1851:
101:
1870:
1888:
5215:
6429:
263:
56:
1452:
1927:
1945:
80:
871:
7526:
1334:
1694:
1033:, 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 that does, it is necessary to treat the colours as part of the tessellation. Here, as many as seven colours may be needed, as demonstrated in the image at left.
1009:
767:
656:
1832:
398:
316:
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959:
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914:, 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. A substitution rule, such as can be used to generate Penrose patterns using assemblies of tiles called rhombs, illustrates scaling symmetry. A
1024:
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
448:
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. Irregular tessellations can also be made from
1193:
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. Voronoi tilings with randomly placed points can be used to
3960:
Figure 1 is part of a tiling of the
Euclidean plane, which we imagine as continued in all directions, and Figure 2 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
1514:"Circle Limit IV" (1960), Escher prepared a pencil and ink study showing the required geometry. 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."
550:
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}. The Schläfli notation makes it possible to describe tilings compactly. For example, a tiling of regular hexagons has three six-sided polygons at each
520:
is a sufficient, but not necessary, set of rules for deciding whether a given shape tiles the plane periodically without reflections: some tiles fail the criterion, but still tile the plane. No general rule has been found for determining whether a given shape can tile the plane or not, which means
461:
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 decorate physical surfaces such as church floors.
624:
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
1819:
is the problem of tiling an integral square (one whose sides have integer length) using only other integral squares. 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.
1072:
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.
417:, 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. The tessellations created by
1180:
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
989:
is a single shape that forces aperiodic tiling. The first such tile, dubbed a "hat", was discovered in 2023 by David Smith, a hobbyist mathematician. The discovery is under professional review and, upon confirmation, will be credited as solving a longstanding
7720:
1655:, allows cracks to form starting from being 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.
1887:
786:; for example, a semi-regular tiling using squares and regular octagons has the vertex configuration 4.8 (each vertex has one square and two octagons). Many non-edge-to-edge tilings of the Euclidean plane are possible, including the family of
909:
are non-periodic, using a rep-tile construction; the tiles appear in infinitely many orientations. It might be thought that a non-periodic pattern would be entirely without symmetry, but this is not so. Aperiodic tilings, while lacking in
330:
proved that every periodic tiling of the plane features one of seventeen different groups of isometries. Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include
1869:
1850:
6192:
682:; 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
782:
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). These can be described by their
652:. 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.
1287:. 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 honeycomb, which has eight
558:, 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 4. The tiling of regular hexagons is noted 6.6.6, or 6.
728:
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
429:
and identical regular corners or vertices, having the same angle between adjacent edges for every tile. There are only three shapes that can form such regular tessellations: the equilateral
5433:
1149:
are examples of tiles that are either convex of non-convex, for which various combinations, rotations, and reflections can be used to tile a plane. For results on tiling the plane with
1926:
1060:
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
4567:
5214:
3760:
1670:
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
6224:
885:, which use two different quadrilateral prototiles, are the best known example of tiles that forcibly create non-periodic patterns. They belong to a general class of
3961:
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
836:. Although this is disputed, the variety and sophistication of the Alhambra tilings have interested modern researchers. Of the three regular tilings two are in the
794:
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.
6219:
5393:
5094:
1944:
1471:
tilings often had geometric patterns. Later civilisations also used larger tiles, either plain or individually decorated. Some of the most decorative were the
100:
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6345:
5373:
2795:
572:
7369:
5033:(open-source software for exploring two-dimensional tilings of the plane, sphere and hyperbolic plane; includes databases containing millions of tilings)
613:
is a shape such as a rectangle that is repeated to form the tessellation. For example, a regular tessellation of the plane with squares has a meeting of
168:
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
6634:
3033:
Dharma-wardana, M. W. C.; MacDonald, A. H.; Lockwood, D. J.; Baribeau, J.-M.; Houghton, D. C. (1987). "Raman scattering in
Fibonacci superlattices".
6567:
106:
An example of non‑periodicity due to another orientation of one tile out of an infinite number of identical tiles
2335:
1903:
1525:
Tessellated designs often appear on textiles, whether woven, stitched in, or printed. Tessellation patterns have been used to design interlocking
943:
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
4725:
7374:
6589:
6323:
1403:
554:
Other methods also exist for describing polygonal tilings. When the tessellation is made of regular polygons, the most common notation is the
5014:
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4888:
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2121:
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affects whether tiles with the same shape, but different colours, are considered identical, which in turn affects questions of symmetry. The
7184:
7019:
4706:
6004:
3001:
1624:
In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including the
844:. Tilings in 2-D with translational symmetry in just one direction may be categorized by the seven frieze groups describing the possible
732:
group of the tiling. If a prototile admits a tiling, but no such tiling is isohedral, then the prototile is called anisohedral and forms
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7334:
7309:
7299:
7269:
7224:
7174:
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6969:
6854:
6164:
1372:
3756:"Ueber diejenigen Fälle in welchen die Gaussichen hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt"
1492:
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4102:
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4452:. Aspects of Australian sandstone landscapes. Special Publication No. 1, Australian and New Zealand Geomorphology. Wollongong, NSW:
3362:
Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; Goodman-Strauss, Chaim (March 2023). "An aperiodic monotile". arXiv:2303.10798
580:
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35:
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7134:
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406:
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used a square tile split into two triangles of contrasting colours. These can tile the plane either periodically or randomly.
951:
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
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3130:
3083:
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1012:
At least seven colors are required if the colours of this tiling are to form a pattern by repeating this rectangle as the
2814:
Lu, Peter J.; Steinhardt (23 February 2007). "Decagonal and quasi-crystalline tilings in medieval
Islamic architecture".
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to the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said to
332:
4357:
Gray, N. H.; Anderson, J. B.; Devine, J. D.; Kwasnik, J. M. (1976). "Topological properties of random crack networks".
2363:
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1932:
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This tessellated, monohedral street pavement uses curved shapes instead of polygons. It belongs to wallpaper group p3.
790:, tessellations that use two (parameterised) sizes of square, each square touching four squares of the other size. An
275:
179:
1993:
The mathematical term for identical shapes is "congruent" – in mathematics, "identical" means they are the same tile.
1143:
With non-convex polygons, there are far fewer limitations in the number of sides, even if only one shape is allowed.
413:
Tessellation in two dimensions, also called planar tiling, is a topic in geometry that studies how shapes, known as
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1540:(paper folding), where pleats are used to connect molecules, such as twist folds, together in a repeating fashion.
1337:
1056:) can be used as a prototile to form a monohedral tessellation, often in more than one way. Copies of an arbitrary
779:
740:
336:
219:
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5182:
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2087:
402:
5056:(how-to guides, Escher tessellation gallery, galleries of tessellations by other artists, lesson plans, history)
2019:
In this context, quasiregular means that the cells are regular (solids), and the vertex figures are semiregular.
7329:
7324:
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7249:
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55:
1643:, also known as random crack networks. The Gilbert tessellation is a mathematical model for the formation of
1375:(that may be regular, quasiregular, or semiregular) is an edge-to-edge filling of the hyperbolic plane, with
300:
made an early documented study of tessellations. He wrote about regular and semiregular tessellations in his
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3805:
Margenstern, Maurice (4 January 2011). "Coordinates for a new triangular tiling of the hyperbolic plane".
3411:
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is undecidable, the problem of deciding whether a Wang domino set can tile the plane is also undecidable.
911:
817:
4252:
3387:
500:. These tiles may be polygons or any other shapes. Many tessellations are formed from a finite number of
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4196:"Controlled mud-crack patterning and self-organized cracking of polydimethylsiloxane elastomer surfaces"
2929:
2412:
1856:
1804:
1353:
947:. A suitable set of Wang dominoes can tile the plane, but only aperiodically. This is known because any
679:
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184:
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1936:
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894:
824:, of which 17 exist. It has been claimed that all seventeen of these groups are represented in the
497:
290:
235:
188:, also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions.
126:
2045:
The Math Book: From
Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics
625:
edge-to-edge because the long side of each rectangular brick is shared with two bordering bricks.
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2010:, which means bizarre shapes with holes, dangling line segments, or infinite areas are excluded.
1502:; he was inspired by the Moorish use of symmetry in places such as the Alhambra when he visited
897:
is a method of generating aperiodic tilings. One class that can be generated in this way is the
547:
531:
308:; he was possibly the first to explore and to explain the hexagonal structures of honeycomb and
1467:
In architecture, tessellations have been used to create decorative motifs since ancient times.
1256:, and may possess between 4 and 38 faces. Naturally occurring rhombic dodecahedra are found as
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4658:(May 1963). "On 'Rep-tiles,' Polygons that can make larger and smaller copies of themselves".
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2483:
2462:
2117:
2052:
1837:
1796:
1747:
1712:
1693:
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1384:
1126:
1104:
849:
791:
756:
665:
649:
602:
593:
203:
42:
31:
5050:" (extensive information on substitution tilings, including drawings, people, and references)
3755:
3289:(1987). "The tiling patterns of Sebastian Truchet and the topology of structural hierarchy".
2191:
1690:, is a rare sedimentary rock formation where the rock has fractured into rectangular blocks.
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1783:
are figures of regular triangles and squares, often used in tiling puzzles. Authors such as
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Mathematically, tessellations can be extended to spaces other than the
Euclidean plane. The
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5142:
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4336:
Schreiber, Tomasz; Soja, Natalia (2010). "Limit theory for planar
Gilbert tessellations".
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Coxeter, Harold Scott
Macdonald; Sherk, F. Arthur; Canadian Mathematical Society (1995).
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366:
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How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
4413:; Rivier, N. (1984). "Soap, cells and statistics: Random patterns in two dimensions".
4180:
3236:
3217:
2787:
2720:
Kirby, Matthew; Umble, Ronald (2011). "Edge Tessellations and Stamp Folding Puzzles".
1727:
with very slightly curved faces. In 1993, Denis Weaire and Robert Phelan proposed the
751:, all of the same shape. There are only three regular tessellations: those made up of
620:
The sides of the polygons are not necessarily identical to the edges of the tiles. An
605:
tiling is a tiling where every vertex point is identical; that is, the arrangement of
465:
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2003:
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1731:, which uses less surface area to separate cells of equal volume than Kelvin's foam.
1719:. Such foams present a problem in how to pack cells as tightly as possible: in 1887,
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is the point of intersection of three or more bordering tiles. Using these terms, an
386:
238:, for artistic effect. Tessellations are sometimes employed for decorative effect in
4903:
4845:
3974:
3202:
2851:
808:
698:
using irregular pentagons: regular pentagons cannot tile the Euclidean plane as the
7577:
7483:
6665:
6402:
6047:
5946:
5797:
5708:
5703:
5601:
5522:
5438:
5261:
5234:
5132:
5039:(good bibliography, drawings of regular, semiregular and demiregular tessellations)
4976:
4898:
4829:
4410:
3575:
Zeitschrift für Kristallographie, Kristallgeometrie, Kristallphysik, Kristallchemie
3459:
3340:
2567:
1499:
1253:
1165:
963:
958:
919:
478:
458:
418:
319:
227:
207:
90:
5030:
3594:
2486:
1279:
Illustration of a Schmitt–Conway biprism, also called a Schmitt–Conway–Danzer tile
1076:
5036:
3528:
3438:(1991). "Voronoi Diagrams – A Survey of a Fundamental Geometric Data Structure".
2869:
2243:
7695:
7653:
7572:
7468:
7463:
6651:
6096:
5869:
5807:
5718:
5713:
5313:
5276:
4395:(1967). "Random plane networks and needle-shaped crystals". In Noble, B. (ed.).
4132:
Thouless, M. D. (1990). "Crack Spacing in Brittle Films on Elastic Substrates".
3944:
3056:
2007:
1720:
1639:
are formed by cracks in sheets of materials. These patterns can be described by
1625:
1553:
1480:
1426:
1288:
660:
591:
is the intersection between two bordering tiles; it is often a straight line. A
525:
454:
211:
138:
17:
3573:
Engel, Peter (1981). "Über Wirkungsbereichsteilungen von kubischer Symmetrie".
1348:
278:(about 4000 BC) in building wall decorations formed by patterns of clay tiles.
7642:
7637:
7555:
7458:
6720:
5917:
5822:
5772:
5352:
5147:
4955:
4434:
4266:
3849:
Coxeter, H.S.M. (1999). "Chapter 10: Regular honeycombs in hyperbolic space".
3272:
2735:
2265:
Flächenschluss: System der Formen lückenlos aneinanderschliessender Flächteile
1776:
1561:
1292:
1225:
1190:
543:
309:
86:
64:
4849:
4298:(6th ed.). United Kingdom: Oxford University Press. 2007. p. 3804.
3783:
3774:
1310:
is a convex polyhedron with the property of tiling space only aperiodically.
7744:
7647:
6740:
6725:
6641:
6617:
5881:
5468:
5448:
5117:
4471:
3707:
2874:
2835:
2633:
2491:
1780:
1775:, to more modern puzzles that often have a mathematical basis. For example,
1739:
1699:
1659:
1621:
is a well-known example of tessellation in nature with its hexagonal cells.
1618:
1610:
1576:
1261:
1150:
1146:
940:
930:
890:
645:
539:
501:
489:
450:
251:
68:
4237:
3064:
2843:
2225:
Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva
4792:
Gardner, Martin; Tutte, William T. (November 1958). "Mathematical Games".
