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| [[File:Penrose Tiling (Rhombi).svg|250px|thumb|right|A Penrose tiling|alt=]]
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| A '''Penrose tiling''' is a non-periodic [[tessellation|tiling]] generated by an [[aperiodic set of prototiles|aperiodic set]] of [[prototile]]s. Penrose tilings are named after mathematician and physicist [[Roger Penrose]] who investigated these sets in the 1970s. The aperiodicity of the Penrose prototiles implies that a shifted copy of a Penrose tiling will never match the original. A Penrose tiling may be constructed so as to exhibit both [[reflection symmetry]] and fivefold [[rotational symmetry]], as in the diagram at the right.
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| A Penrose tiling has many remarkable properties, most notably:
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| *It is non-periodic, which means that it lacks any [[translational symmetry]].
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| * It is [[self-similarity|self-similar]], so the same patterns occur at larger and larger scales. Thus, the tiling can be obtained through "inflation" (or "deflation") and any finite patch from the tiling occurs infinitely many times.
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| *It is a [[quasicrystal]]: implemented as a physical structure a Penrose tiling will produce [[Bragg diffraction]] and its diffractogram reveals both the fivefold symmetry and the underlying long range order.
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| Various methods to construct Penrose tilings have been discovered, including matching rules, [[substitutions|substitution tiling]] or [[subdivision rules|subdivision rule]], cut and project schemes and coverings.
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| [[File:RogerPenroseTileTAMU2010.jpg|250px|right|thumb|[[Roger Penrose]] in the foyer of the Mitchell Institute for Fundamental Physics and Astronomy, [[Texas A&M University]], standing on a floor with a Penrose tiling|alt=]]
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| ==Background and history==
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| ===Periodic and aperiodic tilings===
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| {{anchor|Figure 1}}[[File:RegularRhombs.svg|right|thumb|150px|Figure 1. Part of a periodic tiling|alt=]]
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| Penrose tilings are simple examples of [[aperiodic tiling]]s of the [[plane (geometry)|plane]].<ref>General references for this article include {{Harvnb|Gardner|1997|pp=1–30}}, {{Harvnb|Grünbaum|Shephard|1987|pp=520–548 & 558–579}}, and {{Harvnb|Senechal|1996|pp=170–206}}.</ref> A [[tessellation|tiling]] is a covering of the plane by tiles with no overlaps or gaps; the tiles normally have a [[finite set|finite]] number of shapes, called ''[[prototile]]s'', and [[set (mathematics)|a set]] of prototiles is said to ''admit a tiling'' or ''tile the plane'' if there is a tiling of the plane using only tiles [[congruence (geometry)|congruent]] to these prototiles.<ref>{{Harvnb|Grünbaum|Shephard|1987|pp=20, 23}}</ref> The most familiar tilings (e.g., by squares or triangles) are [[periodic tiling|periodic]]: a perfect copy of the tiling can be obtained by [[translation (mathematics)|translating]] all of the tiles by a fixed distance in a given direction. Such a translation is called a ''period'' of the tiling; more informally, this means that a finite region of the tiling repeats itself in periodic intervals. If a tiling has no periods it is said to be ''non-periodic''. A set of prototiles is said to be ''aperiodic'' if it tiles the plane, but every such tiling is non-periodic; tilings by aperiodic sets of prototiles are called aperiodic tilings.<ref>{{Harvnb|Grünbaum|Shephard|1987|p=520}}</ref>
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| ===Earliest aperiodic tilings===
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| [[File:Wang tiles.svg|thumb|left|150px|An [[aperiodic set]] of [[Wang domino]]es.<ref>{{Harvnb|Culik|Kari|1997}}</ref>|alt=]]
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| The subject of aperiodic tilings received new interest in the 1960s when logician [[Hao Wang (academic)|Hao Wang]] noted connections between [[decision problem]]s and tilings.<ref>{{Harvnb|Wang|1961}}</ref> In particular, he introduced tilings by square plates with colored edges, now known as ''[[Wang domino]]es'' or ''tiles'', and posed the "''[[Domino Problem]]''": to determine whether a given set of Wang dominoes could tile the plane with matching colors on adjacent domino edges. He observed that if this problem were [[recursive set|undecidable]], then there would have to exist an aperiodic set of Wang dominoes. At the time, this seemed implausible, so Wang conjectured no such set could exist.
