The Penrose tilings are aperiodic tilings.

In geometry, an aperiodic tiling is a tiling which never repeats itself, by a (finite) set of prototiles not admitting any tiling that does repeat itself. A shifted copy of such a tiling matches only locally with its original. The set of prototiles may have matching conditions or other local rules restricting how they may be fitted together; those local rules must force global aperiodicity. The best known examples of aperiodic tilings are the Penrose tilings.^{[1] [2]} Aperiodic tilings, typically quasiperiodic, can be obtained by different methods using grids, projections, substitutions or colorings. 'Weakly disordered' systems related to aperiodic tilings appear in physics as quasicrystals. Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... A Penrose tiling A Penrose tiling is an aperiodic tiling of the plane discovered by Roger Penrose in 1973. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... A tessellated plane seen in street pavement. ... In a given set S={A} of shapes (e. ... A Penrose tiling A Penrose tiling is an aperiodic tiling of the plane discovered by Roger Penrose in 1973. ... See Penrose tiling for a mathematical viewpoint. ... Quasicrystals are aperiodic structures which produce diffraction. ...

Terminology

A tiling in Euclidean n-space is said to be: periodic, if it admits n linearly independent translations as symmetries; subperiodic, if it admits k linearly independent translations where 0 < k < n; and nonperiodic, if it admits no translations as symmetries. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...

A set of prototiles is said to be aperiodic, if it can tile the whole space, but any tiling by that set (including tilings by a proper subset) is nonperiodic. A tiling by an aperiodic protoset is said to be an aperiodic tiling.^{[3] [4]} A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X &#8838; Y; Y is a superset of (or includes) X; Y...

History

The second part of Hilbert's eighteenth problem asked for a single polyhedron tiling Euclidean 3-space but such that no tiling by it is isohedral. The problem as stated was solved by Reinhardt in 1928, but aperiodic tilings have been considered as a natural extension.^[5] Hilberts eighteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. ... In geometry, a polyhedron is isohedral or face-transitive when all its faces are the same. ...

The specific question of aperiodic tiling first arose in 1961, when logician Hao Wang tried to determine whether the tiling problem was decidable: i.e. whether there exists an algorithm for deciding if a given finite set of prototiles admits a tiling of the plane. Wang was able to show that such an algorithm exists if it could be shown that every finite set of prototiles that admits a tiling of the plane also tiles it periodically. Wang Hao (Chinese: 王浩; pinyin: ; 1921 – 1995) was a Chinese-American logician, philosopher and mathematician. ...

The above Wang tiles will yield only aperiodic tilings of the plane.

Hence, when in 1966 Robert Berger demonstrated that the tiling problem is in fact not decidable,^[6] it followed that there must exist an aperiodic set of prototiles. The first such set, presented by Berger and used in his proof of undecidability, consisted of 20,426 Wang tiles. Berger reduced his set to size 104, and Hans Läuchli found an aperiodic set of 40 Wang tiles.^[7] The first compact aperiodic set, of six tiles, was discovered by Raphael M. Robinson in 1971.^[8] Roger Penrose discovered three more sets in 1973 and 1974, reducing the number of tiles needed to two, and Robert Ammann discovered several new sets in 1977. Image File history File links Wang_tiles. ... Image File history File links Wang_tiles. ... Wang tiles (or Wang dominoes), first proposed by Hao Wang in 1961, are equal-sized squares with a color on each edge which give rise to a simple undecidable decision problem. ... 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. ... Raphael Mitchel Robinson (November 2, 1911, National City California - January 27, 1995. ... 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. ... 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. ...

In 1988, Peter Schmitt discovered a single aperiodic prototile in 3-dimensional Euclidean space. While no tiling by this prototile admits a translation as a symmetry, it has tilings with a screw symmetry, the combination of a translation and a rotation through an irrational multiple of π. This was subsequently extended by John Horton Conway and Danzer to a convex aperiodic prototile, the Schmitt-Conway-Danzer tile. Because of the screw axis symmetry, this resulted in a reevaluation of the requirements for periodicity.^[9] Chaim Goodman-Strauss suggested that a protoset be considered strongly aperiodic if it admits no tiling with an infinite cyclic group of symmetries, and that other aperiodic protosets (such as the SCD tile) be called weakly aperiodic.^[10] In crystallography, a screw axis is a symmetry operation describing how a combination of rotation about an axis and a translation parallel to that axis leaves a crystal unchanged. ... John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... Look up Convex set in Wiktionary, the free dictionary. ... In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ...

