A Penrose tiling

A Penrose tiling is an aperiodic tiling of the plane discovered by Roger Penrose in 1973. Being aperiodic, it has no translational symmetry - it never repeats itself exactly, but nevertheless it has a fivefold rotational symmetry. The Penrose tiling is also a prime example of a quasicrystal as it produces a sharply outlined diffractogram. There are two popular variants of the Penrose tiling which use different sets of tiles. Robert Ammann independently discovered the tiling. A similarity with decorative patterns used in the Middle East has been frequently noted and in February 2007 a paper by Steinhard and Lu offered evidence that a Penrose tiling underlies some examples of islamic medieval art^[1]. Roger Penrose acknowledges inspiration from the work of Johannes Kepler. Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... An aperiodic tiling is a tiling of the plane by a set of prototiles that can only be tiled in a non-repeating (non-periodic) pattern. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... A translation slides an object by a vector a: Ta(p) = p + a. ... Quasicrystals are aperiodic structures which produce diffraction. ... Robert Ammann (October 1, 1946-May, 1994) was an amateur mathematician who made several significant and groundbreaking contributions to the theory of quasicrystals and aperiodic tilings. ... Johannes Kepler (December 27, 1571 – November 15, 1630) was a German Lutheran mathematician, astronomer and astrologer, and a key figure in the 17th century astronomical revolution. ...

In 1982 Dan Shechtman reported that a sample of aluminium-manganese alloy produced a sharp diffractogram with fivefold symmetry. At that time it was assumed that such symmetry is incompatible with the ability to diffract. The combination of these two features is possible only in an aperiodic structure. The full three-dimensional arrangement, which exhibits icosahedral symmetry, had been worked out by Robert Ammann. The atoms in the planes corresponding to the unusual symmetry are arranged in the pattern of a Penrose tiling. De Bruijn has shown that it was possible to obtain the Penrose tiling as a projection from a five-dimensional cubic lattice, which explains its crystal-like ability to diffract. The Penrose Tiling has become the most studied - and most popular - quasicrystal. Dan Shechtman is the Philip Tobias Professor of Materials Science at the Israel Institute of Technology. ... A Dutch mathematician, especially noted for the invention of the de Bruijn Sequence. External links About the de Bruijn sequence ...

Construction principles

Hao Wang proved that it must be possible to tile the plane aperiodically and shortly after Robert Berger proposed the first set of 20426 distinct tile shapes which plane completely only in an aperiodic pattern. The set was rapidly reduced, reaching the number of two. The rules which produce aperiodicity may be embodied in dents, arrows or colors on the edges of the tiles. Using plain geometric forms is also possible but the invisible rules are still observed (to avoid trivial tilings). The Penrose tiling, as it was discovered, uses two triangles glued together to form a pair of rhombuses or a pair of shapes known as a kite and a dart. With these two sets an equivalent tiling is produced. The basic triangles are also known as Robinson triangles which have been used to produce aperiodic tilings by substitutions. Hao Wang 王浩 (1921 – 1995) was a Chinese-American logician, philosopher and mathematician. ... Robert Berger invented the first aperiodic set of tiles consisting of 20426 distinct tile shapes by using the rules of Penrose Tilling and the Golden Rule in 1966. ...

Rhombus tiling

The Penrose rhombuses have equal sides and angles which are multiples of one tenth of a circle (36 degrees).

• The first tile, known as the thick rhombus T, has four corners with the angles {72, 108, 72, 108} degrees.
• The second tile is the thin rhombus t with angles of {36, 144, 36, 144} degrees.

There are 23 sets of angles which would add up to 360 degrees at a vertex and most of them admit different orderings, but the rules of the tiling allow only 7 distinct types of vertices. The two tiles appear in the tiling with equal constant frequencies in just ten different orientations and the tiling has a statistical tenfold symmetry [1]. There are many local fivefold centers and a unique center point of global fivefold symmetry where five mirror lines cross. As the tiling is aperiodic, there is no translational symmetry: the pattern never repeats exactly. However, given a bounded region, no matter how large, that region will be repeated an infinite number of times within the tiling.

The tiles are put together with one general rule: no two tiles can be touching so as to form a single parallelogram. Given this rule, there is an uncountably infinite combination of ways to tile the plane without gaps. It is easy to check that some of the compact patches consisting of three tiles admit two different arrangements and thus variations are possible.