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| [[File:Robinson tiles.svg|thumb|right|Robinson's six prototiles|alt=]]
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| Wang's student [[Robert Berger (mathematician)|Robert Berger]] proved that the Domino Problem was undecidable (so Wang's conjecture was incorrect) in his 1964 thesis,<ref>{{MathGenealogy|id=114475|title=Robert Berger}}</ref> and obtained an aperiodic set of 20426 Wang dominoes.<ref name="AMSa">{{Harvnb|Austin|2005a}}</ref> He also described a reduction to 104 such prototiles; the latter did not appear in his published monograph,<ref>{{Harvnb|Berger|1966}}</ref> but in 1968, [[Donald Knuth]] detailed a modification of Berger's set requiring only 92 dominoes.<ref>{{Harvnb|Grünbaum|Shephard|1987|p=584}}</ref>
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| The color matching required in a tiling by Wang dominoes can easily be achieved by modifying the edges of the tiles like [[jigsaw puzzle]] pieces so that they can fit together only as prescribed by the edge colorings.<ref>{{Harvnb|Gardner|1997|p=5}}</ref> [[Raphael Robinson]], in a 1971 paper<ref>{{Harvnb|Robinson|1971}}</ref> which simplified Berger's techniques and undecidability proof, used this technique to obtain an aperiodic set of just six prototiles.<ref>{{Harvnb|Grünbaum|Shephard|1987|p=525}}</ref>
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| ===Development of the Penrose tilings===
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| {{anchor|Figure 2}}[[File:Penrose Tiling (P1 over P3).svg|250px|thumb|left|Figure 2. The pentagonal Penrose tiling (P1) drawn in black on a colored rhombus tiling (P3) with yellow edges.<ref name="S:173-174">{{Harvnb|Senechal|1996|pp=173–174}}</ref>|alt=]]
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| The first Penrose tiling (tiling P1 below) is also an aperiodic set of six prototiles, introduced by [[Roger Penrose]] in a 1974 paper,<ref>{{Harvnb|Penrose|1974}}</ref> but is based on pentagons rather than squares. Any attempts to tile the plane with regular pentagons will necessarily leave gaps, but [[Johannes Kepler]] showed, in his 1619 work ''[[Harmonices Mundi]]'', that these gaps could be filled using [[pentagram]]s (viewed as [[star polygon]]s), [[decagon]]s and related shapes.<ref>{{Harvnb|Grünbaum|Shephard|1987|loc=section 2.5}}</ref> Acknowledging inspiration from Kepler, Penrose was able to find matching rules (which can be imposed by decorations of the edges) for these shapes, in order to obtain an aperiodic set; his tiling can be viewed as a completion of Kepler's finite ''Aa'' pattern,<ref name="S:171">{{Harvnb|Senechal|1996|p=171}}</ref> and other traces of these ideas can be found in [[Albrecht Dürer]]'s work.<ref>{{Harvnb|Luck|2000}}</ref>
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| Penrose subsequently reduced the number of prototiles to two, discovering the kite and dart tiling (tiling P2 below) and the rhombus tiling (tiling P3 below).<ref name="G:6">{{Harvnb|Gardner|1997|p=6}}</ref> The rhombus tiling was independently discovered by [[Robert Ammann]] in 1976.<ref name="G:19">{{Harvnb|Gardner|1997|p=19}}</ref> Penrose and [[John H. Conway]] investigated the properties of Penrose tilings, and discovered that a substitution property explained their hierarchical nature; their findings were publicized by [[Martin Gardner]] in his January 1977 "Mathematical Games" column in ''[[Scientific American]]''.<ref name="G1">{{Harvnb|Gardner|1997|loc=chapter 1}}</ref>
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| In 1981, [[Nicolaas Govert de Bruijn|De Bruijn]] explained a method to construct Penrose tilings<ref>{{Harvnb|de Bruijn|1981}}</ref> from five families of parallel lines as well as a "cut and project method", in which Penrose tilings are obtained as two-dimensional projections from a five-dimensional cubic structure. In this approach, the Penrose tiling is viewed as a set of points, its vertices, while the tiles are geometrical shapes obtained by connecting vertices with edges.
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| ==The Penrose tilings==
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| [[File:Penrose Tiling (P1).svg|thumb|right|200px|A P1 tiling using Penrose's original set of six prototiles|alt=]]
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| The three types of Penrose tiling P1–P3 are described individually below.<ref name="P1-P3">The P1–P3 notation is taken from {{Harvnb|Grünbaum|Shephard|1987|loc=section 10.3}}</ref> They have many common features: in each case, the tiles are constructed from shapes related to the pentagon (and hence to the [[golden ratio]]), but the basic tile shapes need to be supplemented by ''matching rules'' in order to tile aperiodically; these rules may be described using labeled vertices or edges, or patterns on the tile faces – alternatively the edge profile can be modified (e.g. by indentations and protrusions) to obtain an aperiodic set of prototiles.<ref name="AMSa" /><ref name="GS10.3">{{Harvnb|Grünbaum|Shephard|1987|loc=section 10.3}}</ref>
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| ===The original pentagonal Penrose tiling (P1)===
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| Penrose's first tiling uses pentagons and three other shapes: a five-pointed "star" (a pentagram), a "boat" (roughly 3/5 of a star) and a "diamond" (a thin rhombus).<ref name="P:32"/> To ensure that all tilings are non-periodic, there are matching rules that specify how tiles may meet each other, and there are three different types of matching rule for the pentagonal tiles. It is common to indicate the three different types of pentagonal tiles using three different colors, as in the figure above right.<ref>"However, as will be explained momentarily, differently colored pentagons will be considered to be different types of tiles." {{Harvnb|Austin|2005a}}; {{Harvnb|Grünbaum|Shephard|1987|loc=figure 10.3.1}}, shows the edge modifications needed to yield an aperiodic set of prototiles.</ref>
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| ===Kite and dart tiling (P2)===
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| Penrose's second tiling uses quadrilaterals called the "kite" and "dart", which may be combined to make a rhombus. However, the matching rules prohibit such a combination.<ref>"The rhombus of course tiles periodically, but we are not allowed to join the pieces in this manner." {{Harvnb|Gardner|1997|pp=6–7}}</ref> Both the kite and dart are composed of two triangles, called ''[[Golden triangle (mathematics)#Golden gnomon|Robinson triangles]]'', after 1975 notes by Robinson.<ref name="GS-R">{{Harvnb|Grünbaum|Shephard|1987|pp=537– 547}}</ref>
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| {{multiple image|align=left|direction=vertical|image1=Kite Dart.svg|image2=Penrose vertex figures.svg|width=225|footer=Kite and dart tiles (top) and the seven possible [[vertex figure]]s in a P2 tiling.}}
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| *The ''kite'' is a quadrilateral whose four interior angles are 72, 72, 72, and 144 degrees. The kite may be bisected along its axis of symmetry to form a pair of acute Robinson triangles (with angles of 36, 72 and 72 degrees).