3451:
1605:
7473:
7438:
6509:
5694:
5296:
5194:
5122:
3473:
Okabe, Atsuyuki; Boots, Barry; Sugihara, Kokichi; Chiu, Sung Nok (2000).
1950:
1918:
1807:
into smaller copies of the same shape. Inspired by Gardner's articles in
1800:
1687:
1644:
1572:
1568:
1557:
1488:
1396:
1269:
944:
898:
825:
744:
729:
687:
528:
474:
430:
239:
223:
199:
178:
uses a small set of tile shapes that cannot form a repeating pattern (an
5047:
4837:
3146:
1517:
7599:
7594:
7582:
7560:
7545:
7453:
5596:
5335:
5249:
5127:
4752:
4370:
3312:
3186:
2960:
2434:
2282:
1772:
1744:
1648:
1537:
1511:
1484:
1472:
1460:
1257:
925:
829:
760:
606:
470:
441:. Any one of these three shapes can be duplicated infinitely to fill a
438:
286:
195:
172:. A tiling that lacks a repeating pattern is called "non-periodic". An
122:
61:
6435:
5072:
4219:
4159:
Xia, Z. C.; Hutchinson, J. W. (2000). "Crack patterns in thin films".
3163:(1971). "Undecidability and nonperiodicity for tilings of the plane".
1252:, among others. Any polyhedron that fits this criterion is known as a
202:, or may have functions such as providing durable and water-resistant
7393:
5301:
3503:
3475:
Spatial Tessellations – Concepts and Applications of Voronoi Diagrams
1656:
1468:
1456:
1322:
1299:
in three dimensions. Uniform honeycombs can be constructed using the
1265:
434:
373:
used to make mosaics. The word "tessella" means "small square" (from
282:
206:, floor, or wall coverings. Historically, tessellations were used in
192:
41:"Mathematical tiling" redirects here. For the building material, see
4744:
3304:
2952:
2426:
1275:
585:
Mathematicians use some technical terms when discussing tilings. An
4021:
Tessellation Quilts: Sensational Designs From Interlocking Patterns
3336:"Mathematicians have finally discovered an elusive 'einstein' tile"
2242:
Shubnikov, Alekseĭ Vasilʹevich; Belov, Nikolaĭ Vasilʹevich (1964).
326:
Some two hundred years later in 1891, the Russian crystallographer
270:
IV (3400–3100 BC), showing a tessellation pattern in coloured tiles
191:
A real physical tessellation is a tiling made of materials such as
7632:
5286:
4342:
3811:
3500:
Delaunay Triangulation and Meshing: Application to Finite Elements
1738:
1692:
1604:
1530:
1516:
1503:
1450:
1347:
1332:
1274:
1207:
1164:
1081:
1075:
1007:
957:
924:
869:
833:
807:
765:
694:, published by Michael D. Hirschhorn and D. C. Hunt in 1985, is a
654:
464:
396:
370:
314:
261:
2395:
2393:
2391:
2389:
2387:
2385:
852:
can be used to describe wallpaper groups of the Euclidean plane.
7510:
7448:
7443:
6193:
Viewpoints: Mathematical Perspective and Fractal Geometry in Art
5308:
5291:
5053:
4103:"Reducing yield losses: using less metal to make the same thing"
1708:
1233:
1208:
1088:
If only one shape of tile is allowed, tilings exist with convex
362:
267:
7397:
6697:
6547:
6447:
6343:
6305:
6301:
5076:
3377:, the New York Times, March 28, 2023, with image of the pattern
385:, which refers to applications of tessellations, often made of
4319:
Purdy, Kathy (2007). "Colchicums: autumn's best-kept secret".
2662:
4046:
Designing tessellations: the secrets of interlocking patterns
3374:
Elusive 'Einstein' Solves a Longstanding Mathematical Problem
2415:(September 1980). "Will It Tile? Try the Conway Criterion!".
1413:. In three-dimensional (3-D) hyperbolic space there are nine
972:
are square tiles decorated with patterns so they do not have
1510:" drawings of tilings that use hyperbolic geometry. For his
1295:
at each polyhedron vertex. However, there are many possible
1068:
we have the quadrilateral. Equivalently, we can construct a
2927:
Radin, C. (May 1994). "The Pinwheel Tilings of the Plane".
889:, which use tiles that cannot tessellate periodically. The
521:
there are many unsolved problems concerning tessellations.
457:
and in fact almost any kind of geometric shape. The artist
377:, square, which in turn is from the Greek word τέσσερα for
4873:(1973). "Section IV : Tessellations and Honeycombs".
2691:
The Penguin Dictionary of Curious and Interesting Geometry
1224:
Tessellation can be extended to three dimensions. Certain
763:. All three of these tilings are isogonal and monohedral.
4448:
Branagan, D.F. (1983). Young, R.W.; Nanson, G.C. (eds.).
2161:
2159:
2157:
1552:
to reduce the wastage of material (yield losses) such as
1459:
floor panel of stone, tile, and glass, from a villa near
1232:
to fill (or tile) three-dimensional space, including the
659:
An example of a non-edge‑to‑edge tiling: the 15th convex
5069:(list of web resources including articles and galleries)
4397:
Applications of Undergraduate Mathematics in Engineering
1878:, dual to a semiregular tiling and one of 15 monohedral
1498:
Tessellations frequently appeared in the graphic art of
3554:
Grünbaum, Branko (1994). "Uniform tilings of 3-space".
3259:
Browne, Cameron (2008). "Truchet curves and surfaces".
918:
can be used to build an aperiodic tiling, and to study
546:
in spaces with more dimensions. He further defined the
409:, Spain, using square, triangle, and hexagon prototiles
71:, forming edge‑to‑edge, regular and other tessellations
4681:
Aha! A Two Volume Collection: Aha! Gotcha Aha! Insight
4601:
Polyominoes: A guide to puzzles and problems in tiling
3081:(1961). "Proving theorems by pattern recognition—II".
878:, with several symmetries, but no periodic repetitions
2507:
Emmer, Michele; Schattschneider, Doris (8 May 2007).
2307:
Conway, R.; Burgiel, H.; Goodman-Strauss, G. (2008).
1771:(with irregular pieces of wood or cardboard) and the
1283:
Tessellations in three or more dimensions are called
3133:(1966). "The undecidability of the domino problem".
1897:
by disregarding their colors and ornaments
1045:, tilings by other polygons have also been studied.
820:
in two independent directions can be categorized by
198:
squares or hexagons. Such tilings may be decorative
152:
has a repeating pattern. Some special kinds include
7732:
7625:
7533:
7431:
6853:
6780:
6749:
6711:
6248:
6202:
6156:
6063:
6016:
5899:
5862:
5855:
5687:
5540:
5486:
5477:
5424:
5366:
5222:
5110:
3828:"Tiling the Hyperbolic Plane with Regular Polygons"
2788:"What symmetry groups are present in the Alhambra?"
2591:"Equilateral convex pentagons which tile the plane"
1953:square tiling, isohedrally distorted into I shapes
6090:The Drawing of Geometric Patterns in Saracenic Art
4980:
4954:
4902:
4627:
4598:
3669:Kaleidoscopes: Selected Writings of H.S.M. Coxeter
3666:
3618:
2758:
2688:
266:A temple mosaic from the ancient Sumerian city of
230:often made use of tessellations, both in ordinary
6225:Goudreau Museum of Mathematics in Art and Science
3945:"Introduction to Hyperbolic and Automatic Groups"
1651:, and similar structures. The model, named after
1185:for each defining point is a convex polygon. The
4664:. Vol. 208, no. May. pp. 154–164.
4518:. American Jigsaw Puzzle Society. Archived from
3007:. University of London and EPSRC. Archived from
2399:
1678:, a characteristic example of which is found at
1212:Tessellating three-dimensional (3-D) space: the
3761:Journal für die reine und angewandte Mathematik
3498:George, Paul Louis; Borouchaki, Houman (1998).
3113:(November 1965). "Games, logic and computers".
1763:Tessellations have given rise to many types of
1248:, and triangular, quadrilateral, and hexagonal
1173:, in which the cells are always convex polygons
1029:states that for every tessellation of a normal
30:"Tessellate" redirects here. For the song, see
2661:NRICH (Millennium Maths Project) (1997–2012).
2510:M.C. Escher's Legacy: A Centennial Celebration
2421:. Vol. 53, no. 4. pp. 224–233.
2192:"Dynamic Coverage Problems in Sensor Networks"
2038:
2036:
1521:A quilt showing a regular tessellation pattern
496:, such that the tiles intersect only on their
421:do not obey this rule. Among those that do, a
226:palace. In the twentieth century, the work of
7409:
6317:
6220:European Society for Mathematics and the Arts
5394:Mathematica: A World of Numbers... and Beyond
5088:
4807:Henle, Frederick V.; Henle, James M. (2008).
1815:found four new tessellations with pentagons.
1723:proposed a packing using only one solid, the
922:, which are structures with aperiodic order.
484:More formally, a tessellation or tiling is a
8:
5374:List of works designed with the golden ratio
3654:(2nd ed.). Blandford. pp. 138–139.
3135:Memoirs of the American Mathematical Society
2796:Notices of the American Mathematical Society
2532:
2530:
2513:. Berlin Heidelberg: Springer. p. 325.
2190:Djidjev, Hristo; Potkonjak, Miodrag (2012).
1933:Alternated octagonal or tritetragonal tiling
1556:when cutting out shapes for objects such as
1356:, one of four regular compact honeycombs in
1216:is one of the solids that can be stacked to
573:Euclidean tilings by convex regular polygons
285:tilings made of small squared blocks called
4726:"Tiling the Plane with Congruent Pentagons"
2107:
2105:
504:in which all tiles in the tessellation are
301:
293:, sometimes displaying geometric patterns.
34:. For the computer graphics technique, see
7416:
7402:
7394:
6708:
6694:
6544:
6444:
6340:
6324:
6310:
6302:
5859:
5483:
5095:
5081:
5073:
5065:"The Geometry Junkyard: Hyperbolic Tiling"
2541:. Woodhead Publishing. pp. 172, 175.
2539:Geometric Symmetry in Patterns and Tilings
2076:. Cambridge University Press. p. 280.
780:semi-regular (or Archimedean) tessellation
632:is a tessellation for which every tile is
542:. These are the analogues to polygons and
89:celebrating the artistic tessellations of
6635:Dividing a square into similar rectangles
5434:Cathedral of Saint Mary of the Assumption
4630:Hinged Dissections: Swinging and Twisting
4341:
4227:
3810:
3773:
3235:
2942:
2899:
2608:
2566:. University of Wisconsin. Archived from
1329:Tessellations in non-Euclidean geometries
678:is a tessellation in which all tiles are
640:, the intersection of any two tiles is a
551:vertex, so its Schläfli symbol is {6,3}.
477:in Spain that attracted the attention of
3723:Senechal, Marjorie (26 September 1996).
3530:Lectures on Random Voronoi Tessellations
2564:"Some Special Radial and Spiral Tilings"
2165:
5980:Vier Bücher von Menschlicher Proportion
4605:. Mathematical Association of America.
2914:
2648:
2589:Hirschhorn, M. D.; Hunt, D. C. (1985).
2340:(2nd ed.). Oxford. pp. 61–62.
2321:
2177:
2032:
1986:
1893:All tiling elements are
1827:
1791:have made many uses of tessellation in
1735:In puzzles and recreational mathematics
1601:Patterns in nature § Tessellations
1536:Tessellations are also a main genre in
1194:construct random tilings of the plane.
381:). It corresponds to the everyday term
4006:
3994:
3930:
2350:
1373:uniform tiling in the hyperbolic plane
4983:Mathematics From the Birth of Numbers
4707:"The Importance of Recreational Math"
4253:"How honeycombs can build themselves"
3879:"Mathematics in Art and Architecture"
3851:The Beauty of Geometry: Twelve Essays
2114:Geometric Patterns from Roman Mosaics
2002:The tiles are usually required to be
1632:, are characteristically tessellate.
1404:uniform honeycomb in hyperbolic space
538:, which mathematicians nowadays call
473:tessellations of glazed tiles at the
141:, tessellation can be generalized to
7:
4909:(New Concise NAL ed.). Abrams.
2695:. New York: Penguin Books. pp.
2687:(1991). "two squares tessellation".
2223:(1891). "Simmetrija na ploskosti ".
2074:Mosaics of the Greek and Roman world
1795:. For example, Dudeney invented the
1399:mapping any vertex onto any other).
1084:-shaped non-convex 12-sided polygons
686:has a unit tile that is a nonconvex
7702:The Chemical Basis of Morphogenesis
6165:Journal of Mathematics and the Arts
3218:"An aperiodic set of 13 Wang tiles"
2892:"Two-Dimensional Symmetry Mutation"
2786:Grünbaum, Branko (June–July 2006).
2088:"The Brantingham Geometric Mosaics"
1613:is a natural tessellated structure.
609:about each vertex is the same. The
137:, with no overlaps and no gaps. In
6179:Making Mathematics with Needlework
4146:10.1111/j.1151-2916.1990.tb05290.x
3881:. National University of Singapore
3097:10.1002/j.1538-7305.1961.tb03975.x
2072:Dunbabin, Katherine M. D. (2006).
