT Cambridge | accessdate=29 May 2015}}
70. ^{{cite journal | last1 = Thouless | first1 = M. D. | year = 1990 | title = Crack Spacing in Brittle Films on Elastic Substrates | url = | journal = J. Am. Chem. Soc. | volume = 73 | issue = 7| pages = 2144–2146 | doi = 10.1111/j.1151-2916.1990.tb05290.x }}
71. ^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}}
72. ^{{cite journal | last1 = Seghir | first1 = R. | last2 = Arscott | first2 = S. | year = 2015 | title = Controlled mud-crack patterning and self-organized cracking of polydimethylsiloxane elastomer surfaces | url = | journal = Sci. Rep. | volume = 5 | issue = | page = 14787 | doi = 10.1038/srep14787 | pmid = 26437880 | bibcode = 2015NatSR...514787S | pmc = 4594096 }}
73. ^{{cite web | last1=Ball | first1=Philip | title=How honeycombs can build themselves | url=http://www.nature.com/news/how-honeycombs-can-build-themselves-1.13398 | work=Nature | accessdate=7 November 2014}}
74. ^{{cite book | title=Shorter Oxford English dictionary | edition=6th | year=2007 | publisher=Oxford University Press | location=United Kingdom |isbn=978-0199206872 |page=3804}}
75. ^{{cite journal | last1=Purdy | first1=Kathy | title=Colchicums: autumn's best-kept secret | journal=American Gardener | date=2007 | issue=September/October | pages=18–22}}
76. ^{{cite arXiv | last1=Schreiber | first1=Tomasz | last2=Soja | first2=Natalia | eprint=1005.0023 | title=Limit theory for planar Gilbert tessellations | year=2010| class=math.PR }}
77. ^{{cite journal | last1=Gray | first1=N. H. | last2=Anderson | first2=J. B. | last3=Devine | first3=J. D. | last4=Kwasnik | first4=J. M. | doi=10.1007/BF01031092 | issue=6 | journal=Mathematical Geology | pages=617–626 | title=Topological properties of random crack networks | volume=8 | year=1976}}
78. ^{{cite book | last=Gilbert | first=E. N. | authorlink=Edgar Gilbert | editor-last=Noble | editor-first=B. | contribution=Random plane networks and needle-shaped crystals | location=New York | publisher=Macmillan | title=Applications of Undergraduate Mathematics in Engineering | year=1967}}
79. ^{{cite journal |last1=Weaire |first1=D. |last2=Rivier |first2=N. |title=Soap, cells and statistics: Random patterns in two dimensions |journal=Contemporary Physics |date=1984|volume=25 |issue=1 |pages=59–99 | authorlink=Denis Weaire |doi=10.1080/00107518408210979}}
80. ^{{cite book | author=Branagan, D.F. | date=1983 | title=Tesselated pavements |editor1=Young, R.W. |editor2=Nanson, G.C. | pages=11–20 | work=Aspects of Australian sandstone landscapes. Special Publication No. 1, Australian and New Zealand Geomorphology | publisher=University of Wollongong | isbn=978-0-864-18001-8}}
81. ^{{cite book | author=Ball, Philip | title=Shapes | publisher=Oxford University Press | date=2009 | pages=73–76 | isbn=978-0-199-60486-9}}
82. ^{{cite web | last1=McAdam | first1=Daniel | title=History of Jigsaw Puzzles | url=http://www.jigsaw-puzzle.org/jigsaw-puzzle-history.html | publisher=American Jigsaw Puzzle Society | accessdate=28 May 2015 |archiveurl=https://web.archive.org/web/20140211155914/http://www.jigsaw-puzzle.org/jigsaw-puzzle-history.html |archivedate=11 February 2014}}
83. ^{{cite book | title=The Tao of Tangram | last=Slocum | first=Jerry | year=2001 |publisher=Barnes & Noble | isbn=978-1-4351-0156-2 | page=9}}
84. ^{{cite book | last1=Golomb | first1=Solomon W. | title=Polyominoes | date=1994 | publisher=Princeton University Press | edition=2nd | isbn=978-0-691-02444-8}}
85. ^{{cite book | last1=Martin | first1=George E.|title=Polyominoes: A guide to puzzles and problems in tiling | date=1991 | publisher=Mathematical Association of America}}
86. ^{{cite book | last=Frederickson | first=Greg N. | date=2002 | title=Hinged Dissections: Swinging and Twisting | url=https://books.google.com/books?id=pW_0EisSP4IC | publisher=Cambridge University Press |isbn=978-0521811927}}
87. ^{{cite magazine | author=Gardner, Martin | date=May 1963 | title=On 'Rep-tiles,' Polygons that can make larger and smaller copies of themselves |magazine=Scientific American | volume=208 | issue=May | pages=154–164}}
88. ^{{cite book | last=Gardner | first=Martin | title=Aha! A Two Volume Collection: Aha! Gotcha Aha! Insight | url=https://books.google.com/books?id=3HzwNVMQOZkC&pg=RA1-PA48 | date=14 December 2006 | publisher=MAA | isbn=978-0-88385-551-5 | page=48}}
89. ^{{cite news | url=https://www.nytimes.com/2015/10/12/opinion/the-importance-of-recreational-math.html?ref=opinion&_r=0 | title=The Importance of Recreational Math | last1=Suri | first1=Mani | date=12 October 2015 | website=New York Times}}
90. ^{{cite journal |last1=Schattschneider |first1=Doris |title=Tiling the Plane with Congruent Pentagons |url=http://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1979/0025570x.di021103.02p0247f.pdf |publisher=MAA |journal=Mathematics Magazine |date=1978 |volume=51 |issue=1 |pages=29–44 |doi=10.2307/2689644|jstor=2689644 }}
91. ^{{cite web | last1=Tutte | first1=W. T. | title=Squaring the Square | url=http://www.squaring.net/history_theory/brooks_smith_stone_tutte_II.html | website=Squaring.net | accessdate=29 May 2015}}
92. ^{{cite magazine | last1=Gardner | first1=Martin| last2=Tutte | first2=William T. | title=Mathematical Games |magazine=Scientific American | date=November 1958}}
93. ^{{cite journal | first1=Frederick V. | last1=Henle | first2=James M. | last2=Henle | url=http://maven.smith.edu/~jhenle/stp/stp.pdf | title=Squaring the plane | journal=American Mathematical Monthly | volume=115 | issue=1 | year=2008 | pages=3–12 | doi=10.1080/00029890.2008.11920491 | jstor=27642387 | deadurl=yes | archiveurl=https://web.archive.org/web/20060620150515/http://maven.smith.edu/%7Ejhenle/stp/stp.pdf | archivedate=2006-06-20 | df= }}