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| *The ''dart'' is a non-convex quadrilateral whose four interior angles are 36, 72, 36, and 216 degrees. The dart may be bisected along its axis of symmetry to form a pair of obtuse Robinson triangles (with angles of 36, 36 and 108 degrees), which are smaller than the acute triangles.
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| The matching rules can be described in several ways. One approach is to color the vertices (with two colors, e.g., black and white) and require that adjacent tiles have matching vertices.<ref name="S:173">{{Harvnb|Senechal|1996|p=173}}</ref> Another is to use a pattern of circular arcs (as shown above left in green and red) to constrain the placement of tiles: when two tiles share an edge in a tiling, the patterns must match at these edges.<ref name="G:6"/>
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| These rules often force the placement of certain tiles: for example, the [[Convex and concave polygons|concave]] vertex of any dart is necessarily filled by two kites. The corresponding figure (center of the top row in the lower image on the left) is called an "ace" by Conway; although it looks like an enlarged kite, it does not tile in the same way.<ref name="G:8">{{Harvnb|Gardner|1997|p=8}}</ref> Similarly the concave vertex formed when two kites meet along a short edge is necessarily filled by two darts (bottom right). In fact, there are only seven possible ways for the tiles to meet at a vertex; two of these figures – namely, the "star" (top left) and the "sun" (top right) – have 5-fold [[dihedral symmetry]] (by rotations and reflections), while the remainder have a single axis of reflection (vertical in the image).<ref name="G:10=11">{{Harvnb|Gardner|1997|pp=10–11}}</ref> All of these vertex figures, apart from the ace and the sun, force the placement of additional tiles.<ref name="G:12">{{Harvnb|Gardner|1997|p=12}}</ref>
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| ===Rhombus tiling (P3)===
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| [[File:Penrose rhombs matching rules.svg|right|thumb|Matching rule for Penrose rhombs using circular arcs or edge modifications|alt=]]
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| The third tiling uses a pair of [[rhombus]]es (often referred to as "[[wikt:rhomb|rhombs]]" in this context) with equal sides but different angles.<ref name="AMSa"/> Ordinary rhombus-shaped tiles can be used to tile the plane periodically, so restrictions must be made on how tiles can be assembled: no two tiles may form a parallelogram, as this would allow a periodic tiling, but this constraint is not sufficient to force aperiodicity, as [[#Figure 1|figure 1 above]] shows.
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| There are two kinds of tile, both of which can be decomposed into Robinson triangles.<ref name="GS-R"/>
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| *The thin rhomb '''t''' has four corners with angles of 36, 144, 36, and 144 degrees. The '''t''' rhomb may be bisected along its short diagonal to form a pair of acute Robinson triangles.
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| *The thick rhomb '''T''' has angles of 72, 108, 72, and 108 degrees. The '''T''' rhomb may be bisected along its long diagonal to form a pair of obtuse Robinson triangles; in contrast to the P2 tiling, these are larger than the acute triangles.
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| The matching rules distinguish sides of the tiles, and entail that tiles may be juxtaposed in certain particular ways but not in others. Two ways to describe these matching rules are shown in the image on the right. In one form, tiles must be assembled such that the curves on the faces match in color and position across an edge. In the other, tiles must be assembled such that the bumps on their edges fit together.<ref name="AMSa"/>
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| There are 54 cyclically ordered combinations of such angles that add up to 360 degrees at a vertex, but the rules of the tiling allow only seven of these combinations to appear (although one of these arises in two ways).<ref name="S:178">{{Harvnb|Senechal|1996|p=178}}</ref>
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| The various combinations of angles and facial curvature allow construction of arbitrarily complex tiles, such as the ''[http://www.murderousmaths.co.uk/books/BKMM7xpc.htm Penrose chickens]''. | |
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| ==Features and constructions== | |
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| ===The golden ratio and local pentagonal symmetry===
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| Several properties and common features of the Penrose tilings involve the [[golden ratio]] ''φ'' = (1+√5)/2 (approximately 1.618).<ref name="GS-R" /><ref name="S:173"/> This is the ratio of [[Chord (geometry)|chord]] lengths to side lengths in a [[regular pentagon]], and satisfies ''φ'' = 1 + 1/''φ''.
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| [[File:Rhomb pentagon.svg|left|thumb|Pentagon with an inscribed thick rhomb (light), acute Robinson triangles (lightly shaded) and a small obtuse Robinson triangle (darker). Dotted lines give additional edges for inscribed kites and darts.|alt=]]
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| Consequently, the ratio of the lengths of long sides to short sides in the ([[isosceles]]) Robinson triangles is ''φ'':1. It follows that the ratio of long side lengths to short in both kite and dart tiles is also ''φ'':1, as are the length ratios of sides to the short diagonal in the thin rhomb '''t''', and of long diagonal to sides in the thick rhomb '''T'''. In both the P2 and P3 tilings, the ratio of the [[Triangle#Computing the area of a triangle|area]] of the larger Robinson triangle to the smaller one is ''φ'':1, hence so are the ratios of the areas of the kite to the dart, and of the thick rhomb to the thin rhomb. (Both larger and smaller obtuse Robinson triangles can be found in the pentagon on the left: the larger triangles at the top – the halves of the thick rhomb – have linear dimensions scaled up by ''φ'' compared to the small shaded triangle at the base, and so the ratio of areas is ''φ''<sup>2</sup>:1.)
