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Tessellations were used by the [[Architecture of Mesopotamia|Sumerians]] (about 4000 BC) in building wall decorations formed by patterns of clay tiles.<ref name="Pickover2009">{{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 Publishing Company, Inc|isbn=9781402757969|page=372}}</ref> |
Tessellations were used by the [[Architecture of Mesopotamia|Sumerians]] (about 4000 BC) in building wall decorations formed by patterns of clay tiles.<ref name="Pickover2009">{{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 Publishing Company, Inc|isbn=9781402757969|page=372}}</ref> |
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In 1619 [[Johannes Kepler]] made one of the first documented studies of tessellations when he wrote about regular and semiregular tessellation, which are coverings of a plane with regular polygons, in his ''[[Harmonices Mundi]]''.<ref>[[Johannes Kepler|Kepler, Johannes]] (1619). ''[[Harmonices Mundi]] (Harmony of the Worlds)''.</ref> 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.<ref>{{cite web | url=http://public.lanl.gov/djidjev/papers/coverage_chapter.pdf | title=Dynamic Coverage Problems in Sensor Networks | publisher=Los Alamos National Laboratory (USA) | date=2012 | accessdate=6 April 2013 | author=Djidjev, Hristo; Potkonjak, Miodrag | pages=2}}</ref><ref>E. Fedorov (1891) "Simmetrija na ploskosti" [Symmetry in the plane], ''Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva'' [Proceedings of the Imperial St. Petersburg Mineralogical Society], series 2, volume 28, pages 245-291 (in Russian).</ref> Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov (1951); and [[Heinrich Heesch]] and Otto Kienzle (1963). |
In 1619 [[Johannes Kepler]] made one of the first documented studies of tessellations when he wrote about regular and semiregular tessellation, which are coverings of a plane with regular polygons, in his ''[[Harmonices Mundi]]''.<ref>[[Johannes Kepler|Kepler, Johannes]] (1619). ''[[Harmonices Mundi]] (Harmony of the Worlds)''.</ref> 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.<ref>{{cite web | url=http://public.lanl.gov/djidjev/papers/coverage_chapter.pdf | title=Dynamic Coverage Problems in Sensor Networks | publisher=Los Alamos National Laboratory (USA) | date=2012 | accessdate=6 April 2013 | author=Djidjev, Hristo; Potkonjak, Miodrag | pages=2}}</ref><ref>E. Fedorov (1891) "Simmetrija na ploskosti" [Symmetry in the plane], ''Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva'' [Proceedings of the Imperial St. Petersburg Mineralogical Society], series 2, volume 28, pages 245-291 (in Russian).</ref> Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. no one cares Other prominent contributors include Shubnikov and Belov (1951); and [[Heinrich Heesch]] and Otto Kienzle (1963). |
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===Etymology=== |
===Etymology=== |
Revision as of 15:01, 9 May 2013
A Tessellation 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.
Some special kinds of tessellations include regular, with tiles all of the same shape; semi-regular, with tiles of more than one shape; and aperiodic tilings, which use tiles that cannot form a repeating pattern. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups.
In computer graphics, tessellation is used to organize the information needed to render the appearance of surfaces as realistic three-dimensional objects.
In the real world, a tessellation is a tiling made of physical 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 tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher often made use of tessellations 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]
In 1619 Johannes Kepler made one of the first documented studies of tessellations when he wrote about regular and semiregular tessellation, which are coverings of a plane with regular polygons, in his Harmonices Mundi.[2] 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.[3][4] Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. no one cares Other prominent contributors include Shubnikov and Belov (1951); and Heinrich Heesch and Otto Kienzle (1963).
Etymology
In Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics.[5] The word "tessella" means "small square" (from "tessera", square, which in its turn is from the Greek word "τέσσερα" for "four"). It corresponds with the everyday term tiling which refers to applications of tessellations, often made of glazed clay.
Overview
Tessellation or tiling is the branch of mathematics that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps. There are only three "regular" tessellations using exactly one kind of identical regular polygons arranged edge-to-edge, but many other types of tessellations are possible, differing in the constraints that are chosen to apply. For example, there are nine types of tessellations made with more than one kind of regular polygon, but having the same arrangement of polygons at every corner. These were described by the Swiss geometer Ludwig Schläfli in the 1850s. Tessellations can also be made from other shapes such as rectangles, 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. If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, and these can be used to form physical surfaces such as church floors.[6]
More formally, a tessellation or tiling is a partition of the Euclidean plane into 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.[a] Many tessellations are formed from a finite number of prototiles; all tiles in the tessellation are congruent to one of 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. Mathematicians have found no general rule for determining if a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations.[7] For example, the types of convex pentagons that can tile the plane remains an unsolved problem.