The picture shows a variant tiling constructed with the same rhombuses and with the same proportion of thick and thin tiles. The underlying symmetry is also fivefold but this tiling is not a quasicrystal. It can be obtained either by 'decorating' the rhombuses of the original tiling with smaller ones or directly by the substitutions T->3T+t, t->T+2t, but not by de Bruijn's cut-and-project method. [2] Image File history File links VarPenrT.jpg‎ Template:Pd-sf File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

Drawing the Penrose tiling

L-System approach

The Penrose tiling can be drawn using the following L-system: See L-system for information on Lindenmayer systems. ...

variables:	1 6 7 8 9 [ ]
constants:	+ −;
start:	[7]++[7]++[7]++[7]++[7]
rules:	6 → 81++91−−−−71[−81−−−−61]++
	7 → +81−−91[−−−61−−71]+
	8 → −61++71[+++81++91]−
	9 → −−81++++61[+91++++71]−−71
	1 → (eliminated at each iteration)
angle:	36º

Where, 1 means "draw forward", + means "turn left by angle", and − means "turn right by angle" (see turtle graphics). The [ means save the present position and direction to restore them when corresponding ] is executed. The symbols 6, 7, 8 and 9 do not correspond to any action; they are there only to produce the correct curve evolution. Logo turtle graphic The Logo programming language is a functional programming language. ...

Evolution of L-system for n=1, n=2, n=3

Image File history File links Penrose_tiling_3_iterations. ...

Deflation approach

The penrose tiling can also be generated by a deflation algorithm, which relies on the matching rule.

Matching rule

In order to enforce the parallelogram rule, one can draw a pattern on the two types of tiles that occur in the penrose tiling. In the examples below, the two tiles are the so-called 'Kite' and 'Dart' which are alternatives for the two rhombs in the introduction.

Image File history File links Kile_Dart. ...

The green and the red arcs in the tiles indicate how the tiles should be connected: When two tiles share an edge in a tiling, the patterns must match at these edges.

Deflation

Based on the matching rule, one can define a recursive substitution (i.e. deflation) that yields the penrose tiling. The algorithm starts with an axiom. This is an initial finite part of the penrose tiling that obeys the matching rules. An axiom can be as simple a single tile. Consequently, the deflation is a process that consists of several generations. During one generation, each tile is systematically substituted by one or more new tiles that exactly cover the area of the original tile. A set of substitution rules is only valid if the pattern on each tile after substitution results in equivalent matching conditions.

An example of such substitution rules is given in the table below. The tiles are half darts and half kites.

	Half a kite	Half a dart
Generation i
Generation i+1

Image File history File links Kile_0. ... Image File history File links Dart_0. ... Image File history File links Kile_1. ... Image File history File links Dart_1. ...

Examples

These are four examples of penrose tilings generated with the deflation approach.

Name	Generation 0 (or axiom)	Generation 1	Generation 2	Generation 3
Kite (half)
Dart (half)
Sun
Star

Image File history File links Penrose_kile_0. ... Image File history File links Penrose_kile_1. ... Image File history File links Penrose_kile_2. ... Image File history File links Penrose_kile_3. ... Image File history File links Penrose_dart_0. ... Image File history File links Penrose_dart_1. ... Image File history File links Penrose_dart_2. ... Image File history File links Penrose_dart_3. ... Image File history File links Penrose_sun_0bis. ... Image File history File links Penrose_sun_1. ... Image File history File links Penrose_sun_2. ... Image File history File links Penrose_sun_3. ... Image File history File links Penrose_star_0. ... Image File history File links Penrose_star_1. ... Image File history File links Penrose_star_2. ... Image File history File links Penrose_star_3. ...

Fibonacci and Golden Ratio features

The Penrose Tiling, the Fibonacci sequence and the Golden ratio are intricately related and perhaps they should be considered as different aspects of the same phenomenon.