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| Any Penrose tiling has local pentagonal symmetry, in the sense that there are points in the tiling surrounded by a symmetric configuration of tiles: such configurations have fivefold [[rotational symmetry]] about the center point, as well as five mirror lines of [[reflection symmetry]] passing through the point, a [[dihedral group|dihedral]] symmetry [[group (mathematics)|group]].<ref name="AMSa"/> This symmetry will generally preserve only a patch of tiles around the center point, but the patch can be very large: Conway and Penrose proved that whenever the colored curves on the P2 or P3 tilings close in a loop, the region within the loop has pentagonal symmetry, and furthermore, in any tiling, there are at most two such curves of each color that do not close up.<ref name="G:9">{{Harvnb|Gardner|1997|p=9}}</ref>
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| There can be at most one center point of global fivefold symmetry: if there were more than one, then rotating each about the other would yield two closer centers of fivefold symmetry, which leads to a mathematical contradiction.<ref name="G:27">{{Harvnb|Gardner|1997|p=27}}</ref> There are only two Penrose tilings (of each type) with global pentagonal symmetry: for the P2 tiling by kites and darts, the center point is either a "sun" or "star" vertex.<ref name="GS:543">{{Harvnb|Grünbaum|Shephard|1987|p=543}}</ref>
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| ===Inflation and deflation===
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| [[File:Pentagon with half dodecahedral net.svg|right|thumb|A pentagon decomposed into six smaller pentagons (half a dodecahedral net) with gaps|alt=]]
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| Many of the common features of Penrose tilings follow from a hierarchical pentagonal structure given by ''[[Substitution tiling|substitution rules]]'': this is often referred to as ''inflation'' and ''deflation'', or ''composition'' and ''decomposition'', of tilings or (collections of) tiles.<ref name="AMSa"/><ref name="G1"/><ref name="GS-Terms">In {{Harvnb|Grünbaum|Shephard|1987}}, the term "inflation" is used where other authors would use "deflation" (followed by rescaling). The terms "composition" and "decomposition", which many authors also use, are less ambiguous.</ref> The substitution rules decompose each tile into smaller tiles of the same shape as those used in the tiling (and thus allow larger tiles to be "composed" from smaller ones). This shows that the Penrose tiling has a scaling self-similarity, and so can be thought of as a [[fractal]].<ref>Ramachandrarao P.,
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| ''On the fractal nature of Penrose tiling'', Current Science '''79'''(2000) p364 [http://www.ias.ac.in/currsci/aug102000/rc80.pdf]</ref>
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| Penrose originally discovered the P1 tiling in this way, by decomposing a pentagon into six smaller pentagons (one half of a [[Net (polyhedron)|net]] of a [[dodecahedron]]) and five half-diamonds; he then observed that when he repeated this process the gaps between pentagons could all be filled by stars, diamonds, boats and other pentagons.<ref name="P:32">{{Harvnb|Penrose|1978|p=32}}</ref> By iterating this process indefinitely he obtained one of the two P1 tilings with pentagonal symmetry.<ref name="AMSa"/><ref name="S:171"/>
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| ====Robinson triangle decompositions====
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| [[File:Robinson triangle decompositions.svg|left|thumb|Robinson triangles and their decompositions|alt=]]
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| The substitution method for both P2 and P3 tilings can be described using Robinson triangles of different sizes. The Robinson triangles arising in P2 tilings (by bisecting kites and darts) are called A-tiles, while those arising in the P3 tilings (by bisecting rhombs) are called B-tiles.<ref name="GS-R"/> The smaller A-tile, denoted A<sub>S</sub>, is an obtuse Robinson triangle, while the larger A-tile, A<sub>L</sub>, is acute; in contrast, a smaller B-tile, denoted B<sub>S</sub>, is an acute Robinson triangle, while the larger B-tile, B<sub>L</sub>, is obtuse.
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| Concretely, if A<sub>S</sub> has side lengths (1, 1, ''φ''), then A<sub>L</sub> has side lengths (''φ'', ''φ'', 1). B-tiles can be related to such A-tiles in two ways:
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| * If B<sub>S</sub> has the same size as A<sub>L</sub> then B<sub>L</sub> is an enlarged version ''φ''A<sub>S</sub> of A<sub>S</sub>, with side lengths (''φ'', ''φ'', ''φ''<sup>2</sup>=1+''φ'') – this decomposes into an A<sub>L</sub> tile and A<sub>S</sub> tile joined along a common side of length 1.
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| * If instead B<sub>L</sub> is identified with A<sub>S</sub>, then B<sub>S</sub> is a reduced version (1/''φ'')A<sub>L</sub> of A<sub>L</sub> with side lengths (1/''φ'',1/''φ'',1) – joining a B<sub>S</sub> tile and a B<sub>L</sub> tile along a common side of length 1 then yields (a decomposition of) an A<sub>L</sub> tile.