Mathematically, tessellations can be extended to spaces other than the Euclidean plane.[8] 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}.[9] 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}.[10]
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 66.[7]: 59
In mathematics
Kinds of 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. For example, a regular tessellation of the plane with squares has a meeting of four squares at every vertex.[7]
The sides of the polygons are not necessarily identical to the edges of the tiles. An edge-to-edge tessellation 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 tessellation, 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.[7]
A normal tiling is a tessellation for which (1) every tile is topologically equivalent to a disk, (2) the intersection of any two tiles is a single connected set or the empty set, and (3) all tiles are uniformly bounded.[11]: 172 A uniformly bounded tile is one in which a finite circle can be circumscribed around the tile and a finite circle can be inscribed within the tile; the condition disallows tiles that are pathologically long or thin.
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, with the Voderberg tiling having a unit tile that is a nonconvex enneagon.[1] The Hirschhorn tiling, published by Michael D. Hirschhorn and D. C. Hunt in 1985, has a unit tile that is an irregular pentagon.[12][13]
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.[11]: 175 If a prototile admits a tiling, but no such tiling is isohedral, then the prototile is call 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.[14]
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).[15] 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).[16]
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, though they have surprising self-replicating properties using the recursive process of substitution tiling.[17]
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.)[18][19] 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.[20]
Wallpaper groups
Tilings with translational symmetry in two independent directions can be categorized by wallpaper groups, of which 17 exist.[21] It has been claimed that all seventeen of these groups are represented in the Alhambra palace in Granada, Spain. Though this is disputed,[22][23] the variety and sophistication of the Alhambra tilings have surprised modern researchers.[24] 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.[25]
Tessellations and colour
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. Note that the colouring guaranteed by the four-colour theorem will not in general respect the symmetries of the tessellation. To produce a colouring which does, as many as seven colours may be needed, as in the picture at right.[26]
Tessellations with triangles and quadrilaterals
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 2-fold rotational centres at the midpoints of all sides, and translational symmetry whose basis vectors are the diagonal of the quadrilateral or, equivalently, one of these and the sum or difference of the two. 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.[27]
Tessellations in higher dimensions
Tessellation can be extended to three or more dimensions. Certain polyhedra can be stacked in a regular crystal pattern to fill (or tile) three-dimensional space, including the cube (the only regular polyhedron to do so); the rhombic dodecahedron; and the truncated octahedron.[28] Some crystals including Andradite (a kind of Garnet) and Fluorite can take the form of rhombic dodecahedra.[29][30]
The Schmitt-Conway biprism is a convex polyhedron which has the property of tiling space only aperiodically. John Horton Conway discovered it in 1993.[31]
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[b] honeycomb, which has eight tetrahedra and six octahedra at each polyhedron vertex. However there are many possible semiregular honeycombs in three dimensions.[32]
In computer models
In computer graphics, tessellation is used to manage datasets of polygons (sometimes called vertex sets) presenting objects in a scene and divide them into suitable structures for rendering. Especially for real-time rendering, data are tessellated into triangles, for example in DirectX 11 and OpenGL.[33][34]
In Direct3D 11 pipeline (a part of DirectX 11), the graphics primitive is the patch.[35] The tessellator generates a triangle-based tessellation of the patch according to tessellation parameters such as the TessFactor, which controls the degree of fineness of the mesh. The tessellation, along with shaders such as a Phong shader, allows for producing smoother surfaces than would be generated by the original mesh.[35]. By offloading the tessellation process onto the GPU hardware, smoothing can be performed in real time. Tessellation can also be used for implementing subdivision surfaces, level of detail scaling and fine displacement mapping.[36] OpenGL uses a similar pipeline, where tessellation into triangles is controlled by the Tessellation Control Shader and a set of four tessellation parameters.[37]
In computer-aided design the constructed design is represented by a boundary representation topological model, where analytical 3D surfaces and curves, limited to faces, edges, and vertices, constitute a continuous boundary of a 3D body. Arbitrary 3D bodies are often too complicated to analyze directly. So they are approximated (tessellated) with a mesh of small, easy-to-analyze pieces of 3D volume—usually either irregular tetrahedra, or irregular hexahedra. The mesh is used for finite element analysis.
The mesh of a surface is usually generated per individual faces and edges (approximated to polylines) so that original limit vertices are included into mesh. To ensure that approximation of the original surface suits the needs of further processing, three basic parameters are usually defined for the surface mesh generator:
- The maximum allowed distance between the planar approximation polygon and the surface (known as "sag"). This parameter ensures that mesh is similar enough to the original analytical surface (or the polyline is similar to the original curve).