1198:Tessellations in higher dimensions
25:
6005:I quattro libri dell'architettura
4929:Penrose Tiles to Trapdoor Ciphers
4545:. Barnes & Noble. p. 9.
4296:Shorter Oxford English dictionary
3673:. John Wiley & Sons. p.
2976:"Penrose Tiles Talk Across Miles"
1799:, while Gardner wrote about the "
1463:in Roman Syria. second century AD
747:, edge-to-edge tiling made up of
581:List of Euclidean uniform tilings
162:tiles all of the same shape, and
7524:
6434:
6427:
6284:
6283:
5529:Self-portrait in a Convex Mirror
5213:
4194:Seghir, R.; Arscott, S. (2015).
3334:Conover, Emily (24 March 2023).
2263:Heesch, H.; Kienzle, O. (1963).
2006:(topologically equivalent) to a
1943:
1925:
1902:
1886:
1868:
1849:
1830:
1711:; these are packed according to
1707:Other natural patterns occur in
1567:Tessellation is apparent in the
1363:It is possible to tessellate in
901:; these tilings have unexpected
242:. Tessellations form a class of
99:
78:
54:
36:Tessellation (computer graphics)
27:Tiling of a plane in mathematics
5037:Wolfram MathWorld: Tessellation
4724:Schattschneider, Doris (1978).
2596:Journal of Combinatorial Theory
1914:
1894:
1487:tiles in buildings such as the
1155:Polyomino § Uses of polyominoes
866:List of aperiodic sets of tiles
407:Archeological Museum of Seville
274:Tessellations were used by the
246:, for example in the arrays of
6240:National Museum of Mathematics
5992:Regole generali d'architettura
4830:10.1080/00029890.2008.11920491
4705:Suri, Mani (12 October 2015).
4634:. Cambridge University Press.
4626:Frederickson, Greg N. (2002).
2562:Dutch, Steven (29 July 1999).
2199:Los Alamos National Laboratory
2043:Pickover, Clifford A. (2009).
1189:is a tessellation that is the
840:wallpaper group and one is in
774:is not an edge‑to‑edge tiling.
492:number of closed sets, called
1:
6660:Regular Division of the Plane
4817:American Mathematical Monthly
4181:10.1016/S0022-5096(99)00081-2
3595:10.1524/zkri.1981.154.3-4.199
3237:10.1016/S0012-365X(96)00118-5
3084:Bell System Technical Journal
2980:American Mathematical Society
2765:. New York: Springer-Verlag.
1406:is a uniform tessellation of
1395:), and isogonal (there is an
1354:{3,5,3} icosahedral honeycomb
1340:in hyperbolic plane, seen in
1096:equal to 3, 4, 5, and 6. For
567:Introduction to tessellations
145:and a variety of geometries.
5765:Garden of Cosmic Speculation
4901:(1974). J. L. Locher (ed.).
3652:Minerals and Rocks in Colour
3477:(2nd ed.). John Wiley.
2610:10.1016/0097-3165(85)90078-0
2400:Grünbaum & Shephard 1987
1811:, the amateur mathematician
1228:can be stacked in a regular
615:four squares at every vertex
488:of the Euclidean plane by a
469:The elaborate and colourful
361:is a small cubical piece of
333:Alexei Vasilievich Shubnikov
6568:Architectonic and catoptric
6466:Aperiodic set of prototiles
5009:. Weidenfeld and Nicolson.
4566:Golomb, Solomon W. (1994).
4516:"History of Jigsaw Puzzles"
4325:(September/October): 18–22.
4023:. F+W Media. pp. 4–8.
3393:Encyclopedia of Mathematics
3057:10.1103/physrevlett.58.1761
1935:is a uniform tiling of the
1725:bitruncated cubic honeycomb
1506:in 1936. Escher made four "
1321:that can be used to tile a
1043:tilings by regular polygons
1037:Tessellations with polygons
1016:; more generally, at least
534:pioneered this by defining
220:decorative geometric tiling
180:aperiodic set of prototiles
7797:
5282:Islamic geometric patterns
5007:What Shape Is a Snowflake?
4953:; Shephard, G. C. (1987).
4933:Cambridge University Press
4597:Martin, George E. (1991).
4574:Princeton University Press
4019:Porter, Christine (2006).
3900:Whittaker, Andrew (2008).
3726:Quasicrystals and Geometry
2368:Section: Tessellated floor
2334:Cundy and Rollett (1961).
1895:identical pseudo‑triangles
1752:
1598:
1444:
1338:Rhombitriheptagonal tiling
1201:
1001:
859:
801:
570:
40:
29:
7682:D'Arcy Wentworth Thompson
7522:
6707:
6693:
6554:
6543:
6456:
6443:
6425:
6352:
6339:
6279:
6103:A Mathematician's Apology
5211:
4905:The World of M. C. Escher
4435:10.1080/00107518408210979
4267:10.1038/nature.2013.13398
3625:. Firefly Books. p.
3617:Oldershaw, Cally (2003).
3353:with image of the pattern
3273:10.1016/j.cag.2007.10.001
2890:Huson, Daniel H. (1991).
2736:10.4169/math.mag.84.4.283
2665:. University of Cambridge
2090:. Hull City Council. 2008
1419:convex uniform honeycombs
933:that tile the plane only
403:rhombitrihexagonal tiling
349:and Otto Kienzle (1963).
6215:The Bridges Organization
4454:University of Wollongong
3903:Speak the Culture: Spain
3775:10.1515/crll.1873.75.292
3650:Kirkaldy, J. F. (1968).
3261:Computers & Graphics
3216:Culik, Karel II (1996).
3166:Inventiones Mathematicae
2894:. Princeton University.
2757:Armstrong, M.A. (1988).
2663:"Schläfli Tessellations"
2537:Horne, Clare E. (2000).
2461:. pp. 14, 69, 149.
2353:, pp. 11–12, 15–16.
2309:The Symmetries of Things
1876:Floret pentagonal tiling
1793:recreational mathematics
1759:recreational mathematics
1697:Tessellate pattern in a
1548:Tessellation is used in
998:Tessellations and colour
289:were widely employed in
6077:The Grammar of Ornament
6029:Nature's Harmonic Unity
5939:De prospectiva pingendi
4491:Oxford University Press
3754:Schwarz, H. A. (1873).
3527:Moller, Jesper (1994).
3417:The Grammar of Ornament
3036:Physical Review Letters
2836:10.1126/science.1135491
2629:"Regular Tessellations"
2370:. Basilica di San Marco
2364:"Basilica di San Marco"
1803:", a shape that can be
1729:Weaire–Phelan structure
721:, is not a divisor of 2
702:of a regular pentagon,
6230:Institute For Figuring
6142:The 'Life' of a Carpet
5967:A Treatise on Painting
4541:Slocum, Jerry (2001).
4399:. New York: Macmillan.
3703:"Wythoff construction"
2904:– via CiteSeerX.
2413:Schattschneider, Doris
2287:Merriam-Webster Online
2112:Field, Robert (1988).
1750:
1704:
1628:, and some species of
1614:
1550:manufacturing industry
1522:
1464:
1360:
1345:
1308:Schmitt-Conway biprism
1297:semiregular honeycombs
1280:
1221:
1187:Delaunay triangulation
1174:
1085:
1021:
966:
937:
912:translational symmetry
879:
818:translational symmetry
813:
775:
669:
481:
410:
323:
302:
271:
7516:Widmanstätten pattern
6111:George David Birkhoff
6085:Ernest Hanbury Hankin
5953:De divina proportione
5933:Piero della Francesca
5912:Leon Battista Alberti
5499:Piero della Francesca
5138:Hyperboloid structure
4768:"Squaring the Square"
4485:Ball, Philip (2009).
4251:Ball, Philip (2013).
4161:J. Mech. Phys. Solids
4075:Origami Tessellations
4073:Gjerde, Eric (2008).
4044:Beyer, Jinny (1999).
3621:Firefly Guide to Gems
3452:10.1145/116873.116880
3440:ACM Computing Surveys
3388:"Four-colour problem"
2930:Annals of Mathematics
2148:Harmony of the Worlds
1981:Explanatory footnotes
1857:Snub hexagonal tiling
1742:
1696:
1674:in Northern Ireland.
1641:Gilbert tessellations
1608:
1583:being observed using
1520:
1454:
1445:Further information:
1425:, and represented by
1423:Wythoff constructions
1351:
1336:
1278:
1211:
1168:
1079:
1011:
1002:Further information:
961:
928:
873:
811:
769:
753:equilateral triangles
658:
571:Further information:
468:
449:other shapes such as
405:: tiled floor in the
400:
318:
265:
216:Moroccan architecture
185:tessellation of space
121:is the covering of a
6036:Frederik Macody Lund
5907:Filippo Brunelleschi
5788:Hamid Naderi Yeganeh
5650:La condition humaine
5048:Tilings Encyclopedia
4957:Tilings and Patterns
4809:"Squaring the plane"
4733:Mathematics Magazine
4678:(14 December 2006).
4450:Tesselated pavements
4415:Contemporary Physics
4359:Mathematical Geology
3950:. University of Utah
3908:Thorogood Publishing
3857:. pp. 212–213.
3287:Smith, Cyril Stanley
3223:Discrete Mathematics
3161:Robinson, Raphael M.
2723:Mathematics Magazine
2418:Mathematics Magazine
1970:Honeycomb (geometry)
1965:Discrete global grid
1676:Tessellated pavement
1477:Islamic architecture
1417:families of compact
1301:Wythoff construction
1246:truncated octahedron
1242:rhombic dodecahedron
1214:rhombic dodecahedron
1204:Honeycomb (geometry)
1178:Voronoi or Dirichlet
1041:Next to the various
992:mathematical problem
784:vertex configuration
741:regular tessellation
668:, discovered in 2015
648:, and all tiles are
556:vertex configuration
423:regular tessellation
129:, using one or more
85:A wall sculpture in
7750:Mathematics and art
7740:Pattern recognition
7710:Aristid Lindenmayer
6261:Mathematical beauty
6186:Rhythm of Structure
6129:Gödel, Escher, Bach
5925:De re aedificatoria
5556:The Ancient of Days
5175:Projective geometry
5104:Mathematics and art
4794:Scientific American
4684:. MAA. p. 48.
4661:Scientific American
4522:on 11 February 2014
4427:1984ConPh..25...59W
4212:2015NatSR...514787S
4173:2000JMPSo..48.1107X
3997:, pp. 142–143.
3975:"Hyperbolic Escher"
3587:1981ZK....154..199E
3179:1971InMat..12..177R
3116:Scientific American
3049:1987PhRvL..58.1761D
2828:2007Sci...315.1106L
2761:Groups and Symmetry
2337:Mathematical Models
1840:, one of the three
1817:Squaring the square
1809:Scientific American
1767:, from traditional
1579:– with a degree of
1529:of patch shapes in
1447:Mathematics and art
1369:hyperbolic geometry
1367:geometries such as
1342:Poincaré disk model
1238:Platonic polyhedron
1080:Tessellation using
1027:four colour theorem
1004:Four colour theorem
974:rotational symmetry
895:substitution tiling
788:Pythagorean tilings
734:anisohedral tilings
622:edge-to-edge tiling
425:has both identical
291:classical antiquity
236:hyperbolic geometry
165:semiregular tilings
7688:On Growth and Form
7588:Logarithmic spiral
7425:Patterns in nature
6266:Patterns in nature
6123:Douglas Hofstadter
5749:Desmond Paul Henry
5739:Bathsheba Grossman
5671:The Swallow's Tail
5592:Giorgio de Chirico
5464:Sydney Opera House
5319:Croatian interlace
5042:Dirk Frettlöh and
4881:Dover Publications
4543:The Tao of Tangram
4493:. pp. 73–76.
4456:. pp. 11–20.
4371:10.1007/BF01031092
4167:(6–7): 1107–1131.
4079:Taylor and Francis
4052:. pp. Ch. 7.
3973:Leys, Jos (2015).
3855:Dover Publications
3700:Weisstein, Eric W.
3506:. pp. 34–35.
3436:Aurenhammer, Franz
3420:(folio ed.).
3371:Roberts, Soibhan,
3187:10.1007/bf01418780
3119:. pp. 98–106.
3002:"Aperiodic Tiling"
2867:Weisstein, Eric W.
2626:Weisstein, Eric W.
2484:Weisstein, Eric W.
2227:. 2 (in Russian).
1975:Space partitioning
1861:semiregular tiling
1751:
1705:
1637:patterns in nature
1615:
1523:
1465:
1408:uniform polyhedral
1361:
1358:hyperbolic 3-space
1346:
1319:spherical triangle
1281:
1222:
1218:fill space exactly
1175:
1086:
1066:fundamental domain
1062:wallpaper group p2
1022:
1014:fundamental domain
967:
938:
880:
814:
776:
772:Pythagorean tiling
670:
611:fundamental region
482:
411:
324:
272:
244:patterns in nature
232:Euclidean geometry
7758:
7757:
7715:Benoît Mandelbrot
7615:Self-organization
7551:Natural selection
7541:Pattern formation
7391:
7390:
7387:
7386:
7383:
7382:
6689:
6688:
6580:Computer graphics
6539:
6538:
6423:
6422:
6299:
6298:
6152:
6151:
6116:Aesthetic Measure
5987:Sebastiano Serlio
5961:Leonardo da Vinci
5851:
5850:
5843:Margaret Wertheim
5504:Leonardo da Vinci
5054:Tessellations.org
5016:978-0-297-60723-6
4994:978-0-393-04002-9
4968:978-0-7167-1193-3
4961:. W. H. Freeman.