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| In these decompositions, there appears to be an ambiguity: Robinson triangles may be decomposed in two ways, which are mirror images of each other in the (isosceles) axis of symmetry of the triangle. In a Penrose tiling, this choice is fixed by the matching rules – furthermore, the matching rules ''also'' determine how the smaller triangles in the tiling compose to give larger ones.<ref name="GS-R"/>
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| {{multiple image|align=right|direction=vertical|image1=Penrose tilings P2 and P3.svg|image2=Penrose tilings P3 and P2.svg|width=200|footer=Partial inflation of star to yield rhombs, and of a collection of rhombs to yield an ace.}}
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| It follows that the P2 and P3 tilings are ''mutually locally derivable'': a tiling by one set of tiles can be used to generate a tiling by another – for example a tiling by kites and darts may be subdivided into A-tiles, and these can be composed in a canonical way to form B-tiles and hence rhombs.<ref name="S:173-174"/> The P2 and P3 tilings are also both mutually locally derivable with the P1 tiling (see [[#Figure 2|figure 2 above]]).<ref name="GS:546">{{Harvnb|Grünbaum|Shephard|1987|p=546}}</ref>
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| The decomposition of B-tiles into A-tiles may be written
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| : B<sub>S</sub> = A<sub>L</sub>, B<sub>L</sub> = A<sub>L</sub> + A<sub>S</sub>
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| (assuming the larger size convention for the B-tiles), which can be summarized in a ''substitution [[matrix (mathematics)|matrix]]'' equation:<ref name="S:157-158">{{Harvnb|Senechal|1996|pp=157–158}}</ref>
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| : <math>\begin{pmatrix} B_L \\ B_S\end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} A_L \\ A_S \end{pmatrix}\, .</math>
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| Combining this with the decomposition of enlarged ''φ''A-tiles into B-tiles yields the substitution
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| : <math>\begin{pmatrix} \varphi A_L \\ \varphi A_S\end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} B_L \\ B_S\end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}\begin{pmatrix} A_L \\ A_S \end{pmatrix}\, ,</math>
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| so that the enlarged tile ''φ''A<sub>L</sub> decomposes into two A<sub>L</sub> tiles and one A<sub>S</sub> tiles. The matching rules force a particular substitution: the two A<sub>L</sub> tiles in a ''φ''A<sub>L</sub> tile must form a kite – thus a kite decomposes into two kites and a two half-darts, and a dart decomposes into a kite and two half-darts.<ref name="AMSb"/><ref name="S:183">{{Harvnb|Senechal|1996|p=183}}</ref> Enlarged ''φ''B-tiles decompose into B-tiles in a similar way (via ''φ''A-tiles).
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| Composition and decomposition can be iterated, so that, for example
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| :<math>\varphi^n\begin{pmatrix} A_L \\ A_S\end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}^n\begin{pmatrix} A_L \\ A_S \end{pmatrix}\, .</math>
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| The number of kites and darts in the ''n''th iteration of the construction is determined by the ''n''th power of the substitution matrix:
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| :<math>\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}^n = \begin{pmatrix} F_{2n+1} & F_{2n} \\ F_{2n} & F_{2n-1} \end{pmatrix}\, ,</math>
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| where ''F''<sub>''n''</sub> is the ''n''th [[Fibonacci number]]. The ratio of numbers of kites to darts in any sufficiently large P2 Penrose tiling pattern therefore approximates to the golden ratio ''φ''.<ref name="G:7">{{Harvnb|Gardner|1997|p=7}}</ref> A similar result holds for the ratio of the number of thick rhombs to thin rhombs in the P3 Penrose tiling.<ref name="AMSb"/>
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| ====Deflation for P2 and P3 tilings====
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| Starting with a collection of tiles from a given tiling (which might be a single tile, a tiling of the plane, or any other collection), deflation proceeds with a sequence of steps called generations. In one generation of deflation, each tile is replaced with two or more new tiles that are scaled-down versions of tiles used in the original tiling. The [[substitution rules|substitution tiling]] guarantee that the new tiles will be arranged in accordance with the matching rules.<ref name="AMSb"/> Repeated generations of deflation produce a tiling of the original axiom shape with smaller and smaller tiles. <!--Given sufficiently many generations, the tiling will contain a scaled-down version of the axiom that does not touch the boundary of the tiling. The axiom can then be surrounded by full-size tiles corresponding to tiles that appear in the scaled-down version. This extended tiling can be used as a new axiom, producing larger and larger extended tilings and ultimately covers the entire plane.-->
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| <!--This is an example of successive generations of deflation starting from different axioms. In the case of the 'Sun' and 'Star', the scaled-down interior version of the axiom appears in generation 2. The 'Sun' also appears in the interior of its generation 3.-->
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| This rule for dividing the tiles is a [[subdivision rule]].
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| {| class="wikitable"
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| |-
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| ! Name
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| ! Initial tiles
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| ! Generation 1
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| ! Generation 2
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| ! Generation 3
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| |-
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| | Half-kite
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| | [[File:penrose kile 0.svg|120px]]
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| | [[File:penrose kile 1.svg|120px]]
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| | [[File:penrose kile 2.svg|120px]]
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| | [[File:penrose kile 3.svg|120px]]
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| |-
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| | Half-dart
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| | [[File:penrose dart 0.svg|120px]]
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| | [[File:penrose dart 1.svg|120px]]
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| | [[File:penrose dart 2.svg|120px]]
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| | [[File:penrose dart 3.svg|120px]]
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| |-
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| | Sun
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| | [[File:penrose sun 0bis.svg|120px]]
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| | [[File:penrose sun 1.svg|120px]]
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| | [[File:penrose sun 2.svg|120px]]
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| | [[File:penrose sun 3.svg|120px]]
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| |-
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| | Star
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| | [[File:penrose star 0.svg|120px]]
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| | [[File:penrose star 1.svg|120px]]
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| | [[File:penrose star 2.svg|120px]]
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| | [[File:penrose star 3.svg|120px]]
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| |}
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| ====Consequences and applications====
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| Inflation and deflation yield a method for constructing kite and dart (P2) tilings, or rhombus (P3) tilings, known as ''up-down generation''.<ref name="G:8"/><ref name="AMSb">{{Harvnb|Austin|2005b}}</ref><ref name="S:183"/>
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| The Penrose tilings, being non-periodic, have no translational symmetry – the pattern cannot be shifted to match itself over the entire plane. However, any bounded region, no matter how large, will be repeated an infinite number of times within the tiling. Therefore, a finite patch cannot differentiate between the uncountably many Penrose tilings, nor even determine which position within the tiling is being shown.<ref>"... any finite patch that we choose in a tiling will lie inside a single inflated tile if we continue moving far enough up in the inflation hierarchy. This means that anywhere that tile occurs at that level in the hierarchy, our original patch must also occur in the original tiling. Therefore, the patch will occur infinitely often in the original tiling and, in fact, in every other tiling as well." {{Harvnb|Austin|2005a}}</ref>
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| This shows in particular that the number of distinct Penrose tilings (of any type) is uncountably infinite. Up-down generation yields one method to parameterize the tilings, but other methods use Ammann bars, pentagrids, or cut and project schemes.<ref name="AMSb"/>
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| ==Related tilings and topics==
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| ===Decagonal coverings and quasicrystals===
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| [[File:Gummelt decagon.svg|300px|thumb|right|Gummelt's decagon (left) with the decomposition into kites and darts indicated by dashed lines; the thicker darker lines bound an inscribed ace and thick rhomb; possible overlaps (right) are by one or two red aces.<ref name="LR">{{Harvnb|Lord|Ranganathan|2001}}</ref>|alt=]]
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| In 1996, German mathematician Petra Gummelt demonstrated that a covering (so called to distinguish it from a non-overlapping tiling) equivalent to the Penrose tiling can be constructed using a single decagonal tile if two kinds of overlapping regions are allowed.<ref>{{Harvnb|Gummelt|1996}}</ref> The decagonal tile is decorated with colored patches, and the covering rule allows only those overlaps compatible with the coloring. A suitable decomposition of the decagonal tile into kites and darts transforms such a covering into a Penrose (P2) tiling. Similarly, a P3 tiling can be obtained by inscribing a thick rhomb into each decagon; the remaining space is filled by thin rhombs.