In 1996 Petra Gummelt showed that a single marked decagonal tile with two kinds of overlapping allowed can force aperiodicity;^[11] this overlapping goes beyond the normal notion of tiling. The existence of an aperiodic protoset consisting of just one tile in the Euclidean plane, with no overlapping allowed, or of a strongly aperiodic protoset consisting of just one tile in any dimension, is an unsolved problem. In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

Constructions

Random tilings are a trivial example of nonperiodic tilings, but true aperiodic tilings are harder to find. If the columns in a square grid are colored in black or white according to the rule of some aperiodic sequence (e.g. Thue-Morse or Binary Fibonacci), the obtained tiling will be aperiodic only in one direction, conventionally the horizontal. Using different colored aperiodic sequences in different directions will produce a fully nonperiodic tiling. Aperiodic sets may be found that force some such nonperiodic tiling structures through the use of matching rules. In mathematics and its applications, the Thue-Morse sequence, or Prouhet-Thue-Morse sequence, is a certain binary sequence whose initial segments alternate (in a certain sense). ... Leonardo of Pisa (1170s or 1180s – 1250), also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some the most talented mathematician of the Middle Ages. ...

A sample of a Binary Fibonacci tiling

Instead of colors, an aperiodic grid may be constructed using sets of parallel lines with spacings that appear in aperiodic sequences in two different directions. This method is easily extended to higher dimensions and more complicated sets of lines. Such grids are also known as Ammann bars, named after Robert Ammann, who observed that they may be generated by suitable decorations of the prototiles. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... 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. ... In a given set S={A} of shapes (e. ...

Substitutions

Main articles: Substitution tiling and L-system

Many aperiodic sequences can be obtained from substitutions and the corresponding grids and tilings retain this property. The Ammann bars for a Penrose tiling form Fibonacci sequences which are generated by substitutions and also the Penrose tiling can be constructed by substituting smaller copies of its tiles according to the rules: T − > 2T + t and t − > T + t, whereT and t denote its two types of tiles. Similar rules are known for the Ammann-Beenker tiling. Goodman-Strauss showed that local matching rules can be found to force any substitution tiling structure, subject to some mild conditions.^[12] A tile-substitution is a simple way to generate highly ordered tilings without any translational symmetry, in other words: aperiodic tilings. ... See L-system for information on Lindenmayer systems. ... A Penrose tiling A Penrose tiling is an aperiodic tiling of the plane discovered by Roger Penrose in 1973. ... Leonardo of Pisa (1170s or 1180s – 1250), also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some the most talented mathematician of the Middle Ages. ... A Penrose tiling A Penrose tiling is an aperiodic tiling of the plane discovered by Roger Penrose in 1973. ... 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. ...

Cut and Project method

Aperiodic tiling are also obtained by projection of higher dimensional structures into spaces with lower dimensionality. Most famously the Penrose tiling is obtained as a projection of a five dimensional cubic lattice into a suitably chosen plane. This approach, which offers an interesting perspective in the understanding of aperiodic tilings, has been substantially developed after the pioneering work of de Bruijn. A Dutch mathematician, especially noted for the invention of the de Bruijn Sequence. External links About the de Bruijn sequence ...

Symmetry

Main article: Symmetry

A notable feature of many aperiodic tilings is that they exhibit unusual symmetries. This feature guarantees that they are aperiodic but it is not mandatory. Simple aperiodic grids form tilings with only the most common symmetries. By definition all aperiodic tilings lack translational symmetry. Their symmetry is rotational and it includes cases which were considered impossible by classical crystallography. The term 'statistical symmetry' is also used for weakly disordered systems. The Penrose tiling and the Ammann-Beenker tilings are constructed from rhombuses which have just a two- or four- fold symmetry, but the overall patterns have a fivefold and eightfold symmetries. Sphere symmetry group o. ... Crystallography (from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein = write) is the experimental science of determining the arrangement of atoms in solids. ... A Penrose tiling A Penrose tiling is an aperiodic tiling of the plane discovered by Roger Penrose in 1973. ...

Physics of aperiodic tilings

Main article: Quasicrystal

Aperiodic tilings were considered as mathematical artefacts until 1984, when physicist Dan Shechtman announced the discovery of a phase of an aluminium-manganese alloy which produced a sharp diffractogram with a unambiguous fivefold symmetry, so it had to be a crystalline substance with icosahedral symmetry. In 1975 Robert Ammann had already extended the Penrose construction a three dimensional icosahedral equivalent. In such cases the term 'tiling' is taken to mean 'filling the space'. Photonic devices are currently built as aperiodical sequences of different layers, being thus aperiodic in one direction and periodic in the other two. Quasicrytal structures of Cd-Te appear to consist of atomic layers in which the atoms are arranged in a planar aperiodic pattern. Sometimes an energetical minimum or a maximum of entropy occur for such aperiodic structures. Steinhardt has shown that Gummelt's overlapping decagons allow the application of an extremal principle and thus provide the link between the mathematics of aperiodic tiling and the structure of quasicrystals^[13] . Faraday waves have been observed to form large patches of aperiodic patterns ^[14]. The physics of this discovery has revived the interest in incommensurate structures and frequencies suggesting to link aperiodic tilings with interference phenomena ^[15]. . Quasicrystals are aperiodic structures which produce diffraction. ... Dan Shechtman is the Philip Tobias Professor of Materials Science at the Israel Institute of Technology. ... 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. ... Faraday waves are nonlinear standing waves that appear on liquids enclosed by a vibrating receptacle. ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ...

