Penrose tiling

A Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and a tiling is aperiodic if it does not contain arbitrarily large periodic regions or patches. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s. There are se…

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Jump to content Main menu Main menu move to sidebar hide Navigation Main page Contents Current events Random article About Wikipedia Contact us Contribute Help Learn to edit Community portal Recent changes Upload file Special pages Search Search Appearance Donate Create account Log in Personal tools Donate Create account Log in Contents move to sidebar hide (Top) 1 Background and history Toggle Background and history subsection 1.1 Periodic and aperiodic tilings 1.2 Earliest aperiodic tilings 1.3 Development of the Penrose tilings 2 Penrose tilings Toggle Penrose tilings subsection 2.1 Original pentagonal Penrose tiling (P1) 2.2 Kite and dart tiling (P2) 2.3 Rhombus tiling (P3) 3 Features and constructions Toggle Features and constructions subsection 3.1 Golden ratio and local pentagonal symmetry 3.2 Inflation and deflation 3.2.1 Robinson triangle decompositions 3.2.2 Deflation for P2 and P3 tilings 3.2.3 Consequences and applications 4 Related tilings and topics Toggle Related tilings and topics subsection 4.1 Decagonal coverings and quasicrystals 4.2 Related tilings 5 Art and architecture 6 See also 7 Notes 8 References Toggle References subsection 8.1 Primary sources 8.2 Secondary sources 9 External links Toggle the table of contents Penrose tiling 26 languages العربية Беларуская Català Čeština Deutsch Esperanto Español Suomi Français עברית Hrvatski Magyar Italiano 日本語 한국어 Nederlands Polski Português Русский Srpskohrvatski / српскохрватски Simple English Српски / srpski Svenska Türkçe Українська 中文 Edit links Article Talk English Read Edit View history Tools Tools move to sidebar hide Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Switch to legacy parser Print/export Download as PDF Printable version In other projects Wikimedia Commons Wikidata item Appearance move to sidebar hide From Wikipedia, the free encyclopedia This is an old revision of this page, as edited by Geometry guy ( talk | contribs ) at 20:33, 21 November 2023 (Like "variants" (spelled correctly) - dislike "multiple") . The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision . Revision as of 20:33, 21 November 2023 by Geometry guy ( talk | contribs ) (Like "variants" (spelled correctly) - dislike "multiple") ( diff ) ← Previous revision | Latest revision ( diff ) | Newer revision → ( diff ) Non-periodic tiling of the plane A Penrose tiling with rhombi exhibiting fivefold symmetry A Penrose tiling is an example of an aperiodic tiling . Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and a tiling is aperiodic if it does not contain arbitrarily large periodic regions or patches. However, despite their lack of translational symmetry , Penrose tilings may have both reflection symmetry and fivefold rotational symmetry . Penrose tilings are named after mathematician and physicist Roger Penrose , who investigated them in the 1970s. There are several variants of Penrose tilings with different tile shapes. The original form of Penrose tiling used tiles of four different shapes, but this was later reduced to only two shapes: either two different rhombi , or two different quadrilaterals called kites and darts. The Penrose tilings are obtained by constraining the ways in which these shapes are allowed to fit together in a way that avoids periodic tiling. This may be done in several different ways, including matching rules, substitution tiling or finite subdivision rules , cut and project schemes, and coverings. Even constrained in this manner, each variation yields infinitely many different Penrose tilings. 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 Penrose tilings are self-similar : they may be converted to equivalent Penrose tilings with different sizes of tiles, using processes called inflation and deflation . The pattern represented by every finite patch of tiles in a Penrose tiling occurs infinitely many times throughout the tiling. They are quasicrystals : implemented as a physical structure a Penrose tiling will produce diffraction patterns with Bragg peaks and five-fold symmetry, revealing the repeated patterns and fixed orientations of its tiles. 1 The study of these tilings has been important in the understanding of physical materials that also form quasicrystals. 2 Penrose tilings have also been applied in architecture and decoration, as in the floor tiling shown. Background and history Periodic and aperiodic tilings Figure 1. Part of a periodic tiling with two prototiles Covering a flat surface ("the plane") with some pattern of geometric shapes ("tiles"), with no overlaps or gaps, is called a tiling . The most familiar tilings, such as covering a floor with squares meeting edge-to-edge, are examples of periodic tilings . If a square tiling is shifted by the width of a tile, parallel to the sides of the tile, the result is the same pattern of tiles as before the shift. A shift (formally, a translation ) that preserves the tiling in this way is called a period of the tiling. A tiling is called periodic when it has periods that shift the tiling in two different directions. 3 The tiles in the square tiling have only one shape, and it is common for other tilings to have only a finite number of shapes. These shapes are called prototiles , and a set of prototiles is said to admit a tiling or tile the plane if there is a tiling of the plane using only these shapes. That is, each tile in the tiling must be congruent to one of these prototiles. 4 A tiling that has no periods is non-periodic . A set of prototiles is said to be aperiodic if all of its tilings are non-periodic, and in this case its tilings are also called aperiodic tilings . 5 Penrose tilings are among the simplest known examples of aperiodic tilings of the plane by finite sets of prototiles. 3 Earliest aperiodic tilings An aperiodic set of Wang dominoes . 6 The subject of aperiodic tilings received new interest in the 1960s when logician Hao Wang noted connections between decision problems and tilings. 7 In particular, he introduced tilings by square plates with colored edges, now known as Wang dominoes 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 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. Robinson's six prototiles Wang's student Robert Berger proved that the Domino Problem was undecidable (so Wang's conjecture was incorrect) in his 1964 thesis, 8 and obtained an aperiodic set of 20,426 Wang dominoes. 9 He also described a reduction to 104 such prototiles; the latter did not appear in his published monograph, 10 but in 1968, Donald Knuth detailed a modification of Berger's set requiring only 92 dominoes. 