4942:978-0-88385-521-8
4916:978-0-451-79961-6
4890:978-0-486-61480-9
4876:Regular Polytopes
4871:Coxeter, H. S. M.
4739:(1). MAA: 29–44.
4691:978-0-88385-551-5
4641:978-0-521-81192-7
4612:978-0-88385-501-0
4583:978-0-691-02444-8
4552:978-1-4351-0156-2
4500:978-0-199-60486-9
4463:978-0-864-18001-8
4322:American Gardener
4305:978-0-19-920687-2
4220:10.1038/srep14787
4088:978-1-568-81451-3
4059:978-0-8092-2866-9
4050:Contemporary Book
4030:978-0-7153-1941-3
3933:, pp. 5, 17.
3917:978-1-85418-605-8
3864:978-0-486-40919-1
3740:978-0-521-57541-6
3684:978-0-471-01003-6
3636:978-1-55297-814-6
3540:978-1-4612-2652-9
3513:978-2-86601-692-0
3484:978-0-471-98635-5
3147:10.1090/memo/0066
3043:(17): 1761–1765.
3014:on 29 August 2017
2822:(5815): 1106–10.
2772:978-3-540-96675-3
2706:978-0-14-011813-1
2548:978-1-85573-492-0
2520:978-3-540-28849-7
2468:978-0-486-61480-9
2454:Regular Polytopes
2449:Coxeter, H. S. M.
2123:978-0-906-21263-9
2058:978-1-4027-5796-9
1838:Triangular tiling
1797:hinged dissection
1748:dissection puzzle
1664:columnar jointing
1581:self-organisation
1437:for each family.
1385:vertex-transitive
1127:Heptagonal tiling
1105:Pentagonal tiling
978:Sébastien Truchet
891:recursive process
887:aperiodic tilings
856:Aperiodic tilings
850:Orbifold notation
792:edge tessellation
692:Hirschhorn tiling
675:monohedral tiling
666:pentagonal tiling
650:uniformly bounded
603:vertex-transitive
160:regular polygonal
143:higher dimensions
43:Mathematical tile
32:Tessellate (song)
16:(Redirected from
7788:
7566:Sexual selection
7528:
7418:
7411:
7404:
7395:
6709:
6695:
6647:Conway criterion
6574:Circle Limit III
6545:
6478:Einstein problem
6445:
6438:
6431:
6367:Schwarz triangle
6341:
6326:
6319:
6312:
6303:
6287:
6286:
6137:Nikos Salingaros
5860:
5828:Hiroshi Sugimoto
5778:Robert Longhurst
5724:Helaman Ferguson
5679:Crockett Johnson
5608:Circle Limit III
5577:Danseuse au café
5484:
5454:Pyramid of Khufu
5217:
5097:
5090:
5083:
5074:
5068:
5020:
4998:
4986:
4972:
4960:
4951:Grünbaum, Branko
4946:
4920:
4908:
4894:
4857:
4856:
4855:on 20 June 2006.
4854:
4848:. Archived from
4813:
4804:
4798:
4797:
4789:
4783:
4782:
4780:
4778:
4763:
4757:
4756:
4730:
4721:
4715:
4714:
4702:
4696:
4695:
4672:
4666:
4665:
4652:
4646:
4645:
4633:
4623:
4617:
4616:
4604:
4594:
4588:
4587:
4572:(2nd ed.).
4563:
4557:
4556:
4538:
4532:
4531:
4529:
4527:
4514:McAdam, Daniel.
4511:
4505:
4504:
4482:
4476:
4475:
4445:
4439:
4438:
4407:
4401:
4400:
4389:
4383:
4382:
4354:
4348:
4347:
4345:
4333:
4327:
4326:
4316:
4310:
4309:
4292:
4286:
4285:
4283:
4281:
4248:
4242:
4241:
4231:
4191:
4185:
4184:
4156:
4150:
4149:
4140:(7): 2144–2146.
4134:J. Am. Chem. Soc
4129:
4123:
4122:
4120:
4118:
4109:. Archived from
4099:
4093:
4092:
4070:
4064:
4063:
4041:
4035:
4034:
4016:
4010:
4004:
3998:
3992:
3986:
3985:
3983:
3981:
3970:
3964:
3963:
3957:
3955:
3949:
3940:
3934:
3928:
3922:
3921:
3897:
3891:
3890:
3888:
3886:
3875:
3869:
3868:
3846:
3840:
3839:
3837:
3835:
3826:Zadnik, Gašper.
3823:
3817:
3816:
3814:
3802:
3796:
3795:
3777:
3751:
3745:
3744:
3720:
3714:
3713:
3712:
3695:
3689:
3688:
3672:
3662:
3656:
3655:
3647:
3641:
3640:
3624:
3614:
3608:
3606:
3581:(3–4): 199–215.
3570:
3564:
3563:
3551:
3545:
3544:
3524:
3518:
3517:
3495:
3489:
3488:
3470:
3464:
3463:
3432:
3426:
3425:
3422:Bernard Quaritch
3408:
3402:
3401:
3384:
3378:
3369:
3363:
3360:
3354:
3352:
3350:
3348:
3331:
3325:
3324:
3283:
3277:
3276:
3256:
3250:
3249:
3239:
3230:(1–3): 245–251.
3213:
3207:
3206:
3157:
3151:
3150:
3127:
3121:
3120:
3107:
3101:
3100:
3075:
3069:
3068:
3030:
3024:
3023:
3021:
3019:
3013:
3006:
2997:
2991:
2990:
2988:
2986:
2971:
2965:
2964:
2946:
2924:
2918:
2917:, pp. 1–18.
2912:
2906:
2905:
2903:
2887:
2881:
2880:
2879:
2862:
2856:
2855:
2811:
2805:
2804:
2792:
2783:
2777:
2776:
2764:
2754:
2748:
2747:
2717:
2711:
2710:
2694:
2681:
2675:
2674:
2672:
2670:
2658:
2652:
2646:
2640:
2639:
2638:
2621:
2615:
2614:
2612:
2586:
2580:
2579:
2577:
2575:
2559:
2553:
2552:
2534:
2525:
2524:
2504:
2498:
2497:
2496:
2479:
2473:
2472:
2445:
2439:
2438:
2409:
2403:
2397:
2380:
2379:
2377:
2375:
2360:
2354:
2348:
2342:
2341:
2331:
2325:
2319:
2313:
2312:
2304:
2298:
2297:
2295:
2293:
2279:
2273:
2272:
2260:
2254:
2253:
2245:Colored Symmetry
2239:
2233:
2232:
2217:
2211:
2210:
2208:
2206:
2196:
2187:
2181:
2175:
2169:
2163:
2152:
2151:
2143:Harmonices Mundi
2138:Kepler, Johannes
2134:
2128:
2127:
2109:
2100:
2099:
2097:
2095:
2084:
2078:
2077:
2069:
2063:
2062:
2040:
2020:
2017:
2011:
2000:
1994:
1991:
1947:
1937:hyperbolic plane
1929:
1911:Voderberg tiling
1906:
1890:
1880:pentagon tilings
1872:
1853:
1834:
1717:minimal surfaces
1715:, which require
1684:Tasman Peninsula
1672:Giant's Causeway
1589:nanotechnologies
1544:In manufacturing
1475:wall tilings of
1435:Coxeter diagrams
1377:regular polygons
1315:Schwarz triangle
1138:octagonal tiling
1135:
1124:
1116:Hexagonal tiling
1113:
1102:
1048:Any triangle or
907:Pinwheel tilings
903:self-replicating
862:Aperiodic tiling
822:wallpaper groups
798:Wallpaper groups
749:regular polygons
724:
720:
718:
717:
714:
711:
710:
684:Voderberg tiling
677:
636:equivalent to a
518:Conway criterion
437:and the regular
419:bonded brickwork
342:Colored Symmetry
328:Yevgraf Fyodorov
322:geometric mosaic
307:
304:Harmonices Mundi
175:aperiodic tiling
170:wallpaper groups
131:geometric shapes
103:
82:
58:
21:
18:Tiling the plane
7796:
7795:
7791:
7790:
7789:
7787:
7786:
7785:
7761:
7760:
7759:
7754:
7728:
7621:
7529:
7520:
7427:
7422:
7392:
7379:
6856:
6849:
6782:
6776:
6745:
6703:
6685:
6550:
6535:
6452:
6439:
6433:
6432:
6419:
6410:Wallpaper group
6348:
6335:
6330:
6300:
6295:
6275:
6271:Sacred geometry
6244:
6210:Ars Mathematica
6198:
6148:
6059:
6012:
5999:Andrea Palladio
5895:
5888:De architectura
5847:
5803:Antoine Pevsner
5783:Jeanette McLeod
5734:Susan Goldstine
5683:
5542:
5536:
5473:
5459:Sagrada Família
5420:
5362:
5230:Algorithmic art
5218:
5209:
5205:Wallpaper group
5143:Minimal surface
5106:
5101:
5061:Eppstein, David
5059:
5027:
5017:
5001:
4995:
4975:
4969:
4949:
4943:
4925:Gardner, Martin
4923:
4917:
4897:
4891:
4869:
4866:
4861:
4860:
4852:
4811:
4806:
4805:
4801:
4791:
4790:
4786:
4776:
4774:
4765:
4764:
4760:
4745:10.2307/2689644
4728:
4723:
4722:
4718:
4704:
4703:
4699:
4692:
4676:Gardner, Martin
4674:
4673:
4669:
4656:Gardner, Martin
4654:
4653:
4649:
4642:
4625:
4624:
4620:
4613:
4596:
4595:
4591:
4584:
4565:
4564:
4560:
4553:
4540:
4539:
4535:
4525:
4523:
4513:
4512:
4508:
4501:
4484:
4483:
4479:
4464:
4447:
4446:
4442:
4409:
4408:
4404:
4391:
4390:
4386:
4356:
4355:
4351:
4335:
4334:
4330:
4318:
4317:
4313:
4306:
4294:
4293:
4289:
4279:
4277:
4250:
4249:
4245:
4193:
4192:
4188:
4158:
4157:
4153:
4131:
4130:
4126:
4116:
4114:
4101:
4100:
4096:
4089:
4072:
4071:
4067:
4060:
4043:
4042:
4038:
4031:
4018:
4017:
4013:
4005:
4001:
3993:
3989:
3979:
3977:
3972:
3971:
3967:
3953:
3951:
3947:
3943:Gersten, S. M.
3942:
3941:
3937:
3929:
3925:
3918:
3910:. p. 153.
3899:
3898:
3894:
3884:
3882:
3877:
3876:
3872:
3865:
3848:
3847:
3843:
3833:
3831:
3825:
3824:
3820:
3804:
3803:
3799:
3768:(75): 292–335.
3753:
3752:
3748:
3741:
3729:. CUP Archive.
3722:
3721:
3717:
3698:
3697:
3696:
3692:
3685:
3664:
3663:
3659:
3649:
3648:
3644:
3637:
3616:
3615:
3611:
3572:
3571:
3567:
3553:
3552:
3548:
3541:
3526:
3525:
3521:
3514:
3497:
3496:
3492:
3485:
3472:
3471:
3467:
3434:
3433:
3429:
3410:
3409:
3405:
3386:
3385:
3381:
3370:
3366:
3361:
3357:
3346:
3344:
3333:
3332:
3328:
3305:10.2307/1578535
3285:
3284:
3280:
3258:
3257:
3253:
3215:
3214:
3210:
3159:
3158:
3154:
3129:
3128:
3124:
3109:
3108:
3104:
3077:
3076:
3072:
3032:
3031:
3027:
3017:
3015:
3011:
3004:
3000:Harriss, E. O.
2999:
2998:
2994:
2984:
2982:
2974:Austin, David.
2973:
2972:
2968:
2953:10.2307/2118575
2926:
2925:
2921:
2913:
2909:
2889:
2888:
2884:
2865:
2864:
2863:
2859:
2813:
2812:
2808:
2790:
2785:
2784:
2780:
2773:
2756:
2755:
2751:
2719:
2718:
2714:
2707:
2683:
2682:
2678:
2668:
2666:
2660:
2659:
2655:
2647:
2643:
2624:
2623:
2622:
2618:
2588:
2587:
2583:
2573:
2571:
2570:on 4 April 2013
2561:
2560:
2556:
2549:
2536:
2535:
2528:
2521:
2506:
2505:
2501:
2482:
2481:
2480:
2476:
2469:
2447:
2446:
2442:
2427:10.2307/2689617
2411:
2410:
2406:
2398:
2383:
2373:
2371:
2362:
2361:
2357:
2349:
2345:
2333:
2332:
2328:
2320:
2316:
2306:
2305:
2301:
2291:
2289:
2281:
2280:
2276:
2262:
2261:
2257:
2241:
2240:
2236:
2219:
2218:
2214:
2204:
2202:
2194:
2189:
2188:
2184:
2176:
2172:
2164:
2155:
2136:
2135:
2131:
2124:
2111:
2110:
2103:
2093:
2091:
2086:
2085:
2081:
2071:
2070:
2066:
2059:
2051:. p. 372.
