PENROSE TILING

A 'Penrose tiling' is an aperiodic tiling of the plane using tiles discovered by Roger Penrose in 1973.^[1] Being aperiodic, every tiling of the plane with these tiles is nonperiodic. A nonperiodic tiling lacks translational symmetry—that is, there is no part of such a tiling that can be repeated at regular intervals to tile the plane. Although every tiling with Penrose tiles is nonperiodic, there are two tilings that possess both mirror symmetry and fivefold rotational symmetry, as in the diagram at the right. A Penrose tiling is also a two-dimensional quasicrystal, in that 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 at approximately the same time as Penrose.
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, because fivefold diffraction is possible only in a nonperiodic structure. The full three-dimensional arrangement, which exhibits icosahedral symmetry, was identified 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. The Penrose tiling has become the most studied—and most popular—quasicrystal. The physicists' interest led to another approach which connected the Penrose tiling to extremal problems and proved it to be a model for the state with minimum energy in some systems. This development came after Petra Gummelt's demonstration that it is possible to realize the Penrose tiling as a covering with a single decagonal prototile, if the tiles are allowed to overlap in specific ways.^[2]

Contents

Construction principles

Penrose tile shapes

Rhombus tiling

Kite and Dart tiling

Drawing the Penrose tiling

L-system approach

Deflation approach

Deflation

Extension to a tiling of the plane

Examples

Decagonal covering

Fibonacci and golden ratio features

Related tilings in art

Trivia

References and notes

External links

Construction principles

In 1961, Hao Wang found conditions required to produce an aperiodic tiling from a finite set of prototiles, and Robert Berger used these conditions to find the first aperiodic tiling in 1966. Berger's original set of 20426 distinct tile shapes was reduced over the years, culminating in Penrose's discovery of two prototiles that tile aperiodically. The prototiles are based on geometric shapes that would tile periodically, but with rules to enforce aperiodicity embodied in dents, arrows or colors on the edges of the tiles. These tilings are often presented as the plain geometric forms that observe the aperiodicity rules.

Penrose tile shapes

There are two pairs of shapes that form Penrose tilings, as they were first discovered. One pair consists of two rhombuses, while the other pair consists of two quadrilateral shapes called the 'Kite' and the 'Dart'. Either of these pairs can be cut in half to form a pair of triangles, called Robinson triangles, which can be used to produce the Penrose tilings as a substitution tiling. The Robinson triangles are the isosceles 36º-36º-108º and 72º-72º-36º triangles. Each of these triangles has edges in the ratio of (1+√5):2, the golden ratio. The rules that enforce aperiodicity on a Robinson triangle tiling make the triangles asymmetric, and each triangle appears in conjunction with its mirror image to form a rhombus, kite, or dart.

Rhombus tiling

The Penrose rhombuses are a pair of rhombuses with equal sides but different shapes.


★ The thin rhombus 't' has four corners with angles of 36, 144, 36, and 144 degrees. The 't' rhombus may be bisected along its short diagonal to form a pair of acute Robinson triangles.

★ The thick rhombus 'T' has angles of 72, 108, 72, and 108 degrees. The 'T' rhombus may be bisected along its long diagonal to form a pair of obtuse Robinson triangles.
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 7 of these combinations to appear. Each tile appears in the tiling with a constant frequency, evenly divided into ten different orientations, and the tiling has a statistical tenfold symmetry.^[3]
Ordinary rhombus-shaped tiles can be used to tile the plane periodically, so restrictions must be made on how tiles can be assembled. The simplest rule, prohibiting two tiles to be put together to form a single parallelogram, is insufficient to ensure aperiodicity.^[4] Instead, rules are made that distinguish sides of the tiles and require that only particular sides can be put together with each other. An examples of appropriate matching rules is shown in the upper part of the diagram to the left. Tiles must be assembled so that the curves across their edges match in color and position. An equivalent condition is that tiles must be assembled so that the bumps on their edges fit together. The same rules can be specified with other formulations.
There are arbitrarily large finite patches with tenfold symmetry and at most one center point of global tenfold symmetry where ten mirror lines cross. As the tiling is aperiodic, there is 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. The only way to distinguish the two symmetric Penrose tilings from the others is that their symmetry continues to infinity.

Kite and Dart tiling

The quadrilaterals called the 'Kite' and 'Dart' can also be used to form a Penrose tiling.


★ The 'Kite' is a quadrilateral whose four corners have angles of 72, 72, 72, and 144 degrees. The Kite may be bisected along its axis of symmetry to form a pair of acute Robinson triangles.

★ 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.
The green and the red arcs in the tiles constrain the placement of tiles: When two tiles share an edge in a tiling, the patterns must match at these edges. For example, the concave vertex of a Dart cannot be filled with a single Kite, but must be filled with a pair of Kites.

Drawing the Penrose tiling

L-system approach

The rhombus Penrose tiling can be drawn using the following L-system:

'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

here '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.

Deflation approach

The Penrose tiling can also be generated by a deflation algorithm, an example of a tile substitution system. The following description of deflation uses the Kite-and-Dart tiling.

