Tessellation

Tessellation - Wikipedia, the free encyclopedia
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Tessellation
From Wikipedia, the free encyclopedia
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, the term "tessellation" is used
to describe the organization of information needed to
render to give the appearance of the surfaces of
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.
Ceramic Tiles in Marrakech, forming edge-toedge, regular and other tessellations
A wall sculpture at Leeuwarden celebrating the
artistic tessellations of M. C. Escher
Contents
◾ 1 History
◾ 1.1 Etymology
◾ 2 Overview
◾ 3 In mathematics
◾ 3.1 Kinds of tessellations
◾ 3.2 Wallpaper groups
◾ 3.3 Tessellations and colour
◾ 3.4 Tessellations with triangles and quadrilaterals
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◾ 3.5 Tessellations in higher dimensions
4 In computer graphics
5 In art
6 In nature
7 Examples
8 See also
9 Footnotes
10 References
11 Sources
12 External links
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. Other prominent contributors include Shubnikov
and Belov (1951); and Heinrich Heesch and Otto Kienzle (1963).
A temple mosaic from the
ancient Sumerian city of
Uruk IV (3400–3100 BC)
showing a tessellation
pattern in the tile colours.
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
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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]
A semi-regular tessellation:
tiled floor of a church in
Seville, Spain, using square,
triangle and hexagon
prototiles
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 63.[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]
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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.
The 3.4.6.4 semi-regular
tessellation is made with
three prototiles: a triangle, a
square and a hexagon. Every
vertex has a triangle, square,
hexagon, square around it, in
that order.
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
Main article: Wallpaper group
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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]
This tessellated, monohedral
street pavement uses curved
shapes instead of polygons.
It belongs to wallpaper
group p3m1.
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.
If the colours of this tiling are to form
a repeat pattern, at least seven colours
are required. If the colouring is
allowed to be aperiodic, then at least
four colours are needed. This tiling
can be used on the surface of a torus.
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 will not in general 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.[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
Main article: honeycomb (geometry)
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Illustration of a SchmittConway biprism, also called
a Schmitt–Conway–Danzer
tile.
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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]
Tessellating threedimensional space: the
rhombic dodecahedron is
one of the solids that can be
stacked to fill space exactly.
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 graphics
Main article: Tessellation (computer graphics)
In computer graphics, tessellation has a variety of usages. It 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 art
In architecture, tessellations have been used to create decorative motifs
since ancient times. Mosaic tilings were used by the Romans, often
with geometric patterns.[35] Later civilisations also used larger tiles,
either plain or individually decorated. Some of the most decorative
were the Moorish wall tilings of buildings such as the Alhambra and
the Córdoba, Andalusia mosque of La Mezquita.
A quilt showing a regular
tessellation pattern.
Tessellated designs also often appear on textiles, either woven or
stitched in or printed. In the context of quilting, tessellation refers to
regular[36] and semiregular[37] of tessellation of either patch shapes or
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the overall design. Tessellation patterns have been used to design
interlocking motifs of patch shapes.[38][39] The repeating motif is
sometimes called a block design.[36]
In graphic art, tessellations frequently appeared in the works of M. C.
Escher, who was inspired by studying the Moorish use of symmetry in
the tilings he saw during a visit to Spain in 1936.[40]
In nature
Main article: Patterns in nature
The honeycomb provides a well-known
example of tessellation in nature.
Tessellate pattern in a
Colchicum flower
Roman mosaic floor panel of
stone, tile and glass, from a
villa near Antioch in Roman
Syria. 2nd century A.D.
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.
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.
Examples
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Isogonal periodic tiling, Regular 36 tessellation. Semi-regular tiling
where any tile adjoins
Color is mathematically 6.3.3.3.3.
by any edge exactly one unimportant here.
tile of another size.
A Penrose tiling, with
several symmetries.
But it can never repeat
periodically.
The Voderberg tiling, a
spiral, monohedral
tiling made of
enneagons.
A honeycomb is a
natural tessellated
structure.