2042:
2041:
2034:
2029:
2024:
2023:
2018:
2014:
2001:
1997:
1992:
1988:
1983:
1961:
1954:
1948:
1939:
1930:
1921:
1917:tiling made of
1907:
1898:
1891:
1882:
1873:
1864:
1854:
1845:
1842:regular tilings
1835:
1826:
1761:
1753:Main articles:
1737:
1666:as a result of
1603:
1597:
1546:
1449:
1443:
1421:, generated as
1331:
1240:to do so), the
1230:crystal pattern
1206:
1200:
1163:
1161:Voronoi tilings
1130:
1119:
1108:
1097:
1039:
1031:Euclidean plane
1006:
1000:
953:halting problem
883:Penrose tilings
868:
860:Main articles:
858:
846:frieze patterns
806:
804:Wallpaper group
800:
722:
715:
712:
708:
706:
705:
703:
696:pentagon tiling
673:
583:
569:
564:
548:Schläfli symbol
532:Ludwig Schläfli
395:
355:
347:Heinrich Heesch
298:Johannes Kepler
260:
248:hexagonal cells
214:such as in the
155:regular tilings
150:periodic tiling
111:
110:
109:
108:
107:
104:
95:
94:
93:
83:
74:
73:
72:
59:
46:
39:
28:
23:
22:
15:
12:
11:
5:
7794:
7792:
7784:
7783:
7778:
7773:
7763:
7762:
7756:
7755:
7753:
7752:
7747:
7742:
7736:
7734:
7730:
7729:
7727:
7726:
7725:
7724:
7712:
7707:
7706:
7705:
7693:
7692:
7691:
7679:
7677:Wilson Bentley
7674:
7672:Joseph Plateau
7669:
7664:
7659:
7658:
7657:
7645:
7640:
7635:
7629:
7627:
7623:
7622:
7620:
7619:
7618:
7617:
7612:
7610:Plateau's laws
7607:
7605:Fluid dynamics
7602:
7592:
7591:
7590:
7585:
7580:
7570:
7569:
7568:
7563:
7558:
7553:
7543:
7537:
7535:
7531:
7530:
7523:
7521:
7519:
7518:
7513:
7508:
7503:
7498:
7497:
7496:
7491:
7486:
7481:
7471:
7466:
7461:
7456:
7451:
7446:
7441:
7435:
7433:
7429:
7428:
7423:
7421:
7420:
7413:
7406:
7398:
7389:
7388:
7385:
7384:
7381:
7380:
7378:
7377:
7372:
7367:
7362:
7357:
7352:
7347:
7342:
7337:
7332:
7327:
7322:
7317:
7312:
7307:
7302:
7297:
7292:
7287:
7282:
7277:
7272:
7267:
7262:
7257:
7252:
7247:
7242:
7237:
7232:
7227:
7222:
7217:
7212:
7207:
7202:
7197:
7192:
7187:
7182:
7177:
7172:
7167:
7162:
7157:
7152:
7147:
7142:
7137:
7132:
7127:
7122:
7117:
7112:
7107:
7102:
7097:
7092:
7087:
7082:
7077:
7072:
7067:
7062:
7057:
7052:
7047:
7042:
7037:
7032:
7027:
7022:
7017:
7012:
7007:
7002:
6997:
6992:
6987:
6982:
6977:
6972:
6967:
6962:
6957:
6952:
6947:
6942:
6937:
6932:
6927:
6922:
6917:
6912:
6907:
6902:
6897:
6892:
6887:
6882:
6877:
6872:
6867:
6861:
6859:
6851:
6850:
6848:
6847:
6842:
6837:
6832:
6827:
6822:
6817:
6812:
6807:
6802:
6797:
6792:
6786:
6784:
6778:
6777:
6775:
6774:
6769:
6764:
6759:
6753:
6751:
6747:
6746:
6744:
6743:
6738:
6733:
6728:
6723:
6717:
6715:
6705:
6704:
6698:
6691:
6690:
6687:
6686:
6684:
6683:
6678:
6673:
6668:
6663:
6656:
6655:
6654:
6649:
6639:
6638:
6637:
6632:
6627:
6622:
6621:
6620:
6607:
6602:
6597:
6592:
6587:
6582:
6577:
6570:
6565:
6555:
6552:
6551:
6548:
6541:
6540:
6537:
6536:
6534:
6533:
6528:
6523:
6522:
6521:
6507:
6502:
6497:
6492:
6487:
6486:
6485:
6483:Socolar–Taylor
6475:
6474:
6473:
6463:
6461:Ammann–Beenker
6457:
6454:
6453:
6448:
6441:
6440:
6426:
6424:
6421:
6420:
6418:
6417:
6412:
6407:
6406:
6405:
6400:
6395:
6384:Uniform tiling
6381:
6380:
6379:
6369:
6364:
6359:
6353:
6350:
6349:
6344:
6337:
6336:
6331:
6329:
6328:
6321:
6314:
6306:
6297:
6296:
6294:
6293:
6280:
6277:
6276:
6274:
6273:
6268:
6263:
6258:
6252:
6250:
6246:
6245:
6243:
6242:
6237:
6232:
6227:
6222:
6217:
6212:
6206:
6204:
6200:
6199:
6197:
6196:
6189:
6182:
6175:
6168:
6160:
6158:
6154:
6153:
6150:
6149:
6147:
6146:
6145:
6144:
6134:
6133:
6132:
6120:
6119:
6118:
6108:
6107:
6106:
6094:
6093:
6092:
6082:
6081:
6080:
6067:
6065:
6061:
6060:
6058:
6057:
6056:
6055:
6053:The Greek Vase
6045:
6044:
6043:
6033:
6032:
6031:
6020:
6018:
6014:
6013:
6011:
6010:
6009:
6008:
5996:
5995:
5994:
5984:
5983:
5982:
5975:Albrecht Dürer
5972:
5971:
5970:
5958:
5957:
5956:
5944:
5943:
5942:
5930:
5929:
5928:
5921:
5909:
5903:
5901:
5897:
5896:
5894:
5893:
5892:
5891:
5879:
5878:
5877:
5866:
5864:
5857:
5853:
5852:
5849:
5848:
5846:
5845:
5840:
5838:Roman Verostko
5835:
5830:
5825:
5820:
5815:
5813:Alba Rojo Cama
5810:
5805:
5800:
5795:
5790:
5785:
5780:
5775:
5770:
5769:
5768:
5759:Charles Jencks
5756:
5751:
5746:
5744:George W. Hart
5741:
5736:
5731:
5726:
5721:
5716:
5711:
5706:
5697:
5691:
5689:
5685:
5684:
5682:
5681:
5676:
5675:
5674:
5667:
5655:
5654:
5653:
5641:
5640:
5639:
5632:
5625:
5618:
5611:
5599:
5594:
5589:
5588:
5587:
5580:
5571:Jean Metzinger
5568:
5567:
5566:
5559:
5546:
5544:
5538:
5537:
5535:
5534:
5533:
5532:
5520:
5518:Albrecht Dürer
5515:
5514:
5513:
5501:
5496:
5490:
5488:
5481:
5475:
5474:
5472:
5471:
5466:
5461:
5456:
5451:
5446:
5441:
5436:
5430:
5428:
5422:
5421:
5419:
5418:
5411:
5404:
5397:
5390:
5383:
5376:
5370:
5368:
5364:
5363:
5361:
5360:
5355:
5350:
5345:
5344:
5343:
5333:
5328:
5327:
5326:
5321:
5316:
5306:
5305:
5304:
5299:
5294:
5289:
5279:
5274:
5269:
5264:
5259:
5258:
5257:
5252:
5247:
5237:
5235:Anamorphic art
5232:
5226:
5224:
5220:
5219:
5212:
5210:
5208:
5207:
5202:
5197:
5192:
5191:
5190:
5185:
5177:
5172:
5167:
5166:
5165:
5163:Camera obscura
5160:
5150:
5145:
5140:
5135:
5130:
5125:
5120:
5114:
5112:
5108:
5107:
5102:
5100:
5099:
5092:
5085:
5077:
5071:
5070:
5057:
5051:
5044:Edmund Harriss
5040:
5034:
5026:
5025:External links
5023:
5022:
5021:
5015:
4999:
4993:
4973:
4967:
4947:
4941:
4921:
4915:
4895:
4889:
4865:
4862:
4859:
4858:
4799:
4784:
4758:
4716:
4711:New York Times
4697:
4690:
4667:
4647:
4640:
4618:
4611:
4589:
4582:
4558:
4551:
4533:
4506:
4499:
4477:
4462:
4440:
4402:
4393:Gilbert, E. N.
4384:
4365:(6): 617–626.
4349:
4328:
4311:
4304:
4287:
4243:
4186:
4151:
4124:
4113:on 29 May 2015
4094:
4087:
4065:
4058:
4036:
4029:
4011:
3999:
3987:
3965:
3935:
3923:
3916:
3892:
3870:
3863:
3841:
3818:
3797:
3746:
3739:
3715:
3690:
3683:
3657:
3642:
3635:
3609:
3565:
3556:Geombinatorics
3546:
3539:
3519:
3512:
3490:
3483:
3465:
3446:(3): 345–405.
3427:
3403:
3379:
3364:
3355:
3326:
3299:(4): 373–385.
3278:
3267:(2): 268–281.
3251:
3208:
3173:(3): 177–209.
3152:
3131:Berger, Robert
3122:
3102:
3070:
3025:
2992:
2966:
2944:10.1.1.44.9723
2937:(3): 661–702.
2919:
2907:
2901:10.1.1.30.8536
2882:
2870:"Frieze Group"
2857:
2806:
2778:
2771:
2749:
2712:
2705:
2676:
2653:
2641:
2616:
2581:
2554:
2547:
2526:
2519:
2499:
2487:"Tessellation"
2474:
2467:
2440:
2404:
2381:
2355:
2343:
2326:
2314:
2299:
2274:
2255:
2234:
2212:
2182:
2170:
2168:, p. 395.
2153:
2129:
2122:
2101:
2079:
2064:
2057:
2031:
2030:
2028:
2025:
2022:
2021:
2012:
1995:
1985:
1984:
1982:
1979:
1978:
1977:
1972:
1967:
1960:
1957:
1956:
1955:
1949:
1942:
1940:
1931:
1924:
1922:
1908:
1901:
1899:
1892:
1885:
1883:
1874:
1867:
1865:
1855:
1848:
1846:
1836:
1829:
1825:
1822:
1789:Martin Gardner
1769:jigsaw puzzles
1736:
1733:
1713:Plateau's laws
1680:Eaglehawk Neck
1662:often display
1647:, needle-like
1599:Main article:
1596:
1593:
1545:
1542:
1442:
1439:
1330:
1327:
1202:Main article:
1199:
1196:
1171:Voronoi tiling
1162:
1159:
1038:
1035:
999:
996:
964:Truchet tiling
949:Turing machine
916:Fibonacci word
876:Penrose tiling
857:
854:
802:Main article:
799:
796:
700:internal angle
577:Uniform tiling
568:
565:
563:
562:In mathematics
560:
514:tile the plane
445:with no gaps.