Deflation

Deflation is a substitution method that produces a Penrose tiling of the plane starting with a finite tiling called the 'axiom'. The axiom can be as simple as a single tile. Deflation proceeds with a sequence of steps called 'generations'.
In a single generation of deflation, each tile is replaced with one or more new tiles that exactly cover the area of the original tile. The new tiles are scaled-down versions of the original tiles. The substitution rules guarantee that the new tiles are arranged according to the matching rules. A system of substitution rules is given in the table below. The tiles are half darts and half kites.

	Half a kite	Half a dart
Generation i	Kile_0.svg	Dart_0.svg
Generation i+1	Kile_1.svg	Dart_1.svg

Extension to a tiling of the plane

Repeated generations of deflation produce a tilings 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. Repeating the process an infinite number of times covers the entire plane.

Examples

These are four examples 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.

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

Decagonal covering

In 1996 German mathematician Petra Gummelt demonstrated that a covering equivalent to the Penrose tiling can be built by covering the plane with a single decagonal tile, if two kinds of overlap are allowed. This novel approach is called 'covering' to distinguish it from non-overlapping 'tiling'. The decagonal tile is decorated with colored patches and the covering rule allows only the overlap of these decorations.

Decomposing the decagonal tile into kites and darts transform the aperiodic covering into a Penrose tiling. If a fat 'T' rhombus is inscribed into each decagon, that part of the Penrose tiling corresponding to these shapes is obtained, while places for thin tiles are left unoccupied.
The covering method is taken to be a realistic model for the growth of
quasicrystals. Different atomic clusters 'share' the fragments from
which the aperiodic structure is built. The analogy with crystals constructed from a unit cell is restored when the overlapping decagons are seen as quasi-unit cells.

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 [1]

★ the distances between repeated finite motifs in the tiling grow as Fibonacci numbers when the size of the motif increases

★ the distribution of oscillation frequencies in a Penrose tiling shows bands and gaps whose widths are in proportions expressed by φ.^[5]

★ the substitution scheme $T\; o\; 2T+t;\; t\; o\; T+t$ introduces φ as a scaling factor;its matrix is the square of the Fibonacci substitution matrix; 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 (=φ²) and 2-φ (=1/φ²)

Related tilings in art

Similar tiles have been used in art, although they generally have not been applied with matching rules that enforce aperiodicity.
A similarity with some decorative patterns used in the Middle East has been frequently noted ^{[6] [7]} and in February 2007 a paper by Steinhardt and Lu offered evidence that a Penrose tiling underlies some examples of medieval Islamic art.^[8] Roger Penrose acknowledges inspiration from the work of Johannes Kepler. In 1970 the ''Penrose rhombuses'' were independently investigated in artwork by Drop City artist, Clark Richert.
The picture at right shows a variant rhombus tiling, built with rhombuses similar to those used in the Penrose tiling, but with different matching rules. 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'' → 3''T'' + ''t'', ''t'' → ''T'' + 2''t'', but not by de Bruijn's cut-and-project method.^[9]

Trivia

Pentaplex Ltd., a company in Yorkshire, England controlled by Penrose, owns the licensing rights to Penrose tilings.^[10] 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.[2]

Art historian Martin Kemp has commented a contemporary decoration which used Penrose tiles and observed that Albert Durer has sketched similar motifs of a rhombus tiling^[11]

References and notes

1. Penrose R., Bull. Inst. Maths. Appl. 10 (1974) 266
2. P. Gummelt, Geometriae Dedicata 62, 1 (1996); H.-C. Jeong and P.J. Steinhardt, Phys. Rev. B55, 3520 (1997)
3. Symmetries of Quasicrystals, Charles Radin, , , Journal of Statistical Physics, 1999
4. Statements to the contrary appear are widely published; a counterexample is
5. Maynard J.D., Rev. Mod. Phys. 73(2001)401
6. Minimal chaos, stochastic web and structures with 'quasicrystal' type symmetry (in Russian), Zaslavskiy G. et al, , , Uspekhi Fizicheskih Nauk, 1988
7. E. Makovicky (1992), 800-year-old pentagonal tiling from Maragha, Iran, and the new varieties of aperiodic tiling it inspired. In: I. Hargittai, editor: Fivefold Symmetry, pp.67-86. World Scientific, Singapore-London
8. Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture, Peter J. Lu and Paul J. Steinhardt, , , Science, 2007
9. A simple example of a non-Pisot tiling with five-fold symmetry, C. Godrèche and F. Lançon, , , Journal de Physique I, 1992 , also described at http://tilings.math.uni-bielefeld.de/tilings/substitution_rules/binary
10. Penrose, Roger, "Set of tiles for covering a surface," patent issued January 9, 1979 (expired)
11. Science in culture: A trick of the tiles, Kemp, Martin, , , Nature, 2005


★ Penrose, Roger. (1989) ''The Emperor's New Mind''. ISBN 0-19-851973-7

★ Gardner, Martin. "Penrose Tiles", chapter 7 in his book ''The Colossal Book of Mathematics''. ISBN 0-393-02023-1

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

★ A Penrose tiling features prominently in the painting Santa Fe Ribbon by American artist Connie Simon.