Approach to aperiodic tilings in medieval Islamic architecture

In 2007, Peter J. Lu of Harvard University and Professor Paul J. Steinhardt of Princeton University announced a finding that some medieval Islamic architecture made use of tilings based on standardized girih tiles in the form of decagons, pentagons, diamonds, bow ties, and hexagons. These could be used to construct nonperiodic tilings with fivefold or tenfold rotational symmetry. This finding was supported both by analysis of patterns on surviving structures, and by examination of 15th century Persian scrolls. If correct, it would indicate that Islamic architects came close to discovering aperiodic tilings some five centuries before they were discovered by Western mathematicians. Lu and Steinhardt^[16] are careful to observe that use of aperiodic tilings does not necessarily indicate mathematical understanding of their structure; in particular that "we have no evidence that they ever developed the alternative matching-rule approach". Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era. ... Peter James Lu (born 1978 in Cleveland (Ohio), USA) is a physics student (PhD expected in 2007) at Harvard University, Cambridge Massachusetts. ... Harvard University (incorporated as The President and Fellows of Harvard College) is a private university in Cambridge, Massachusetts, USA and a member of the Ivy League. ... Paul J. Steinhardt is the Albert Einstein Professor of Science at Princeton University and a professor of theoretical physics. ... Princeton University is a private coeducational research university located in Princeton, New Jersey, in the United States of America. ... Girih tiles are a set of five tiles that were used in the creation of tiling patterns for decoration of buildings in Islamic architecture. ... (14th century - 15th century - 16th century - other centuries) As a means of recording the passage of time, the 15th century was that century which lasted from 1401 to 1500. ... This article does not cite any references or sources. ...

References

1. ^ Gardner, Martin (January 1977). "Mathematical Games". Scientific American 236: 111-119. 
2. ^ Gardner, Martin (1988). Penrose Tiles to Trapdoor Ciphers. W H Freeman & Co. ISBN 0-7167-1987-8. 
3. ^ Senechal, Marjorie (1995 (corrected paperback edition, 1996)). Quasicrystals and geometry. Cambridge University Press. ISBN 0-521-57541-9. 
4. ^ Grünbaum, Branko; Geoffrey C. Shephard (1986). Tilings and Patterns. W.H. Freeman & Company. ISBN 0-7167-1194-X. 
5. ^ Senechal, pp 22-24.
6. ^ Berger, Robert (1966). "The undecidability of the domino problem". Memoirs of the American Mathematical Society (66): 1-72. 
7. ^ Grünbaum and Shephard, section 11.1.
8. ^ Robinson, Raphael M. (1971). "Undecidability and Nonperiodicity for Tilings of the Plane". Inventiones Mathematicae 12: 177-209. 
9. ^ Radin, Charles (1995). "Aperiodic tilings in higher dimensions" (fee required). Proceedings of the American Mathematical Society 123 (11): 3543-3548. 
10. ^ Goodman-Strauss, Chaim (2000-01-10). Open Questions in Tiling (PDF). Retrieved on 2007-03-24.
11. ^ Gummelt, Petra (1996). "Penrose Tilings as Coverings of Congruent Decagons". Geometriae Dedicata 62 (1): 1-17. 
12. ^ Goodman-Strauss, Chaim (1998). "Matching rules and substitution tilings". Annals of Mathematics 147 (1): 181-223. 
13. ^ Steinhardt, Paul J.. [http://wwwphy.princeton.edu/~steinh/quasi/ A New Paradigm for the Structure of Quasicrystals]. Retrieved on 2007-03-26.
14. ^ W. S. Edwards and S. Fauve, Parametrically excited quasicrystalline surface waves, Phys. Rev. E 47, (1993)R788 - R791
15. ^ Levy J-C. S., Mercier D., Stable quasicrystals, Acta Phys. Superficium 8(2006)115
16. ^ Peter J. Lu and Paul J. Steinhardt (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture". Science 315: 1106-1110. 

Martin Gardner (b. ... Martin Gardner (b. ... The headquarters of the Cambridge University Press, in Trumpington Street, Cambridge. ... Branko Grünbaum is a mathematician who works mainly in geometry and is considered a founder of discrete geometry. ... Inventiones Mathematicae, often just referred to as Inventiones, is a mathematical journal published monthly by Springer Berlin/Heidelberg. ... Proceedings of the AMS, October 2005 issue. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era. ... is the 83rd day of the year (84th in leap years) in the Gregorian calendar. ... The Annals of Mathematics (ISSN 0003-486X), often just called Annals, is a bimonthly mathematics research journal published by Princeton University and the Institute for Advanced Study. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era. ... March 26 is the 85th day of the year (86th in leap years) in the Gregorian calendar. ... Science is the journal of the American Association for the Advancement of Science (AAAS). ...

External links

• The Geometry Junkyard
• Aperiodic Tilings