394:
391:
354:
351:
339:in their book
259:
256:
105:
98:
97:
96:
84:
77:
76:
75:
60:
53:
52:
51:
50:
49:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7793:
7782:
7779:
7777:
7774:
7772:
7769:
7768:
7766:
7751:
7748:
7746:
7743:
7741:
7738:
7737:
7735:
7731:
7723:
7722:
7718:
7717:
7716:
7713:
7711:
7708:
7704:
7703:
7699:
7698:
7697:
7694:
7690:
7689:
7685:
7684:
7683:
7680:
7678:
7675:
7673:
7670:
7668:
7667:Ernst Haeckel
7665:
7663:
7662:Adolf Zeising
7660:
7656:
7655:
7651:
7650:
7649:
7646:
7644:
7641:
7639:
7636:
7634:
7631:
7630:
7628:
7624:
7616:
7613:
7611:
7608:
7606:
7603:
7601:
7598:
7597:
7596:
7593:
7589:
7586:
7584:
7581:
7579:
7576:
7575:
7574:
7571:
7567:
7564:
7562:
7559:
7557:
7554:
7552:
7549:
7548:
7547:
7544:
7542:
7539:
7538:
7536:
7532:
7527:
7517:
7514:
7512:
7509:
7507:
7506:Vortex street
7504:
7502:
7499:
7495:
7492:
7490:
7487:
7485:
7484:Quasicrystals
7482:
7480:
7477:
7476:
7475:
7472:
7470:
7467:
7465:
7462:
7460:
7457:
7455:
7452:
7450:
7447:
7445:
7442:
7440:
7437:
7436:
7434:
7430:
7426:
7419:
7414:
7412:
7407:
7405:
7400:
7399:
7396:
7376:
7373:
7371:
7368:
7366:
7363:
7361:
7358:
7356:
7353:
7351:
7348:
7346:
7343:
7341:
7338:
7336:
7333:
7331:
7328:
7326:
7323:
7321:
7318:
7316:
7313:
7311:
7308:
7306:
7303:
7301:
7298:
7296:
7293:
7291:
7288:
7286:
7283:
7281:
7278:
7276:
7273:
7271:
7268:
7266:
7263:
7261:
7258:
7256:
7253:
7251:
7248:
7246:
7243:
7241:
7238:
7236:
7233:
7231:
7228:
7226:
7223:
7221:
7218:
7216:
7213:
7211:
7208:
7206:
7203:
7201:
7198:
7196:
7193:
7191:
7188:
7186:
7183:
7181:
7178:
7176:
7173:
7171:
7168:
7166:
7163:
7161:
7158:
7156:
7153:
7151:
7148:
7146:
7143:
7141:
7138:
7136:
7133:
7131:
7128:
7126:
7123:
7121:
7118:
7116:
7113:
7111:
7108:
7106:
7103:
7101:
7098:
7096:
7093:
7091:
7088:
7086:
7083:
7081:
7078:
7076:
7073:
7071:
7068:
7066:
7063:
7061:
7058:
7056:
7053:
7051:
7048:
7046:
7043:
7041:
7038:
7036:
7033:
7031:
7028:
7026:
7023:
7021:
7018:
7016:
7013:
7011:
7008:
7006:
7003:
7001:
6998:
6996:
6993:
6991:
6988:
6986:
6983:
6981:
6978:
6976:
6973:
6971:
6968:
6966:
6963:
6961:
6958:
6956:
6953:
6951:
6948:
6946:
6943:
6941:
6938:
6936:
6933:
6931:
6928:
6926:
6923:
6921:
6918:
6916:
6913:
6911:
6908:
6906:
6903:
6901:
6898:
6896:
6893:
6891:
6888:
6886:
6883:
6881:
6878:
6876:
6873:
6871:
6868:
6866:
6863:
6862:
6860:
6858:
6852:
6846:
6843:
6841:
6838:
6836:
6833:
6831:
6828:
6826:
6823:
6821:
6818:
6816:
6813:
6811:
6808:
6806:
6803:
6801:
6798:
6796:
6793:
6791:
6788:
6787:
6785:
6779:
6773:
6770:
6768:
6765:
6763:
6760:
6758:
6755:
6754:
6752:
6748:
6742:
6739:
6737:
6734:
6732:
6729:
6727:
6724:
6722:
6719:
6718:
6716:
6714:
6710:
6706:
6702:
6696:
6692:
6682:
6679:
6677:
6674:
6672:
6669:
6667:
6664:
6662:
6661:
6657:
6653:
6650:
6648:
6645:
6644:
6643:
6640:
6636:
6633:
6631:
6628:
6626:
6623:
6619:
6616:
6615:
6614:
6611:
6610:
6608:
6606:
6603:
6601:
6598:
6596:
6593:
6591:
6588:
6586:
6583:
6581:
6578:
6576:
6575:
6571:
6569:
6566:
6564:
6560:
6557:
6556:
6553:
6546:
6542:
6532:
6529:
6527:
6524:
6520:
6517:
6516:
6515:
6511:
6508:
6506:
6503:
6501:
6498:
6496:
6493:
6491:
6488:
6484:
6481:
6480:
6479:
6476:
6472:
6469:
6468:
6467:
6464:
6462:
6459:
6458:
6455:
6451:
6446:
6442:
6437:
6430:
6416:
6413:
6411:
6408:
6404:
6401:
6399:
6396:
6394:
6391:
6390:
6389:
6385:
6382:
6378:
6375:
6374:
6373:
6370:
6368:
6365:
6363:
6360:
6358:
6355:
6354:
6351:
6347:
6342:
6338:
6334:
6327:
6322:
6320:
6315:
6313:
6308:
6307:
6304:
6292:
6291:
6282:
6281:
6278:
6272:
6269:
6267:
6264:
6262:
6259:
6257:
6256:Droste effect
6254:
6253:
6251:
6247:
6241:
6238:
6236:
6235:Mathemalchemy
6233:
6231:
6228:
6226:
6223:
6221:
6218:
6216:
6213:
6211:
6208:
6207:
6205:
6203:Organizations
6201:
6195:
6194:
6190:
6188:
6187:
6183:
6181:
6180:
6176:
6174:
6173:
6172:Lumen Naturae
6169:
6167:
6166:
6162:
6161:
6159:
6155:
6143:
6140:
6139:
6138:
6135:
6131:
6130:
6126:
6125:
6124:
6121:
6117:
6114:
6113:
6112:
6109:
6105:
6104:
6100:
6099:
6098:
6095:
6091:
6088:
6087:
6086:
6083:
6079:
6078:
6074:
6073:
6072:
6069:
6068:
6066:
6062:
6054:
6051:
6050:
6049:
6046:
6042:
6039:
6038:
6037:
6034:
6030:
6027:
6026:
6025:
6024:Samuel Colman
6022:
6021:
6019:
6015:
6007:
6006:
6002:
6001:
6000:
5997:
5993:
5990:
5989:
5988:
5985:
5981:
5978:
5977:
5976:
5973:
5969:
5968:
5964:
5963:
5962:
5959:
5955:
5954:
5950:
5949:
5948:
5945:
5941:
5940:
5936:
5935:
5934:
5931:
5927:
5926:
5922:
5920:
5919:
5915:
5914:
5913:
5910:
5908:
5905:
5904:
5902:
5898:
5890:
5889:
5885:
5884:
5883:
5880:
5876:
5873:
5872:
5871:
5868:
5867:
5865:
5861:
5858:
5854:
5844:
5841:
5839:
5836:
5834:
5833:Daina Taimiņa
5831:
5829:
5826:
5824:
5821:
5819:
5818:Reza Sarhangi
5816:
5814:
5811:
5809:
5806:
5804:
5801:
5799:
5796:
5794:
5791:
5789:
5786:
5784:
5781:
5779:
5776:
5774:
5771:
5767:
5766:
5762:
5761:
5760:
5757:
5755:
5752:
5750:
5747:
5745:
5742:
5740:
5737:
5735:
5732:
5730:
5729:Peter Forakis
5727:
5725:
5722:
5720:
5717:
5715:
5712:
5710:
5707:
5705:
5701:
5698:
5696:
5693:
5692:
5690:
5686:
5680:
5677:
5673:
5672:
5668:
5666:
5665:
5661:
5660:
5659:
5658:Salvador Dalí
5656:
5652:
5651:
5647:
5646:
5645:
5644:René Magritte
5642:
5638:
5637:
5633:
5631:
5630:
5626:
5624:
5623:
5619:
5617:
5616:
5615:Print Gallery
5612:
5610:
5609:
5605:
5604:
5603:
5600:
5598:
5595:
5593:
5590:
5586:
5585:
5584:L'Oiseau bleu
5581:
5579:
5578:
5574:
5573:
5572:
5569:
5565:
5564:
5560:
5558:
5557:
5553:
5552:
5551:
5550:William Blake
5548:
5547:
5545:
5539:
5531:
5530:
5526:
5525:
5524:
5521:
5519:
5516:
5512:
5511:
5510:Vitruvian Man
5507:
5506:
5505:
5502:
5500:
5497:
5495:
5494:Paolo Uccello
5492:
5491:
5489:
5485:
5482:
5480:
5476:
5470:
5467:
5465:
5462:
5460:
5457:
5455:
5452:
5450:
5447:
5445:
5442:
5440:
5437:
5435:
5432:
5431:
5429:
5427:
5423:
5417:
5416:
5415:Pi in the Sky
5412:
5410:
5409:
5405:
5403:
5402:
5398:
5396:
5395:
5391:
5389:
5388:
5387:Mathemalchemy
5384:
5382:
5381:
5377:
5375:
5372:
5371:
5369:
5365:
5359:
5356:
5354:
5351:
5349:
5346:
5342:
5339:
5338:
5337:
5334:
5332:
5329:
5325:
5322:
5320:
5317:
5315:
5312:
5311:
5310:
5307:
5303:
5300:
5298:
5295:
5293:
5290:
5288:
5285:
5284:
5283:
5280:
5278:
5275:
5273:
5270:
5268:
5265:
5263:
5260:
5256:
5255:Vastu shastra
5253:
5251:
5248:
5246:
5245:Geodesic dome
5243:
5242:
5241:
5238:
5236:
5233:
5231:
5228:
5227:
5225:
5221:
5216:
5206:
5203:
5201:
5198:
5196:
5193:
5189:
5186:
5184:
5181:
5180:
5178:
5176:
5173:
5171:
5170:Plastic ratio
5168:
5164:
5161:
5159:
5158:Camera lucida
5156:
5155:
5154:
5151:
5149:
5146:
5144:
5141:
5139:
5136:
5134:
5131:
5129:
5126:
5124:
5121:
5119:
5116:
5115:
5113:
5109:
5105:
5098:
5093:
5091:
5086:
5084:
5079:
5078:
5075:
5066:
5062:
5058:
5055:
5052:
5049:
5045:
5041:
5038:
5035:
5032:
5029:
5028:
5024:
5018:
5012:
5008:
5004:
5000:
4996:
4990:
4985:
4984:
4978:
4977:Gullberg, Jan
4974:
4970:
4964:
4959:
4958:
4952:
4948:
4944:
4938:
4934:
4930:
4926:
4922:
4918:
4912:
4907:
4906:
4900:
4899:Escher, M. C.
4896:
4892:
4886:
4882:
4878:
4877:
4872:
4868:
4867:
4863:
4851:
4847:
4843:
4839:
4835:
4831:
4827:
4823:
4819:
4818:
4810:
4803:
4800:
4795:
4788:
4785:
4773:
4769:
4766:Tutte, W. T.
4762:
4759:
4754:
4750:
4746:
4742:
4738:
4734:
4727:
4720:
4717:
4712:
4708:
4701:
4698:
4693:
4687:
4683:
4682:
4677:
4671:
4668:
4663:
4662:
4657:
4651:
4648:
4643:
4637:
4632:
4631:
4622:
4619:
4614:
4608:
4603:
4602:
4593:
4590:
4585:
4579:
4575:
4571:
4570:
4562:
4559:
4554:
4548:
4544:
4537:
4534:
4521:
4517:
4510:
4507:
4502:
4496:
4492:
4488:
4481:
4478:
4473:
4469:
4465:
4459:
4455:
4451:
4444:
4441:
4436:
4432:
4428:
4424:
4420:
4416:
4412:
4406:
4403:
4398:
4394:
4388:
4385:
4380:
4376:
4372:
4368:
4364:
4360:
4353:
4350:
4344:
4339:
4332:
4329:
4324:
4323:
4315:
4312:
4307:
4301:
4297:
4291:
4288:
4276:
4272:
4268:
4264:
4260:
4259:
4254:
4247:
4244:
4239:
4235:
4230:
4225:
4221:
4217:
4213:
4209:
4205:
4201:
4197:
4190:
4187:
4182:
4178:
4174:
4170:
4166:
4162:
4155:
4152:
4147:
4143:
4139:
4135:
4128:
4125:
4112:
4108:
4107:UIT Cambridge
4104:
4098:
4095:
4090:
4084:
4080:
4076:
4069:
4066:
4061:
4055:
4051:
4047:
4040:
4037:
4032:
4026:
4022:
4015:
4012:
4009:, p. 16.
4008:
4003:
4000:
3996:
3991:
3988:
3976:
3969:
3966:
3962:
3946:
3939:
3936:
3932:
3927:
3924:
3919:
3913:
3909:
3905:
3904:
3896:
3893:
3880:
3874:
3871:
3866:
3860:
3856:
3852:
3845:
3842:
3829:
3822:
3819:
3813:
3808:
3801:
3798:
3793:
3789:
3785:
3781:
3776:
3771:
3767:
3763:
3762:
3757:
3750:
3747:
3742:
3736:
3732:
3728:
3727:
3719:
3716:
3710:
3709:
3704:
3701:
3694:
3691:
3686:
3680:
3676:
3671:
3670:
3661:
3658:
3653:
3646:
3643:
3638:
3632:
3628:
3623:
3622:
3613:
3610:
3604:
3600:
3596:
3592:
3588:
3584:
3580:
3576:
3569:
3566:
3561:
3557:
3550:
3547:
3542:
3536:
3532:
3531:
3523:
3520:
3515:
3509:
3505:
3501:
3494:
3491:
3486:
3480:
3476:
3469:
3466:
3461:
3457:
3453:
3449:
3445:
3441:
3437:
3431:
3428:
3423:
3419:
3418:
3413:
3407:
3404:
3399:
3395:
3394:
3389:
3383:
3380:
3376:
3375:
3368:
3365:
3359:
3356:
3343:
3342:
3337:
3330:
3327:
3322:
3318:
3314:
3310:
3306:
3302:
3298:
3294:
3293:
3288:
3282:
3279:
3274:
3270:
3266:
3262:
3255:
3252:
3247:
3243:
3238:
3233:
3229:
3225:
3224:
3219:
3212:
3209:
3204:
3200:
3196:
3192:
3188:
3184:
3180:
3176:
3172:
3168:
3167:
3162:
3156:
3153:
3148:
3144:
3140:
3136:
3132:
3126:
3123:
3118:
3117:
3112:
3106:
3103:
3098:
3094:
3090:
3086:
3085:
3080:
3074:
3071:
3066:
3062:
3058:
3054:
3050:
3046:
3042:
3038:
3037:
3029:
3026:
3010:
3003:
2996:
2993:
2981:
2977:
2970:
2967:
2962:
2958:
2954:
2950:
2945:
2940:
2936:
2932:
2931:
2923:
2920:
2916:
2911:
2908:
2902:
2897:
2893:
2886:
2883:
2877:
2876:
2871:
2868:
2861:
2858:
2853:
2849:
2845:
2841:
2837:
2833:
2829:
2825:
2821:
2817:
2810:
2807:
2803:(6): 670–673.