• the ratio of thick T to thin t rhombuses in the infinite tile is the golden ratio T/t = φ = 1.618..
• the Conway worms, sequences of neighbouring rhombuses with parallel sides, are Fibonacci ordered appearances of T and t and thus the Ammann bars also form Fibonacci ordered grids
• around each 5T − star a segmented Fibonacci spiral is formed by the sides of rhombuses [3]
• the distances between repeated finite motifs in the tiling grow as Fibonacci numbers when the size of the motif increases
• the substitution scheme T − > 2T + t,t − > T + t uses φ as a scaling factor; implemented as a symbol sequence ( e.g. 1->101, 0->10) this substitution produces a series of words with lengths which are the Fibonacci numbers with odd index, F(2n+1) for n=1,2,3.., the limit being the infinite Fibonacci binary sequence
• the eigenvalues of the substitution matrix are φ+1 and 2-φ

Conway can refer to any of the following: // People David Conway Deborah Conway Derek Conway Elias Nelson Conway Gerry Conway Henry Seymour Conway Henry Wharton Conway James Conway James Sevier Conway Jill Ker Conway Jimmy Conway John Conway, mathematician Jon Conway Lynn Conway Moncure Daniel Conway Rob Conway Sean Conway... In general, substitution is the replacement of one thing with another. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ...

Trivia

Pentaplex Ltd., a company in Yorkshire, England controlled by Penrose, owns the licensing rights to Penrose tilings. Penrose and Pentaplex filed a lawsuit against Kimberly-Clark for breach of copyright. Kimberly-Clark had allegedly embossed Penrose tilings on Kleenex quilted toilet paper in the UK. SCA Hygiene Products later came to control Kleenex products and reached an agreement with Penrose and Pentaplex on the Penrose tiling issue. SCA is not involved in the copyright dispute.[4] Look up Yorkshire in Wiktionary, the free dictionary. ... Motto: (French for God and my right) Anthem: God Save the King/Queen Capital London (de facto) Largest city London Official language(s) English (de facto) Unification    - by Athelstan AD 927  Area    - Total 130,395 km² (1st in UK)   50,346 sq mi  Population    - 2006 est. ... Kimberly-Clark Corporation (NYSE: KMB) is an American corporation that produces mostly paper-based consumer products. ... Kleenex logo This article is about the Kleenex brand. ... A roll of toilet paper. ...

Around the same time these quasi-patterns were also being used in artwork (in 1970) by Drop City artist, Clark Richert. Drop City was an artists community that formed in southern Colorado in 1965. ...

Penrose tiles covering the CMS building floor at The University of Western Australia

Image File history File linksMetadata Download high-resolution version (2048x1360, 183 KB) Penrose Tiles in CMS Building, University of Western Australia I, the creator of this work, hereby release it into the public domain. ... Image File history File linksMetadata Download high-resolution version (2048x1360, 183 KB) Penrose Tiles in CMS Building, University of Western Australia I, the creator of this work, hereby release it into the public domain. ...

Notes

1. ^ Peter J. Lu and Paul J. Steinhardt (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture". Science 315: 1106-1110. 

Science is the journal of the American Association for the Advancement of Science (AAAS). ...

References

• Penrose, Roger. (1989) The Emperor's New Mind. ISBN 0-19-851973-7
• Penrose, Roger, U.S. Patent 4133152  "Set of tiles for covering a surface," patent issued January 9, 1979
• Gardner, Martin. "Penrose Tiles", chapter 7 in his book The Colossal Book of Mathematics. ISBN 0-393-02023-1
• Kemp, Martin (2005). "Science in culture: A trick of the tiles". Nature 436: 332. DOI:10.1038/436332a. 

The Emperors New Mind: Concerning Computers, Minds and The Laws of Physics is a 1989 book by mathematical physicist Roger Penrose. ... A patent is a set of exclusive rights granted by a state to a patentee (the inventor or assignee) for a fixed period of time in exchange for the regulated, public disclosure of certain details of a device, method, process or composition of matter (substance) (known as an invention) which... Martin Gardner (b. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ...

External links

• A wealth of information on the Penrose tiling is available on the Internet. Two sites among the best are: John Savard's pages on Pentagonal Tiling and Eric Hwang's Penrose Tiling
• An implementation of the aforementioned L-System as a Scalable Vector Graphic with ECMAScript by Sam Ruby
• A freeware program (for Microsoft Windows) to generate and explore rhombic Penrose tiling. The software was written by Stephen Collins of JKS Software, in collaboration with the Universities of York, UK and Tsuka, Japan.
• Two theories for the formation of quasicrystals resembling Penrose tilings