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From the Back Cover : The definitive book on tiling and geometric patterns, this magnificently illustrated volume features figures and more than tables. Buy Used Condition: Fair A readable copy. All pages are The recursive process of substitution tiling is a method of generating aperiodic tilings.

One class that can be generated in this way is the rep-tiles ; these tilings have surprising self-replicating properties. Aperiodic tilings, while lacking in translational symmetry , do have symmetries of other types, by infinite repetition of any bounded patch of the tiling and in certain finite groups of rotations or reflections of those patches. Wang tiles are squares coloured on each edge, and placed so that abutting edges of adjacent tiles have the same colour; hence they are sometimes called Wang dominoes.

A suitable set of Wang dominoes can tile the plane, but only aperiodically. This is known because any Turing machine can be represented as a set of Wang dominoes that tile the plane if and only if the Turing machine does not halt. Since the halting problem is undecidable, the problem of deciding whether a Wang domino set can tile the plane is also undecidable.

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These can tile the plane either periodically or randomly. 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.

Geometric Symmetry in Patterns and Tilings - 1st Edition

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. 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.

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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.

If only one shape of tile is allowed, tilings exists with convex N -gons for N equal to 3, 4, 5 and 6. 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. 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.

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. Tessellations in three or more dimensions are called honeycombs. In three dimensions there is just one regular honeycomb, which has eight cubes at each polyhedron vertex.

Similarly, in three dimensions there is just one quasiregular [c] honeycomb, which has eight tetrahedra and six octahedra at each polyhedron vertex. However, there are many possible semiregular honeycombs in three dimensions. The Schmitt-Conway biprism is a convex polyhedron with the property of tiling space only aperiodically. A Schwarz triangle is a spherical triangle that can be used to tile a sphere. 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.

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.

In architecture, tessellations have been used to create decorative motifs since ancient times. Mosaic tilings often had geometric patterns. Some of the most decorative were the Moorish wall tilings of Islamic architecture , using Girih and Zellige tiles in buildings such as the Alhambra [66] and La Mezquita. Tessellations frequently appeared in the graphic art of M. Escher ; he was inspired by the Moorish use of symmetry in places such as the Alhambra when he visited Spain in 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. 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.

5 and Penrose Tiling - Numberphile

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. Tessellation is apparent in the mudcrack -like cracking of thin films [77] [78] — with a degree of self-organisation being observed using micro and nanotechnologies. The honeycomb is a well-known example of tessellation in nature with its hexagonal cells. In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit.

Flowers including the fritillary [81] and some species of Colchicum are characteristically tessellate. Many patterns in nature are formed by cracks in sheets of materials.

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These patterns can be described by Gilbert tessellations , [83] also known as random crack networks. 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.

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. 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 , Lord Kelvin proposed a packing using only one solid, the bitruncated cubic honeycomb with very slightly curved faces.

In , 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. Tessellations have given rise to many types of tiling puzzle , from traditional jigsaw puzzles with irregular pieces of wood or cardboard [89] and the tangram [90] to more modern puzzles which often have a mathematical basis. For example, polyiamonds and polyominoes are figures of regular triangles and squares, often used in tiling puzzles.

For example, Dudeney invented the hinged dissection , [93] while Gardner wrote about the rep-tile , a shape that can be dissected into smaller copies of the same shape. Triangular tiling , one of the three regular tilings of the plane. Snub hexagonal tiling , a semiregular tiling of the plane. Floret pentagonal tiling , dual to a semiregular tiling and one of 15 monohedral pentagon tilings.

The Voderberg tiling , a spiral, monohedral tiling made of enneagons. Alternated octagonal or tritetragonal tiling is a uniform tiling of the hyperbolic plane. Topological square tiling, isohedrally distorted into I shapes. From Wikipedia, the free encyclopedia. Tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. For the song by Alt-J, see Tessellate song. For the computer graphics technique, see Tessellation computer graphics. Further information: Euclidean tilings of regular polygons , Uniform tiling , and List of convex uniform tilings. Main article: Wallpaper group.

Main articles: Aperiodic tiling and List of aperiodic sets of tiles. Further information: Four colour theorem. Main article: Honeycomb geometry. Further information: Mathematics and art. Main articles: Tiling puzzle and recreational mathematics. Mosaics of the Greek and Roman world. Cambridge University Press. Hull City Council. Retrieved 26 May Geometric Patterns from Roman Mosaics. Harmonices Mundi [ Harmony of the Worlds ]. Los Alamos National Laboratory. Retrieved 6 April Petersburg Mineralogical Society]. Colored Symmetry.

Merriam-Webster Online. The Symmetries of Things. Mathematical Models 2nd ed. Section: Tessellated floor. Basilica di San Marco. Retrieved 26 April Try the Conway Criterion! Mathematics Magazine. Regular Polytopes. Escher's Legacy: A Centennial Celebration. Berlin Heidelberg: Springer. Geometric Symmetry in Patterns and Tilings. Woodhead Publishing. University of Wisconsin.


Journal of Combinatorial Theory, Series A. University of Cambridge. New York: Penguin Books. Groups and Symmetry. New York: Springer-Verlag. Notices of the American Mathematical Society. Bibcode : Sci May Annals of Mathematics. American Mathematical Society. Retrieved 29 May Physical Review Letters. Bibcode : PhRvL.. Bell System Technical Journal. Scientific American.

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Memoirs of the American Mathematical Society. Inventiones Mathematicae. Bibcode : InMat.. Discrete Mathematics. The Grammar of Ornament folio ed. Bernard Quaritch. ACM Computing Surveys. John Wiley. Lectures on Random Voronoi Tessellations. Bibcode : ZK Firefly Guide to Gems. Firefly Books.