2802:
2798:
2797:
2789:
2782:
2779:
2774:
2768:
2763:
2762:
2753:
2750:
2745:
2741:
2737:
2733:
2730:(4): 283–89.
2729:
2725:
2724:
2716:
2713:
2708:
2702:
2698:
2693:
2692:
2686:
2680:
2677:
2664:
2657:
2654:
2651:, p. 75.
2650:
2645:
2642:
2636:
2635:
2630:
2627:
2620:
2617:
2611:
2606:
2602:
2598:
2597:
2592:
2585:
2582:
2569:
2565:
2558:
2555:
2550:
2544:
2540:
2533:
2531:
2527:
2522:
2516:
2512:
2511:
2503:
2500:
2494:
2493:
2488:
2485:
2478:
2475:
2470:
2464:
2460:
2456:
2455:
2450:
2444:
2441:
2436:
2432:
2428:
2424:
2420:
2419:
2414:
2408:
2405:
2402:, p. 59.
2401:
2396:
2394:
2392:
2390:
2388:
2386:
2382:
2369:
2365:
2359:
2356:
2352:
2347:
2344:
2339:
2338:
2330:
2327:
2323:
2318:
2315:
2310:
2303:
2300:
2288:
2284:
2278:
2275:
2270:
2267:(in German).
2266:
2259:
2256:
2251:
2247:
2246:
2238:
2235:
2230:
2226:
2222:
2216:
2213:
2200:
2193:
2186:
2183:
2180:, p. 13.
2179:
2174:
2171:
2167:
2166:Gullberg 1997
2162:
2160:
2158:
2154:
2149:
2145:
2144:
2139:
2133:
2130:
2125:
2119:
2115:
2108:
2106:
2102:
2089:
2083:
2080:
2075:
2068:
2065:
2060:
2054:
2050:
2046:
2039:
2037:
2033:
2026:
2016:
2013:
2009:
2005:
1999:
1996:
1990:
1987:
1980:
1976:
1973:
1971:
1968:
1966:
1963:
1962:
1958:
1952:
1946:
1941:
1938:
1934:
1928:
1923:
1920:
1916:
1912:
1905:
1900:
1896:
1889:
1884:
1881:
1877:
1871:
1866:
1863:of the plane
1862:
1858:
1852:
1847:
1843:
1839:
1833:
1828:
1823:
1821:
1818:
1814:
1813:Marjorie Rice
1810:
1806:
1802:
1798:
1794:
1790:
1786:
1785:Henry Dudeney
1782:
1778:
1774:
1770:
1766:
1765:tiling puzzle
1760:
1756:
1755:Tiling puzzle
1749:
1746:
1741:
1734:
1732:
1730:
1726:
1722:
1718:
1714:
1710:
1702:
1701:
1695:
1691:
1689:
1685:
1681:
1677:
1673:
1669:
1665:
1661:
1658:
1654:
1653:Edgar Gilbert
1650:
1646:
1642:
1638:
1633:
1631:
1627:
1622:
1620:
1612:
1607:
1602:
1594:
1592:
1590:
1586:
1582:
1578:
1574:
1570:
1565:
1563:
1559:
1555:
1551:
1543:
1541:
1539:
1534:
1532:
1528:
1519:
1515:
1513:
1509:
1505:
1501:
1496:
1494:
1490:
1486:
1482:
1478:
1474:
1470:
1462:
1458:
1453:
1448:
1440:
1438:
1436:
1432:
1428:
1424:
1420:
1416:
1415:Coxeter group
1412:
1409:
1405:
1400:
1398:
1394:
1390:
1386:
1382:
1378:
1374:
1370:
1366:
1365:non-Euclidean
1359:
1355:
1350:
1343:
1339:
1335:
1328:
1326:
1324:
1320:
1316:
1311:
1309:
1304:
1302:
1298:
1294:
1290:
1286:
1277:
1273:
1271:
1267:
1263:
1259:
1255:
1251:
1247:
1243:
1239:
1235:
1231:
1227:
1219:
1215:
1210:
1205:
1197:
1195:
1192:
1188:
1184:
1179:
1172:
1167:
1160:
1158:
1156:
1152:
1148:
1144:
1141:
1139:
1133:
1128:
1122:
1117:
1111:
1106:
1100:
1095:
1091:
1083:
1078:
1074:
1071:
1070:parallelogram
1067:
1063:
1059:
1058:quadrilateral
1055:
1051:
1050:quadrilateral
1046:
1044:
1036:
1034:
1032:
1028:
1019:
1015:
1010:
1005:
997:
995:
993:
988:
987:
986:einstein tile
981:
979:
975:
971:
970:Truchet tiles
965:
960:
956:
954:
950:
946:
942:
936:
935:aperiodically
932:
927:
923:
921:
920:quasicrystals
917:
913:
908:
904:
900:
896:
892:
888:
884:
877:
872:
867:
863:
855:
853:
851:
847:
843:
839:
835:
831:
827:
823:
819:
816:Tilings with
810:
805:
797:
795:
793:
789:
785:
781:
773:
768:
764:
762:
759:, or regular
758:
754:
750:
746:
742:
737:
735:
731:
726:
701:
697:
693:
689:
685:
681:
676:
667:
664:
663:
657:
653:
651:
647:
643:
642:connected set
639:
635:
634:topologically
631:
630:normal tiling
626:
623:
618:
616:
612:
608:
604:
600:
596:
595:
590:
589:
582:
578:
574:
566:
561:
559:
557:
552:
549:
545:
541:
537:
533:
530:
527:
522:
519:
515:
511:
507:
503:
499:
495:
491:
487:
480:
476:
472:
467:
463:
460:
456:
452:
446:
444:
440:
436:
432:
428:
427:regular tiles
424:
420:
416:
408:
404:
399:
392:
390:
388:
384:
380:
376:
372:
368:
364:
360:
352:
350:
348:
344:
343:
338:
337:Nikolai Belov
334:
329:
321:
317:
313:
311:
306:
305:
299:
294:
292:
288:
284:
279:
277:
269:
264:
257:
255:
253:
249:
245:
241:
237:
233:
229:
225:
221:
217:
213:
209:
205:
201:
197:
194:
189:
187:
186:
181:
177:
176:
171:
167:
166:
161:
157:
156:
151:
146:
144:
140:
136:
132:
128:
124:
120:
116:
102:
92:
88:
81:
70:
66:
63:
57:
48:
44:
37:
33:
19:
7771:Tessellation
7719:
7700:
7686:
7652:
7578:Chaos theory
7501:Tessellation
7500:
6671:Substitution
6666:Regular grid
6658:
6572:
6505:Quaquaversal
6403:Kisrhombille
6333:Tessellation
6332:
6288:
6191:
6184:
6177:
6170:
6163:
6157:Publications
6141:
6127:
6115:
6101:
6089:
6075:
6052:
6048:Jay Hambidge
6041:Ad Quadratum
6040:
6028:
6003:
5991:
5979:
5965:
5951:
5947:Luca Pacioli
5937:
5923:
5916:
5886:
5874:
5798:Hinke Osinga
5793:István Orosz
5763:
5754:Anthony Hill
5709:Scott Draves
5704:Erik Demaine
5688:Contemporary
5669:
5662:
5648:
5634:
5627:
5620:
5613:
5606:
5602:M. C. Escher
5582:
5575:
5561:
5554:
5527:
5523:Parmigianino
5508:
5439:Hagia Sophia
5413:
5406:
5399:
5392:
5385:
5378:
5357:
5262:Computer art
5240:Architecture
5200:Tessellation
5199:
5183:Architecture
5133:Golden ratio
5006:
5003:Stewart, Ian
4982:
4956:
4928:
4904:
4875:
4850:the original
4821:
4815:
4802:
4793:
4787:
4775:. Retrieved
4772:Squaring.net
4771:
4761:
4736:
4732:
4719:
4710:
4700:
4680:
4670:
4659:
4650:
4629:
4621:
4600:
4592:
4568:
4561:
4542:
4536:
4524:. Retrieved
4520:the original
4509:
4486:
4480:
4449:
4443:
4421:(1): 59–99.
4418:
4414:
4405:
4396:
4387:
4362:
4358:
4352:
4331:
4320:
4314:
4295:
4290:
4278:. Retrieved
4256:
4246:
4203:
4199:
4189:
4164:
4160:
4154:
4137:
4133:
4127:
4115:. Retrieved
4111:the original
4097:
4074:
4068:
4045:
4039:
4020:
4014:
4002:
3990:
3978:. Retrieved
3968:
3959:
3952:. Retrieved
3938:
3926:
3902:
3895:
3883:. Retrieved
3873:
3850:
3844:
3832:. Retrieved
3821:
3800:
3765:
3759:
3749:
3725:
3718:
3706:
3693:
3677:and passim.
3668:
3660:
3651:
3645:
3620:
3612:
3578:
3574:
3568:
3559:
3555:
3549:
3533:. Springer.
3529:
3522:
3499:
3493:
3474:
3468:
3443:
3439:
3430:
3416:
3406:
3391:
3382:
3372:
3367:
3358:
3345:. Retrieved
3341:Science News
3339:
3329:
3296:
3290:
3281:
3264:
3260:
3254:
3227:
3221:
3211:
3170:
3164:
3155:
3138:
3134:
3125:
3114:
3105:
3088:
3082:
3073:
3040:
3034:
3028:
3016:. Retrieved
3009:the original
2995:
2983:. Retrieved
2969:
2934:
2928:
2922:
2915:Gardner 1989
2910:
2885:
2873:
2860:
2819:
2815:
2809:
2800:
2794:
2781:
2760:
2752:
2727:
2721:
2715:
2690:
2685:Wells, David
2679:
2667:. Retrieved
2656:
2649:Stewart 2001
2644:
2632:
2619:
2600:
2599:. Series A.
2594:
2584:
2572:. Retrieved
2568:the original
2557:
2538:
2509:
2502:
2490:
2477:
2453:
2443:
2416:
2407:
2372:. Retrieved
2367:
2358:
2346:
2336:
2329:
2322:Coxeter 1973
2317:
2308:
2302:
2290:. Retrieved
2286:
2283:"Tessellate"
2277:
2264:
2258:
2244:
2237:
2228:
2224:
2221:Fyodorov, Y.
2215:
2203:. Retrieved
2185:
2178:Stewart 2001
2173:
2147:
2142:
2132:
2113:
2092:. Retrieved
2082:
2073:
2067:
2044:
2015:
2004:homeomorphic
1998:
1989:
1913:, a spiral,
1844:of the plane
1762:
1743:Traditional
1706:
1698:
1634:
1629:
1623:
1616:
1566:
1547:
1535:
1524:
1508:Circle Limit
1500:M. C. Escher
1497:
1466:
1427:permutations
1401:
1383:; these are
1362:
1352:The regular
1312:
1305:
1282:
1254:plesiohedron
1223:
1183:Voronoi cell
1182:
1176:
1145:
1142:
1131:
1120:
1109:
1098:
1093:
1089:
1087:
1047:
1040:
1023:
1018:four colours
984:
982:
968:
939:
929:A set of 13
905:properties.
881:
841:
837:
815:
777:
743:is a highly
738:
727:
691:
674:
671:
661:
629:
627:
621:
619:
598:
592:
586:
584:
553:
535:
523:
513:
509:
493:
483:
479:M. C. Escher
459:M. C. Escher
447:
414:
412:
382:
378:
374:
358:
356:
345:(1964), and
340:
325:
295:
280:
273:
228:M. C. Escher
208:Ancient Rome
190:
183:
173:
163:
153:
149:
147:
134:
118:
115:tessellation
114:
112:
91:M. C. Escher
47:
7696:Alan Turing
7654:Liber Abaci
7573:Mathematics
7479:in crystals
7469:Soap bubble
7464:Phyllotaxis
6701:vertex type
6559:Anisohedral
6514:Self-tiling
6357:Pythagorean
6097:G. H. Hardy
5900:Renaissance
5870:Polykleitos
5808:Tony Robbin
5719:John Ernest
5714:Jan Dibbets
5664:Crucifixion
5487:Renaissance
5341:Mathematics
5314:Celtic knot
5277:Fractal art
5179:Proportion
5153:Perspective
4824:(1): 3–12.
4569:Polyominoes
4007:Escher 1974
3995:Escher 1974
3931:Escher 1974
3562:(2): 49–56.
3412:Jones, Owen
3091:(1): 1–41.
2603:(1): 1–18.
2351:Escher 1974
2201:. p. 2
2116:. Tarquin.
