Penrose tiling

A Penrose tiling is one of Roger Penrose and Robert Ammann discovered in 1973 and 1974, published family of so-called aperiodic tile patterns that a plane without gaps parkettieren without thereby a basic scheme periodically repeated can.

A Penrose tiling	The same tiling as opposite, but with highlighting of complete stars, shows increasing "chaos" when moving away from the center
The ratio of the golden ratio appears in both tiles ${\displaystyle \Phi }$	Tiles with bulges and notches as well as color patterns. Either of these two methods prevents periodic tiling from being possible.

Example of a tiling

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There are several different sets of Penrose tiles; the picture on the right shows a frequently chosen example. It consists of two diamonds that have the same side lengths but different corner angles:

• the first tile, the thin diamond , has corner angles of 36 ° and 144 °,
• the second tile, the thick diamond , has corner angles of 72 ° and 108 °.

So all angles are multiples of 36 °. Both tiles are related to the golden ratio . If the side length is set as 1, the long diagonal of the thick diamond has the length . The length of the short diagonal of the thin diamond is . The area ratio of the two diamonds is also the same , as is the number ratio of the tiles used in total for the parquet. ${\displaystyle \Phi =\left(1+{\sqrt {5}}\right)/2}$ ${\displaystyle \Phi }$ ${\displaystyle 1/\Phi }$ ${\displaystyle \Phi }$

When joining the tiles, it must be ensured that they cannot be joined together at will. Adding bulges and notches to the tiles (as with puzzle pieces) can only ensure correct assembly, alternatively also color samples that may only be assembled appropriately. For aesthetic reasons, the parquet is usually presented with straight edges. The often wrongly named parallelogram rule , which forbids two tiles to be put together so that they together form a parallelogram , is in any case not sufficient to prevent periodic tiling.

A tiling that violates the parallelogram rule.

If you observe this rule, you get many (even infinitely many) different tiling of the level, i.e. H. Coverings "without holes" that can be continued indefinitely. The pictures show two examples, which also have a fivefold rotational symmetry and five mirror symmetries. However, there is no translation symmetry in these patterns ; H. the patterns are aperiodic. However, it can be shown that every finite section of such a pattern can be found infinitely often (and even in every other Penrose tiling consisting of the same tiles). So you can never determine which pattern is present on the basis of a finite section.

The fact that it is possible to cover the level with aperiodic tiling was first proven in 1966 (o. 1964) by Robert Berger, who shortly afterwards was able to give a concrete example with 20426 different tiles. As a result, smaller and smaller sets of tiles were specified for such aperiodic tiling until Penrose was finally able to reduce the number of tiles to two. In addition to the rhombic tiles mentioned, there is another pair of tiles that provide an aperiodic tiling, called "dragon" and "arrow". In all Penrose parquets with kites and arrows, the distance between two identical partial patterns is less than (assumption by Ammann and Penrose, so far unproven), whereby the diameter of the partial pattern is. This means that the same partial patterns are not only contained in every tiling infinitely often, but also "close together". ${\displaystyle d\cdot (1/2+\Phi )}$ ${\displaystyle d}$

Whether a single tile shape ( "Einstein" ) exists, with which only aperiodic tiling can be realized, is still an open problem. The best approximate solution to date for such a tile was found in 2009 by the Australian Joan M. Taylor and published about it with Joshua Socolar. This tile is not connected or, in another version, three-dimensional.

Roger Penrose in the foyer of the Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University , the floor of which is laid out with a Penrose pattern (photo March 2010).

Traditionally, the Penrose parquet consists of a thick and a thin diamond. If one goes into the third dimension, however, one can interpret the thin diamond as a perspective distortion of the thick diamond. The resulting surface is called the Wieringa roof . Due to the similarities with the three-dimensional quasicrystals , one recognizes rhombic triacontahedron and rhombic hexacontahedron in the tiling.