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| These coverings have been considered as a realistic model for the growth of [[quasicrystals]]: the overlapping decagons are 'quasi-unit cells' analogous to the [[unit cell]]s from which crystals are constructed, and the matching rules maximize the density of certain atomic clusters.<ref name="LR"/><ref>{{Harvnb|Steinhardt|Jeong|1996}}; see also {{cite web|separator=, |first=Paul J.|last=Steinhardt|title=A New Paradigm for the Structure of Quasicrystals|url=http://www.physics.princeton.edu/~steinh/quasi/}}</ref>
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| ===Related tilings===
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| [[File:Tie and Navette Tiling.png|300px|left|thumb||Tie and Navette Tiling (in red on a Penrose background)]]
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| The three variants of the Penrose tiling are mutually locally derivable. Selecting some subsets from the vertices of a P1 tiling allows to produce other non-periodic tilings. If the corners of one pentagon in P1 are labeled in succession by ''1,3,5,2,4'' an unambiguous tagging in all the pentagons is established, the order being either clockwise or counterclockwise.
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| Points with the same label define a tiling by Robinson triangles while points with the numbers 3 and 4 on them define the vertices of a Tie-and-Navette tiling.<ref>Luck R., "Penrose Sublattices", ''Journal of Non Crystalline Solids'' 117-8 (90)832-5</ref>
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| There are also other related unequivalent tilings, such as the hexagon-boat-star and Mikulla–Roth tilings. For instance, if the matching rules for the rhombus tiling are reduced to a specific restriction on the angles permitted at each vertex, a binary tiling is obtained.<ref>{{Harvnb|Lançon|Billard|1988}}</ref> Its underlying symmetry is also fivefold but it is not a quasicrystal. It can be obtained either by decorating the rhombs of the original tiling with smaller ones, or by applying substitution rules, but not by de Bruijn's cut-and-project method.<ref>{{Harvnb|Godrèche|Lançon|1992}}; see also {{cite web|separator=, |url=http://tilings.math.uni-bielefeld.de/tilings/substitution_rules/binary|title=Binary|author= E. Harriss and D. Frettlöh|work=Tilings Encyclopedia |publisher= Department of Mathematics, University of Bielefeld}}</ref>
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| [[File:Variable penrose tiling.svg|300px|right|thumb|A variant tiling which is not a quasicrystal. It is not a Penrose tiling because the tile alignment rules are not used.|alt=]]
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| ===Penrose tilings and art===
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| The aesthetic value of tilings has long been appreciated, and remains a source of interest in them; here the visual appearance (rather than the formal defining properties) of Penrose tilings has attracted attention. The similarity with [[girih tiles|some decorative patterns]] used in the Middle East has been noted<ref>{{Harvnb|Zaslavskiĭ|Sagdeev|Usikov|Chernikov|1988}}; {{Harvnb|Makovicky|1992}}</ref><ref>{{Cite news| first=Sebastian R. |last=Prange |coauthors= Peter J. Lu |title=The Tiles of Infinity |date=2009-09-01 |publisher=Aramco Services Company |url =http://www.saudiaramcoworld.com/issue/200905/the.tiles.of.infinity.htm |work =Saudi Aramco World |pages =24–31 |accessdate = 2010-02-22}}</ref> and [[Peter Lu|Lu]] and [[Paul Steinhardt|Steinhardt]] have presented evidence that a Penrose tiling underlies some examples of medieval Islamic art.<ref>{{Harvnb|Lu|Steinhardt|2007}}</ref>
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| [[Drop City]] artist Clark Richert used Penrose rhombs in artwork in 1970. Art historian [[Martin Kemp (art historian)|Martin Kemp]] has observed that [[Albrecht Dürer]] sketched similar motifs of a rhombus tiling.<ref>{{Harvnb|Kemp|2005}}</ref>
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| San Francisco new $4.2 billion [[San Francisco Transbay development|Transbay Transit Center]] is planning to perforate its exterior's undulating white metal skin with the Penrose pattern.<ref>{{citation|url=http://sf.curbed.com/archives/2013/07/11/check_out_the_proposed_skin_for_the_transbay_transit_center.php|title=Check Out the Proposed Skin for the Transbay Transit Center|date=July 11, 2013|first=Sally|last=Kuchar|journal=[[Curbed]]}}.</ref>
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| The floor of the atrium of the Molecular and Chemical Sciences Building at the University of Western Australia is tiled with Penrose Tiles.<ref>http://www.treasures.uwa.edu.au/treasures/66/</ref>
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| The Andrew Wiles Building, the location of the Mathematics Department at the [[University of Oxford]] as of October 2013<ref>{{citation|url =http://www.maths.ox.ac.uk/new-building|title=New Building Project}}</ref> includes a section of Penrose tiling as the paving of its entrance.<ref>{{citation|url =http://www.maths.ox.ac.uk/new-building/time-lapse/penrose|title=Penrose Paving Time Lapse Movie}}</ref>
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| ==See also==
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| * [[List of aperiodic sets of tiles]]
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| * [[Zellige]]
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| * [[Girih tiles]]
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| * [[Patterns in nature]]
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| * [[Pinwheel tiling]]
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| * [[Quaquaversal tiling]]
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| ==Notes==
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| {{Reflist|30em}}
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| ==References==
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| ===Primary sources===
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| * {{Citation|first=R.|last=Berger|title=The undecidability of the domino problem|series=Memoirs of the American Mathematical Society|volume=66| year=1966}}.