- The maximum allowed size of the approximation polygon (for triangulations it can be maximum allowed length of triangle sides). This parameter ensures enough detail for further analysis.
- The maximum allowed angle between two adjacent approximation polygons (on the same face). This parameter ensures that even very small humps or hollows that can have significant effect to analysis will not disappear in mesh.
An algorithm generating a mesh is typically controlled by the above three and other parameters. Some types of computer analysis of a constructed design require an adaptive mesh refinement, which is a mesh made finer (using stronger parameters) in regions where the analysis needs more detail.[33][34]
In art
Mosaic tilings were used by the Romans, often with geometric patterns.[38]
Tessellations frequently appeared in the art of M. C. Escher, who was inspired by studying the Moorish use of symmetry in the Moorish tilings of the Alhambra and the Córdoba, Andalusia mosque of La Mezquita during a visit to Spain in 1936.[39]
In the context of quilting, tessellation refers to regular[40] and semiregular[41] of tessellation of either patch shapes or the overall design. Tessellation patterns have been used to design interlocking motifs of patch shapes.[42][43] The repeating motif is sometimes called a block design.[40]
In nature
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.
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.
In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including the Fritillary and some species of Colchicum are characteristically tessellate.
Examples
-
The Voderberg tiling, a spiral, monohedral tiling made of enneagons.
-
The regular 36 tessellation (Color is not mathematically important here.)
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This real-world tiling has two prototiles, a large square and a small square. It is isogonal, but not edge-to-edge because the large square shares each side with two adjacent tiles.
-
A honeycomb is a natural tessellated structure.
-
A Penrose tiling. It possesses reflection and rotation symmetry, but can never repeat periodically.
See also
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Footnotes
- ^ The tiles are usually required to be topologically equivalent to a closed disk, which means bizarre shapes with holes, dangling line segments or infinite areas are excluded.[7]
- ^ In this context, quasiregular means that the cells are regular (solids), and the vertex figures are semiregular.
References
- ^ a b Pickover, Clifford A. (2009). The math book: from Pythagoras to the 57th dimension, 250 milestones in the history of mathematics. Sterling Publishing Company, Inc. p. 372. ISBN 9781402757969.
- ^ Kepler, Johannes (1619). Harmonices Mundi (Harmony of the Worlds).
- ^ Djidjev, Hristo; Potkonjak, Miodrag (2012). "Dynamic Coverage Problems in Sensor Networks" (PDF). Los Alamos National Laboratory (USA). p. 2. Retrieved 6 April 2013.
{{cite web}}
: CS1 maint: multiple names: authors list (link) - ^ E. Fedorov (1891) "Simmetrija na ploskosti" [Symmetry in the plane], Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva [Proceedings of the Imperial St. Petersburg Mineralogical Society], series 2, volume 28, pages 245-291 (in Russian).
- ^ tessellate, Merriam-Webster Online
- ^ "Basilica di San Marco". Section dedicated to the tessellated floor. Basilica di San Marco, Venice, Italy. Retrieved 26 April 2013.
- ^ a b c d e Grünbaum, Branko (1987). Tilings and Patterns. New York: W. H. Freeman.
{{cite book}}
: Unknown parameter|coauthors=
ignored (|author=
suggested) (help) - ^ Gullberg, 1997. p. 395
- ^ Coxeter, H. S. M. (1948). Regular Polytopes. Methuen. pp. 14, 69, 149.
- ^ Weisstein, Eric W. (1999–2013). "Tessellation". Wolfram MathWorld. Retrieved 26 April 2013.
{{cite web}}
: CS1 maint: date format (link) - ^ a b Horne, Clare E. (2000). Geometric Symmetry in Patterns and Tilings. Woodhead Publishing. ISBN 9781855734920.
- ^ Dutch, Steven (29 July 1999). "Some Special Radial and Spiral Tilings". University of Wisconsin. Retrieved 6 April 2013.
- ^ Hirschhorn, M. D. (1985). "Equilateral convex pentagons which tile the plane". Journal of Combinatorial Theory, Series A. 39 (1): 1–18. doi:10.1016/0097-3165(85)90078-0. Retrieved 29 April 2013.
{{cite journal}}
: Unknown parameter|coauthors=
ignored (|author=
suggested) (help) - ^ MathWorld: Regular Tessellations
- ^ Stewart, 2001. p. 75
- ^ NRICH (Millennium Maths Project) (1997–2012). "Schlafli Tessellations". University of Cambridge. Retrieved 26 April 2013.
{{cite web}}
: CS1 maint: date format (link) - ^ Gardner, 1989.