11 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. 12 Raphael Robinson , in a 1971 paper 13 which simplified Berger's techniques and undecidability proof, used this technique to obtain an aperiodic set of just six prototiles. 14 Development of the Penrose tilings Figure 2. The pentagonal Penrose tiling (P1) drawn in black on a colored rhombus tiling (P3) with yellow edges. 15 The first Penrose tiling (tiling P1 below) is an aperiodic set of six prototiles, introduced by Roger Penrose in a 1974 paper, 16 based on pentagons rather than squares. Any attempt to tile the plane with regular pentagons necessarily leaves gaps, but Johannes Kepler showed, in his 1619 work Harmonices Mundi , that these gaps can be filled using pentagrams ( star polygons ), decagons and related shapes. 17 Kepler extended this tiling by five polygons and found no periodic patterns, and already conjectured that every extension would introduce a new feature 18 hence creating an aperiodic tiling. Traces of these ideas can also be found in the work of Albrecht Dürer . 19 Acknowledging inspiration from Kepler, Penrose found matching rules for these shapes, obtaining an aperiodic set. These matching rules can be imposed by decorations of the edges, as with the Wang tiles. Penrose's tiling can be viewed as a completion of Kepler's finite Aa pattern. 20 A non-Penrose tiling by pentagons and thin rhombs in the early 18th-century Pilgrimage Church of Saint John of Nepomuk at Zelená hora, Czech Republic 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). 21 The rhombus tiling was independently discovered by Robert Ammann in 1976. 22 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 . 23 In 1981, N. G. de Bruijn provided two different methods to construct Penrose tilings. De Bruijn's "multigrid method" obtains the Penrose tilings as the dual graphs of arrangements of five families of parallel lines. In his "cut and project method", Penrose tilings are obtained as two-dimensional projections from a five-dimensional cubic structure. In these approaches, the Penrose tiling is viewed as a set of points, its vertices, while the tiles are geometrical shapes obtained by connecting vertices with edges. 24 Penrose tilings A P1 tiling using Penrose's original set of six prototiles Penrose P1 tiles with overlaid circular arcs and nodes to enforce the tiling rules The three types of Penrose tiling, P1 – P3, are described individually below. 25 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. 9 26 Original pentagonal Penrose tiling (P1) 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). 27 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. Treating these three types as different prototiles gives a set of six prototiles overall. It is common to indicate the three different types of pentagonal tiles using three different colors, as in the figure above right. 28 Kite and dart tiling (P2) Part of the plane covered by Penrose tiling of type P2 (kite and dart). Created by applying several deflations, see section below. 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. 29 Both the kite and dart are composed of two triangles, called Robinson triangles , after 1975 notes by Robinson. 30 Kite and dart tiles (top) and the seven possible vertex figures in a P2 tiling, which are each given nicknames as follows – First row: star, ace, sun. Second row: king, jack, queen, deuce 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). 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. 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. 31 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. 21 These rules often force the placement of certain tiles: for example, the 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. 32 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). 33 Apart from the ace (top middle) and the sun, all of these vertex figures force the placement of additional tiles. 34 Rhombus tiling (P3) Matching rule for Penrose rhombs using circular arcs or edge modifications to enforce the tiling rules Matching rule for Penrose rhombs using parabolic edges to enforce the tiling rules A Penrose Tiling of type P3 using Penrose Rhombuses with parabolic edges The third tiling uses a pair of rhombuses (often referred to as " rhombs " in this context) with equal sides but different angles. 9 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 above shows. There are two kinds of tile, both of which can be decomposed into Robinson triangles. 30 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. 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. 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. 9 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). 35 The various combinations of angles and facial curvature allow construction of arbitrarily complex tiles, such as the Penrose chickens . 36 Features and constructions Golden ratio and local pentagonal symmetry Several properties and common features of the Penrose tilings involve the golden ratio φ = 1 + 5 2 {\displaystyle \varphi =}}}} (approximately 1.618). 30 31 This is the ratio of chord lengths to side lengths in a regular pentagon , and satisfies φ = 1 + 1/ φ . 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. 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 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 2 :1.) 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 symmetry group . 9 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. 37 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. 38 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. 39 Inflation and deflation A pentagon decomposed into six smaller pentagons (half a dodecahedral net) with gaps Many of the common features of Penrose tilings follow from a hierarchical pentagonal structure given by substitution rules : this is often referred to as inflation and deflation , or composition and decomposition , of tilings or (collections of) tiles. 9 23 40 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 , using the same process as the pentaflake . 41 Penrose originally discovered the P1 tiling in this way, by decomposing a pentagon into six smaller pentagons (one half of a 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. 27 By iterating this process indefinitely he obtained one of the two P1 tilings with pentagonal symmetry. 9 20 Robinson triangle decompositions Robinson triangles and their decompositions 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. 30 The smaller A-tile, denoted A S , is an obtuse Robinson triangle, while the larger A-tile, A L , is acute ; in contrast, a smaller B-tile, denoted B S , is an acute Robinson triangl