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| * {{Citation|first=N. G.|last=de Bruijn|authorlink=Nicolaas Govert de Bruijn|journal=Indagationes mathematicae|volume=43|pages=39–66|year=1981|title=Algebraic theory of Penrose's non-periodic tilings of the plane, I, II|url=http://alexandria.tue.nl/repository/freearticles/597566.pdf|format=PDF|issue=1}}.
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| * {{Citation|first=Petra|last= Gummelt|title=Penrose tilings as coverings of congruent decagons|doi= 10.1007/BF00239998|journal= Geometriae Dedicata|volume= 62|issue=1|year=1996}}.
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| * {{Citation|last=Penrose|first=Roger|authorlink=Roger Penrose|title=The role of aesthetics in pure and applied mathematical research|journal= Bulletin of the Institute of Mathematics and its Applications|volume= 10|year=1974|page=266ff}}.
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| * {{Citation| inventor-last = Penrose| inventor-first = Roger| inventorlink = Roger Penrose |publication-date = 1976-06-24 |issue-date =1979-01-09 |title = Set of tiles for covering a surface |country-code = US |patent-number = 4133152}}.
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| * {{Citation|first=R.M.|last=Robinson|authorlink=Raphael M. Robinson|title=Undecidability and non-periodicity for tilings of the plane|journal=Inventiones Mathematicae|volume=12|year=1971|pages=177–190|doi=10.1007/BF01418780|bibcode = 1971InMat..12..177R|issue=3 }}.
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| * {{Citation| last1 = Schechtman|first1 = D.|last2 = Blech|first2 = I.|last3 = Gratias|first3 = D.|last4 = Cahn|first4 = J.W.|title = Metallic Phase with long-range orientational order and no translational symmetry| journal = Physical Review Letters| volume = 53|year = 1984| pages = 1951–1953|doi = 10.1103/PhysRevLett.53.1951| bibcode=1984PhRvL..53.1951S| issue = 20}}
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| * {{Citation|first=H.|last=Wang|authorlink=Hao Wang (academic)|title=Proving theorems by pattern recognition II|journal= Bell Systems Technical Journal|volume=40|year=1961|pages=1–42}}.
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| ===Secondary sources===
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| * {{Citation|first=David|last=Austin|title=Penrose Tiles Talk Across Miles|journal=Feature Column|location=Providence|publisher=American Mathematical Society|year=2005a|url=http://www.ams.org/featurecolumn/archive/penrose.html}}.
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| * {{Citation|first=David|last=Austin|title=Penrose Tilings Tied up in Ribbons|journal=Feature Column|location=Providence|publisher=American Mathematical Society|year=2005b|url=http://www.ams.org/featurecolumn/archive/ribbons.html}}.
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| * {{Citation|first1=Karel|last1= Culik|first2=Jarkko|last2= Kari|author2-link=Jarkko Kari|title= Foundations of Computer Science|series= Lecture Notes in Computer Science|volume=1337|pages=153–162|year= 1997|doi=10.1007/BFb0052084|chapter=On aperiodic sets of Wang tiles|isbn=3-540-63746-X}}
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| * {{Citation|author-link=Martin Gardner|last=Gardner|first= Martin|title=Penrose Tiles to Trapdoor Ciphers|year=1997|publisher=Cambridge University Press|isbn=978-0-88385-521-8}}. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
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| **Chapter 1 (pp. 1–18) is a reprint of {{citation|last=Gardner|first= Martin|title=Extraordinary non-periodic tiling that enriches the theory of tiles|journal=Scientific American|date=January 1977|volume=236|pages=110–121}}.
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| * {{Citation|first1 = C|last1=Godrèche|first2=F.|last2=Lançon|year = 1992|title = A simple example of a non-Pisot tiling with five-fold symmetry|journal = Journal de Physique I|volume = 2|pages = 207–220|url = http://inac.cea.fr/sp2m/L_Sim/Publications/1992/Godreche_Lancon-1992-5_fold_non_Pisot_tiling.pdf|doi = 10.1051/jp1:1992134|bibcode = 1992JPhy1...2..207G|issue = 2 }}.
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| * {{Citation|author1-link=Branko Grünbaum|last1=Grünbaum|first1=Branko|last2= Shephard|first2= G. C.|title=Tilings and Patterns| location=New York |publisher=W. H. Freeman |year=1987 |isbn=0-7167-1193-1}}.
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| * {{Citation|last = Kemp|first= Martin|title = Science in culture: A trick of the tiles|journal = Nature|volume = 436|page = 332|year = 2005|doi = 10.1038/436332a|issue=7049|bibcode = 2005Natur.436..332K }}.