- ^ Franz Aurenhammer (1991). Voronoi Diagrams – A Survey of a Fundamental Geometric Data Structure. ACM Computing Surveys, 23(3):345–405, 1991
- ^ Atsuyuki Okabe, Barry Boots, Kokichi Sugihara & Sung Nok Chiu (2000). Spatial Tessellations – Concepts and Applications of Voronoi Diagrams. 2nd edition. John Wiley, 2000. ISBN 0-471-98635-6
- ^ Paul Louis George and Houman Borouchaki (1998). Delaunay Triangulation and Meshing: Application to Finite Elements. Paris: Hermes. pp. 34–35. ISBN 2-86601-692-0.
- ^ Armstrong, M.A. (1988). Groups and Symmetry. New York: Springer-Verlag. ISBN 978-3-540-96675-3.
- ^ Grünbaum, Branko (2006). Notices of the American Mathematical Society. 53 (6): 670–673.
{{cite journal}}
: Missing or empty|title=
(help); Unknown parameter|month=
ignored (help) - ^ Jaworski, J. "A mathematician's guide to the Alhambra" (PDF). Retrieved September 1, 2011.
- ^ Lu, Peter J. (23 February, 2007). Science. 315: 1106.
{{cite journal}}
: Check date values in:|date=
(help); Missing or empty|title=
(help); Unknown parameter|coauthors=
ignored (|author=
suggested) (help) - ^ Weisstein, Eric W. ""Frieze Group."". MathWorld--A Wolfram Web Resource. Retrieved 29 April 2013.
- ^ Hazewinkel, 2001.
- ^ Jones, 1856.
- ^ Weisstein, Eric W. (1999–2013). "Schmitt-Conway Biprism". Wolfram MathWorld. Retrieved 28 April 2013.
{{cite web}}
: CS1 maint: date format (link) - ^ "Rhodolite Garnet Gemstone Information". AJS Gems. Retrieved 28 April 2013.
- ^ "The mineral Andradite". Amethyst Galleries. 1995–2013. Retrieved 28 April 2013.
{{cite web}}
: CS1 maint: date format (link) - ^ Weisstein, Eric W. (1999–2013). "Schmitt-Conway Biprism". Wolfram MathWorld. Retrieved 28 April 2013.
{{cite web}}
: CS1 maint: date format (link) - ^ Weisstein, Eric W. (1999–2013). "Schmitt-Conway Biprism". Wolfram MathWorld. Retrieved 28 April 2013.
{{cite web}}
: CS1 maint: date format (link) - ^ a b MSDN: Tessellation Overview
- ^ a b The OpenGL® Graphics System: A Specification (Version 4.0 (Core Profile) - March 11, 2010)
- ^ a b Abi-Chahla, Fedy. "Tessellation". Tom's Hardware. Retrieved 27 April 2013.
- ^ Tariq, Sara. "D3D11 Tessellation" (PDF). Nvida. Retrieved 27 April 2013.
- ^ "Tessellation". OpenGL. Retrieved 27 April 2013.
- ^ Field, Robert (1988). Geometric Patterns from Roman Mosaics. Tarquin. ISBN 978-0-906-21263-9.
- ^ Locher, J. L.; Escher, M. C. (1971). Locher, J. L. (ed.). The Work of M. C. Escher. Harry N. Abrams. p. 5.
{{cite book}}
: CS1 maint: multiple names: authors list (link) - ^ a b Beyer, Jinny. "Tessellations". Retrieved 28 April 2013.
- ^ Swanson, Irena. "Quilting semi-regular tessellations" (PDF). Retrieved 28 April 2013.
- ^ 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 Books. pp. Ch. 7. ISBN 9780809228669.
Sources
- Coxeter, H.S.M.. Regular Polytopes, Section IV : Tessellations and Honeycombs. Dover, 1973. ISBN 0-486-61480-8.
- Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
- Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977), "Extraordinary non-periodic tiling that enriches the theory of tiles", Scientific American, 236: 110–121.
- Grünbaum, Branko and G. C. Shephard. Tilings and Patterns. New York: W. H. Freeman & Co., 1987. ISBN 0-7167-1193-1.
- Gullberg, Jan (1997). Mathematics From the Birth of Numbers. Norton. ISBN 0-393-04002-X.
- "Four-colour problem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Jones, Owen (1910 (folio ed.), first published 1856). The Grammar of Ornament. Bernard Quaritch.
{{cite book}}
: Check date values in:|year=
(help) - 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
- Wolfram MathWorld: Tessellation (good bibliography, drawings of regular, semiregular and demiregular tessellations)
- Tilings Encyclopedia (extensive information on substitution tilings, including drawings, people, and references)