2008:closed disk
1951:Topological
1781:polyominoes
1777:polyiamonds
1721:Lord Kelvin
1668:contraction
1554:sheet metal
1493:La Mezquita
1264:(a kind of
1151:polyominoes
1147:Polyominoes
1020:are needed.
976:; in 1704,
536:polyschemes
455:polyominoes
281:Decorative
212:Islamic art
139:mathematics
7765:Categories
7643:Empedocles
7638:Pythagoras
7556:Camouflage
7494:in biology
7489:in flowers
7459:Parastichy
6605:Pentagonal
6071:Owen Jones
5918:De pictura
5823:Oliver Sin
5773:Andy Lomas
5622:Relativity
5353:String art
5267:Fiber arts
5148:Paraboloid
4987:. Norton.
4411:Weaire, D.
4280:7 November
3141:(66): 72.
2231:: 245–291.
2027:References
1915:monohedral
1660:lava flows
1626:fritillary
1577:thin films
1562:drink cans
1389:transitive
1344:projection
1289:tetrahedra
1285:honeycombs
1236:(the only
1191:dual graph
1092:-gons for
1054:non-convex
941:Wang tiles
931:Wang tiles
828:palace in
662:monohedral
510:tessellate
502:prototiles
498:boundaries
357:In Latin,
310:snowflakes
252:honeycombs
125:, often a
87:Leeuwarden
65:terracotta
7745:Emergence
7648:Fibonacci
6713:Spherical
6681:Voderberg
6642:Prototile
6609:Problems
6585:Honeycomb
6563:Isohedral
6450:Aperiodic
6388:honeycomb
6372:Rectangle
6362:Rhombille
5882:Vitruvius
5856:Theorists
5636:Waterfall
5541:19th–20th
5469:Taj Mahal
5449:Parthenon
5426:Buildings
5380:Continuum
5348:Sculpture
5324:Interlace
5118:Algorithm
4379:119949515
4343:1005.0023
4275:138195687
4206:: 14787.
3830:. Wolfram
3812:1101.0530
3792:121698536
3784:0075-4102
3708:MathWorld
3414:(1910) .
3398:EMS Press
3321:192944820
3111:Wang, Hao
3079:Wang, Hao
2939:CiteSeerX
2896:CiteSeerX
2875:MathWorld
2744:123579388
2634:MathWorld
2492:MathWorld
2311:. Peters.
2250:Macmillan
1919:enneagons
1805:dissected
1700:Colchicum
1645:mudcracks
1630:Colchicum
1619:honeycomb
1611:honeycomb
1595:In nature
1558:car doors
1293:octahedra
1262:andradite
1226:polyhedra
899:rep-tiles
745:symmetric
680:congruent
646:empty set
544:polyhedra
540:polytopes
506:congruent
490:countable
451:pentagons
353:Etymology
296:In 1619,
276:Sumerians
250:found in
133:, called
69:Marrakech
67:tiles in
7781:Symmetry
7474:Symmetry
7432:Patterns
6795:V3.4.3.4
6630:Squaring
6625:Heesch's
6590:Isotoxal
6510:Rep-tile
6500:Pinwheel
6393:Coloring
6346:Periodic
6290:Category
6017:Romantic
5695:Max Bill
5629:Reptiles
5444:Pantheon
5401:Octacube
5367:Artworks
5309:Knotting
5297:Muqarnas
5195:Symmetry
5123:Catenary
5111:Concepts
5005:(2001).
4979:(1997).
4927:(1989).
4846:26663945
4838:27642387
4472:12650092
4238:26437880
4200:Sci. Rep
3347:25 March
3292:Leonardo
3203:14259496
3065:10034529
2852:10374218
2844:17322056
2669:26 April
2451:(1948).
2374:26 April
2269:Springer
2140:(1619).
2049:Sterling
1959:See also
1824:Examples
1801:rep-tile
1688:Tasmania
1657:Basaltic
1649:crystals
1573:cracking
1569:mudcrack
1489:Alhambra
1479:, using
1397:isometry
1393:vertices
1291:and six
1270:fluorite
1258:crystals
1129:and for
945:dominoes
826:Alhambra
761:hexagons
730:symmetry
688:enneagon
607:polygons
599:isogonal
529:geometer
475:Alhambra
431:triangle
393:Overview
359:tessella
287:tesserae
240:quilting
224:Alhambra
204:pavement
200:patterns
193:cemented
7733:Related
7600:Crystal
7595:Physics
7583:Fractal
7561:Mimicry
7546:Biology
7454:Meander
7255:6.4.8.4
7210:5.4.6.4
7170:4.12.16
7160:4.10.12
7130:V4.8.10
7105:V4.6.16
7095:V4.6.14
6995:3.6.4.6
6990:3.4.∞.4
6985:3.4.8.4
6980:3.4.7.4
6975:3.4.6.4
6925:3.∞.3.∞
6920:3.4.3.4
6915:3.8.3.8
6910:3.7.3.7
6905:3.6.3.8
6900:3.6.3.6
6895:3.5.3.6
6890:3.5.3.5
6885:3.4.3.∞
6880:3.4.3.8
6875:3.4.3.7
6870:3.4.3.6
6865:3.4.3.5
6820:3.4.6.4
6790:3.4.3.4
6783:regular
6750:Regular
6676:Voronoi
6600:Packing
6531:Truchet
6526:Socolar
6495:Penrose
6490:Gilbert
6415:Wythoff
6249:Related
5863:Ancient
5597:Man Ray
5543:Century
5479:Artists
5336:Origami
5250:Pyramid
5128:Fractal
4864:Sources
4753:2689644
4423:Bibcode
4229:4594096
4208:Bibcode
4169:Bibcode
3603:0598811
3583:Bibcode
3460:4613674
3400:, 2001
3313:1578535
3246:1417576
3195:0297572
3175:Bibcode
3045:Bibcode
2961:2118575
2824:Bibcode
2816:Science
2697:260–261
2574:6 April
2459:Methuen
2435:2689617
2205:6 April
1773:tangram
1745:tangram
1682:on the
1538:origami
1512:woodcut
1485:Zellige
1473:Moorish
1461:Antioch
1433:of the
1391:on its
962:Random
830:Granada
757:squares
719:
704:
644:or the
471:zellige
439:hexagon
375:tessera
258:History
234:and in
222:of the
210:and in
196:ceramic
123:surface
62:Zellige
7776:Mosaic
7626:People
7534:Causes
7145:4.8.16
7140:4.8.14
7135:4.8.12
7125:4.8.10
7100:4.6.16
7090:4.6.14
7085:4.6.12
6855:Hyper-
6840:4.6.12
6613:Domino
6519:Sphinx
6398:Convex
6377:Domino
6064:Modern
5700:Martin
5563:Newton
5358:Tiling
5302:Zellij
5272:4D art
5031:Tegula
5013:
4991:
4965:
4939:
4913:
4887:
4844:
4836:
4777:29 May
4751:
4688:
4638:
4609:
4580:
4549:
4526:28 May
4497:
4487:Shapes
4470:
4460:
4377:
4302:
4273:
4258:Nature
4236:
4226:
4117:29 May
4085:
4056:
4027:
3980:27 May
3954:27 May
3914:
3885:17 May
3861:
3834:27 May
3790:
3782:
3737:
3731:p. 209
3681:
3633:
3601:
3537:
3510:
3504:Hermes
3481:
3458:
3319:
3311:
3244:
3201:
3193:
3063:
3018:29 May
2985:29 May
2959:
2941:
2898:
2850:
2842:
2769:
2742:
2703:
2545:
2517:
2465:
2433:
2292:26 May
2150:].
2120:
2094:26 May
2055:
1703:flower
1571:-like
1531:quilts
1527:motifs
1469:Mosaic
1457:mosaic
1455:Roman
1441:In art
1323:sphere
1268:) and
1266:garnet
1250:prisms
1244:, the
1153:, see
1136:, see
1125:, see
1118:, for
1114:, see
1107:, for
1103:, see
1052:(even
690:. The
594:vertex
579:, and
516:. The
512:or to
435:square
389:clay.
387:glazed
383:tiling
283:mosaic
119:tiling
7633:Plato
7439:Crack
7260:(6.8)
7215:(5.6)
7150:4.8.∞
7120:(4.8)
7115:(4.7)
7110:4.6.∞
7080:(4.6)
7075:(4.5)
7045:4.∞.4
7040:4.8.4
7035:4.7.4
7030:4.6.4
7025:4.5.4
7005:(3.8)
7000:(3.7)
6970:(3.4)
6965:(3.4)
6857:bolic
6825:(3.6)
6781:Semi-
6652:Girih
6549:Other
5875:Canon
5331:Music
5287:Girih
5223:Forms
5188:Human
4853:(PDF)
4842:S2CID
4834:JSTOR
4812:(PDF)
4749:JSTOR
4729:(PDF)
4375:S2CID
4338:arXiv
4271:S2CID
3948:(PDF)
3807:arXiv
3788:S2CID
3456:S2CID
3317:S2CID
3309:JSTOR
3199:S2CID
3012:(PDF)
3005:(PDF)
2957:JSTOR
2848:S2CID
2791:(PDF)
2740:S2CID
2431:JSTOR
2195:(PDF)
2146:[
1709:foams
1635:Many
1585:micro
1504:Spain
1481:Girih
1431:rings
1411:cells
1381:faces
1317:is a
1082:Texas
1064:. As
834:Spain
526:Swiss
494:tiles
486:cover
443:plane
415:tiles
371:glass
369:, or
367:stone
320:Roman
182:). A
158:with
135:tiles
127:plane
7511:Wave
7449:Foam
7444:Dune
7345:8.16
7340:8.12
7310:7.14
7280:6.16
7275:6.12
7270:6.10
7230:5.12
7225:5.10
7180:4.16
7175:4.14
7165:4.12
7155:4.10
7015:3.16
7010:3.14
6830:3.12
6815:V3.6
6741:V4.n
6731:V3.n
6618:Wang
6595:List
6561:and
6512:and
6471:List
6386:and
5702:and
5292:Jali
5011:ISBN
4989:ISBN
4963:ISBN
4937:ISBN
4911:ISBN
4885:ISBN
4779:2015
4686:ISBN
4636:ISBN
4607:ISBN
4578:ISBN
4547:ISBN
4528:2015
4495:ISBN
4468:OCLC
4458:ISBN
4300:ISBN
4282:2014
4234:PMID
4119:2015
4083:ISBN
4054:ISBN
4025:ISBN
3982:2015
3956:2015
3912:ISBN
3887:2015
3859:ISBN
3836:2015
3780:ISSN
3766:1873
3735:ISBN
3679:ISBN
3631:ISBN
3535:ISBN
3508:ISBN
3479:ISBN
3349:2023
3061:PMID
3020:2015
2987:2015
2840:PMID
2767:ISBN
2701:ISBN
2671:2013
2576:2013
2543:ISBN
2515:ISBN
2463:ISBN
2376:2013
2294:2015
2207:2013
2118:ISBN
2096:2015
2053:ISBN
1909:The
1859:, a
1787:and
1779:and
1757:and
1617:The
1587:and
1491:and
1483:and
1371:. A
1306:The
1234:cube
864:and
638:disk
588:edge
379:four
363:clay
335:and
268:Uruk
218:and
7375:∞.8
7370:∞.6
7335:8.6
7305:7.8
7300:7.6
7265:6.8
7220:5.8
7185:4.∞
7020:3.∞
6945:3.4
6940:3.∞
6935:3.8
6930:3.7
6845:4.8
6835:4.∞
6810:3.6
6805:3.∞
6800:3.4
6736:4.n
6726:3.n
6699:By
5046:. "
4826:doi
4822:115
4741:doi
4431:doi
4367:doi
4263:doi
4224:PMC
4216:doi
4177:doi
4142:doi
3770:doi
3627:107
3591:doi
3579:154
3448:doi
3301:doi
3269:doi
3232:doi
3228:160
3183:doi
3143:doi
3093:doi
3053:doi
2949:doi
2935:139
2832:doi
2820:315
2732:doi
2605:doi
2423:doi
1686:of
1575:of
1560:or
1429:of
1379:as
1260:of
1134:= 8
1123:= 7
1112:= 6
1101:= 5
983:An
893:of
842:p4m
838:p6m
601:or
117:or
7767::
5408:Pi
5063:.
4935:.
4931:.
4883:.
4879:.
4840:.
4832:.
4820:.
4814:.
4770:.
4747:.
4737:51
4735:.
4731:.
4709:.
4576:.
4489:.
4466:.
4429:.
4419:25
4417:.
4373:.
4361:.
4269:.
4261:.
4255:.
4232:.
4222:.
4214:.
4202:.
4198:.
4175:.
4165:48
4163:.
4138:73
4136:.
4105:.
4081:.
4077:.
4048:.
3958:.
3906:.
3853:.
3786:.
3778:.
3764:.
3758:.
3733:.
3705:.
3629:.
3599:MR
3597:.
3589:.
3577:.
3558:.
3502:.
3454:.
3444:23
3442:.
3396:,
3390:,
3338:.
3315:.
3307:.
3297:20
3295:.
3265:32
3263:.
3242:MR
3240:.
3226:.
3220:.
3197:.
3191:MR
3189:.
3181:.
3171:12
3169:.
3139:66
3137:.
3089:40
3087:.
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