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| * {{Citation|first1 = Frédéric|last1= Lançon|first2=Luc|last2=Billard|year = 1988|title = Two-dimensional system with a quasi-crystalline ground state|journal = Journal de Physique|volume = 49|pages = 249–256|doi = 10.1051/jphys:01988004902024900|url=http://inac.cea.fr/sp2m/L_Sim/Publications/1988/Lancon-Billard-1988-binary_tilings.pdf|issue = 2}}.
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| * {{Citation|journal=Acta Crystallographica|volume=A57|year=2001|pages= 531–539 |title=The Gummelt decagon as a 'quasi unit cell'|first1=E.A.|last1=Lord|first2=S.|last2=Ranganathan|url=http://materials.iisc.ernet.in/~lord/webfiles/actagum.pdf|doi=10.1107/S0108767301007504|issue=5}}
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| * {{Citation|first1=Peter J.|last1= Lu|first2=Paul J.|last2= Steinhardt|year = 2007|title = Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture|journal = [[Science (journal)|Science]]|volume = 315|pages = 1106–1110|url = http://www.peterlu.org/sites/peterlu.org/files/Science_315_1106_2007.pdf|doi = 10.1126/science.1135491|pmid = 17322056|issue=5815|bibcode=2007Sci...315.1106L}}.
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| * {{Citation|last=Luck|first= R.|title=Dürer-Kepler-Penrose: the development of pentagonal tilings|journal=Materials Science and Engineering|volume=294|issue=6|year=2000|pages=263–267}}.
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| * {{Citation|first=E.|last=Makovicky|year=1992|chapter=800-year-old pentagonal tiling from Maragha, Iran, and the new varieties of aperiodic tiling it inspired|editor=I. Hargittai|title=Fivefold Symmetry|pages=67–86|publisher=World Scientific|location=Singapore–London}}.
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| * {{Citation|first=Roger|last=Penrose|authorlink=Roger Penrose|title=Pentaplexity|journal=[[Eureka (University of Cambridge magazine)|Eureka]]|volume= 39|year=1978|pages=16–22}}. (Page numbers cited here are from the reproduction as {{citation|title=Pentaplexity: A class of non-periodic tilings of the plane|journal=[[The Mathematical Intelligencer]]|volume=2|year=1979/80|pages=32–37|doi=10.1007/BF03024384|last1=Penrose|first1=R.}}.)
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| * {{Citation| title=Quasicrystals and geometry| first=Marjorie|last=Senechal|authorlink=Marjorie Senechal|publisher= Cambridge University Press|year=1996|isbn=978-0-521-57541-6}}.
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| * {{Citation|first1=Paul J.|last1=Steinhardt|first2=Hyeong-Chai|last2=Jeong|journal=Nature|issue = 1 August|volume=382|pages=431–433|year=1996|doi=10.1038/382431a0|title=A simpler approach to Penrose tiling with implications for quasicrystal formation|url=http://www.nature.com/nature/journal/v382/n6590/abs/382431a0.html|bibcode = 1996Natur.382..431S}}.
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| * {{Citation|first1= G.M.|last1=Zaslavskiĭ|first2= Roal'd Z.|last2=Sagdeev|first3= D.A.|last3=Usikov|first4= A.A.|last4=Chernikov|year = 1988|title = Minimal chaos, stochastic web and structures of quasicrystal symmetry|journal =Soviet Physics Uspekhi|volume=31|pages=887–915|doi=10.1070/PU1988v031n10ABEH005632|bibcode = 1988SvPhU..31..887Z|issue= 10}}.
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| ==External links==
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| {{Commons category|Penrose tilings}}
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| * {{mathworld|urlname = PenroseTiles |title = Penrose Tiles}}
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| * {{cite web|separator=, |url=http://www.quadibloc.com/math/pen01.htm |title=Penrose Tilings |author=John Savard|publisher=quadibloc.com |accessdate=2009-11-28}}
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| * {{cite web|separator=, |url=http://intendo.net/penrose/info.html|title=Penrose Tiling |author=Eric Hwang|publisher=intendo.net |accessdate=2009-11-28}}
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| * {{cite web|separator=, |url=http://tilings.math.uni-bielefeld.de/substitution_rules/penrose_rhomb |title=Penrose Rhomb |author= E. Harriss and D. Frettlöh |date= |work=Tilings Encyclopedia |publisher= Department of Mathematics, University of Bielefeld |accessdate=2009-11-28}}
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| * {{cite web|separator=, |url=http://www.mathpages.com/home/kmath621/kmath621.htm|title=On de Bruijn Grids and Tilings |author=Kevin Brown|publisher=mathpages.com |accessdate=2009-11-28}}
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| * {{cite web|separator=, |url=http://www.ics.uci.edu/~eppstein/junkyard/penrose.html|title=Penrose Tiles |author=David Eppstein |authorlink = David Eppstein|publisher=www.ics.uci.edu/~eppstein| work=The Geometry Junkyard |accessdate=2009-11-28}} This has a list of additional resources.
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| * {{cite web|separator=, |url=http://www.eschertile.com/penrose.htm |title=Penrose tile in architecture |author=William Chow |accessdate=2009-12-28}}
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| * {{cite web| url = http://www.spacegoo.com/penrose| title = Penrose's tiles viewer}}
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| {{Use dmy dates|date=September 2010}}
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| {{Good article}}
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| {{DEFAULTSORT:Penrose Tiling}}
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| [[Category:Discrete geometry]]
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| [[Category:Tessellation]]
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| [[Category:Aperiodic sets of tiles]]
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| [[Category:Mathematics and art]]
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| {{Link FA|hr}}
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