A Voronoi tiling
See also
Types of tessellation
◾ Convex uniform
honeycomb
◾ List of aperiodic sets
of tiles
◾ List of uniform tilings
◾ Pinwheel tiling – nonperiodic tilings using
Conway
◾ Rep-tile - a type of
substitution tiling
◾ Tilings of regular
polygons
◾ Uniform coloring
Mathematics
◾ Convex uniform
honeycombs in
hyperbolic space
◾ Coxeter groups –
algebraic groups to find
tessellations
◾ Girih tiles – set of 5 tiles
used in Islamic
architecture
◾ Triangulation
(geometry)
◾ Uniform tiling
http://en.wikipedia.org/wiki/Tessellation
Related topics
◾ Jigsaw puzzle
◾ Mathematics and fiber arts
◾ Polyiamond and Polyomino —
figures of regular triangles and
squares, often in tiling puzzles
◾ Quilt block designs and quilt
blocks
◾ "Tessellate" - song by the British
alternative indie pop quartet Alt-J
(∆).
◾ Tiling puzzle
◾ Trianglepoint – needlepoint with
polyiamonds (equilateral triangles)
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◾ Uniform tessellation
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◾ Uniform tilings in
hyperbolic plane
◾ Wang tiles
◾ Wythoff construction
Footnotes
a. ^ 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]
b. ^ In this context, quasiregular means that the cells are regular (solids), and the vertex figures are semiregular.
References
1. ^ 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.
2. ^ Kepler, Johannes (1619). Harmonices Mundi (Harmony of the Worlds).
3. ^ Djidjev, Hristo; Potkonjak, Miodrag (2012). "Dynamic Coverage Problems in Sensor
Networks" (http://public.lanl.gov/djidjev/papers/coverage_chapter.pdf). Los Alamos National Laboratory
(USA). p. 2. Retrieved 6 April 2013.
4. ^ E. Fedorov (1891) "Simmetrija na ploskosti" [Symmetry in the plane], Zapiski Imperatorskogo SantPetersburgskogo Mineralogicheskogo Obshchestva [Proceedings of the Imperial St. Petersburg
Mineralogical Society], series 2, volume 28, pages 245-291 (in Russian).
5. ^ tessellate (http://m-w.com/dictionary/tessellate), Merriam-Webster Online
6. ^ "Basilica di San
Marco" (http://www.basilicasanmarco.it/WAI/eng/basilica/architettura/interne/pavimento.bsm). Section
dedicated to the tessellated floor. Basilica di San Marco, Venice, Italy. Retrieved 26 April 2013.
7. ^ a b c d e Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman.
8. ^ Gullberg, 1997. p. 395
9. ^ Coxeter, H. S. M. (1948). Regular Polytopes (http://books.google.com/books?
id=iWvXsVInpgMC&lpg=PP1). Methuen. pp. 14, 69, 149.
10. ^ Weisstein, Eric W. (1999-2013). "Tessellation" (http://mathworld.wolfram.com/Tessellation.html).
Wolfram MathWorld. Retrieved 26 April 2013.
11. ^ a b Horne, Clare E. (2000). Geometric Symmetry in Patterns and Tilings. Woodhead Publishing.
ISBN 9781855734920.
12. ^ Dutch, Steven (29 July 1999). "Some Special Radial and Spiral
Tilings" (http://www.uwgb.edu/dutchs/symmetry/radspir1.htm). University of Wisconsin. Retrieved 6 April
2013.
13. ^ Hirschhorn, M. D.; D. C. Hunt (1985). "Equilateral convex pentagons which tile the
plane" (http://www.sciencedirect.com/science/article/pii/0097316585900780). Journal of Combinatorial
Theory, Series A 39 (1): 1–18. doi:10.1016/0097-3165(85)90078-0 (http://dx.doi.org/10.1016%2F00973165%2885%2990078-0). Retrieved 29 April 2013.
14. ^ MathWorld: Regular Tessellations (http://mathworld.wolfram.com/RegularTessellation.html)
15. ^ Stewart, 2001. p. 75
16. ^ NRICH (Millennium Maths Project) (1997-2012). "Schlafli Tessellations" (http://nrich.maths.org/1556).
University of Cambridge. Retrieved 26 April 2013.
17. ^ Gardner, 1989.
18. ^ Franz Aurenhammer (1991). Voronoi Diagrams – A Survey of a Fundamental Geometric Data Structure.
ACM Computing Surveys, 23(3):345–405, 1991
19. ^ 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
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20. ^ 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.
21. ^ Armstrong, M.A. (1988). Groups and Symmetry. New York: Springer-Verlag. ISBN 978-3-540-96675-3.
22. ^ Grünbaum, Branko (June/July 2006). Notices of the American Mathematical Society 53 (6): 670–673.
23. ^ Jaworski, J. "A mathematician’s guide to the
Alhambra" (http://people.ucsc.edu/~rha/danm221/Refs/jaworski.pdf). Retrieved September 1, 2011.
24. ^ Lu, Peter J.; Steinhardt (23 February 2007). Science 315: 1106.
25. ^ Weisstein, Eric W. "Frieze Group." (http://mathworld.wolfram.com/FriezeGroup.html). MathWorld--A
Wolfram Web Resource. Retrieved 29 April 2013.
26. ^ Hazewinkel, 2001.
27. ^ Jones, 1856.
28. ^ Weisstein, Eric W. (1999-2013). "Schmitt-Conway
Biprism" (http://mathworld.wolfram.com/Tessellation.html). Wolfram MathWorld. Retrieved 28 April 2013.
29. ^ "Rhodolite Garnet Gemstone Information" (http://www.ajsgem.com/gemstone-information/rhodolitegarnet-51.html). AJS Gems. Retrieved 28 April 2013.
30. ^ "The mineral Andradite" (http://www.galleries.com/Andradite). Amethyst Galleries. 1995-2013. Retrieved
28 April 2013.
31. ^ Weisstein, Eric W. (1999-2013). "Schmitt-Conway Biprism" (http://mathworld.wolfram.com/SchmittConwayBiprism.html). Wolfram MathWorld. Retrieved 28 April 2013.
32. ^ Weisstein, Eric W. (1999-2013). "Schmitt-Conway
Biprism" (http://mathworld.wolfram.com/Honeycomb.html). Wolfram MathWorld. Retrieved 28 April 2013.
33. ^ MSDN: Tessellation Overview (http://msdn.microsoft.com/en-us/library/ff476340%28v=VS.85%29.aspx)
34. ^ The OpenGL® Graphics System: A Specification (Version 4.0 (Core Profile) - March 11, 2010)
(http://www.opengl.org/registry/doc/glspec40.core.20100311.pdf)
35. ^ Field, Robert (1988). Geometric Patterns from Roman Mosaics. Tarquin. ISBN 978-0-906-21263-9.
36. ^ a b Beyer, Jinny. "Tessellations" (http://www.jinnybeyer.com/quilting-with-jinny/tips-lessons/detail.cfm?
instanceId=2141577C-0324-E8C6-1F88CFCEC9808ADD). Retrieved 28 April 2013.
37. ^ Swanson, Irena. "Quilting semi-regular tessellations" (http://people.reed.edu/~iswanson/semireg.pdf).
Retrieved 28 April 2013.
38. ^ Porter, Christine (2006). Tessellation Quilts: Sensational Designs From Interlocking Patterns. F+W
Media. pp. 4–8. ISBN 9780715319413.
39. ^ Beyer, Jinny (1999). Designing tessellations: the secrets of interlocking patterns. Contemporary Books.
pp. Ch. 7. ISBN 9780809228669.
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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 978-07167-1986-1.)
◾ Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977), "Extraordinary nonperiodic 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.
◾ Hazewinkel, Michiel, ed. (2001), "Four-colour
problem" (http://www.encyclopediaofmath.org/index.php?title=p/f040970), Encyclopedia of
Mathematics, Springer, ISBN 978-1-55608-010-4
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◾ Jones, Owen (1910 (folio ed.), first published 1856). The Grammar of Ornament
(http://digital.library.wisc.edu/1711.dl/DLDecArts.GramOrnJones). Bernard Quaritch.
◾ 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 0-29760723-5.
External links
◾ Wolfram MathWorld: Tessellation (http://mathworld.wolfram.com/Tessellation.html) (good
bibliography, drawings of regular, semiregular and demiregular tessellations)
◾ Tilings Encyclopedia (http://tilings.math.uni-bielefeld.de/) (extensive information on substitution
tilings, including drawings, people, and references)
◾ Tessellations.org (http://www.tessellations.org) (how-to guides, Escher tessellation gallery,
galleries of tessellations by other artists, lesson plans, history)
Retrieved from "http://en.wikipedia.org/w/index.php?title=Tessellation&oldid=583272468"
Categories: Symmetry Mosaic Tessellation
◾ This page was last modified on 25 November 2013 at 19:24.
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http://en.wikipedia.org/wiki/Tessellation
4/12/2013