Aperiodic tiling was only considered an interesting mathematical structure at first, but materials have since been found in which the atoms are arranged like in Penrose tiles. These materials cannot form periodic crystals, but quasicrystals because the patterns are "almost" repeated.

Historical precursors

During a trip through Uzbekistan in 2007, Peter Lu from Harvard University , who works in the field of quasicrystals, noticed tile ornaments on a building that reminded him of Penrose tiling. While viewing many photographs in the Darb-e-Imam Shrine in Isfahan , Iran, he came across works from the 15th century that seem to anticipate the results of Penrose.

This tile ornamentation clearly has its beginnings in the sense of non-repeating infinite tiling as early as the 12th century (as Emil Makovicky at Gonbad-e-Kabud in Maragha showed in 1992), with a set of five easy-to-construct basic forms, the so-called Girih -Tile , was used. Unlike z. B. for Celtic knots , in which the construction of the pattern can be traced, there are no indications for the methods for constructive pattern generation in this case. From the 15th century, the explanations were further supplemented by the property of self-similarity , as we know it from fractals , among other things .

There are currently no known finds of stencils that represent the basic shapes mentioned. On the one hand it would have been difficult to recognize them as such in earlier years of archaeological research, on the other hand there is also the possibility that they were not durable enough or that they were even destroyed after the work. The use of such a system at least proves that the application of it was understood and mastered and was used specifically for the ornamentation work. To what extent this is an indication of a more in-depth, mathematical understanding of those involved in the area of ​​structures and patterns is currently open.

literature

• Computer thinking - the emperor's new clothes or the debate about artificial intelligence, consciousness and the laws of nature. With a foreword by Martin Gardner and a foreword to the German edition by Dieter Wandschneider . Heidelberg 1991. ISBN 3-8274-1332-X .
• The Emperor's New Mind. Penguin Books, New York 1991. ISBN 0-14-014534-6 (English original edition).
• Patent US4133152 : Set of tiles for covering a surface. Published January 9, 1979 , inventor: Roger Penrose. ‌
• Martin Gardner : Penrose Tiles. Chapter 7, in: The Colossal Book of Mathematics. Norton, New York NY 2001, ISBN 0-393-02023-1 .
• Roger Penrose: Pentaplexity - A Class of Non-Periodic Tilings of the Plane. In: The Mathematical Intelligencer. Volume 2, No. 1, Springer, New York 1979, ISSN  0343-6993 , pp. 32-37 (reprinted from Eureka No. 39).
• Roger Penrose: The role of aesthetics in pure and applied mathematical research. In: Bulletin of the Institute of Mathematics and Its Applications (Bull. Inst. Math. Appl.). Southend-on-Sea Volume 10, 1974, ISSN  0146-3942 , pp. 266-271.
• Christoph Pöppe: Quasicrystals in a new light. In: Spectrum of Science . No. 7, 1999, ISSN  0170-2971 , pp. 14-17.
• P. Stephens, A. Goldman: The structure of the quasicrystals. In: Spectrum of Science. No. 6, 1991, ISSN  0170-2971 , pp. 48-56.
• Martin Gardner: Mathematical gimmicks. In: Spectrum of Science. Heidelberg 11/1979, ISSN  0170-2971 , pp. 22-33.
• Albrecht Beutelspacher / Bernhard Petri: The golden ratio. Spektrum Akademischer Verlag, Heidelberg, Berlin, Oxford, 2nd edition 1996, ISBN 3-86025-404-9 , pp. 80ff.
• Branko Grünbaum, GC Shephard: Tilings and Patterns WH Freeman and Company, New York, 1987, ISBN 0-7167-1193-1

On patterns similar to the Penrose tiling in Islamic ornaments:

• Peter Lu and Paul Steinhardt : Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture. In: Science . Volume 315, Washington 2007, ISSN  0036-8075 , pp. 1106-1110.
• Emil Makovicky: 800-Year-Old Pentagonal Tiling From Maragha, Iran, and the New Varieties of Aperiodic Tiling it Inspired. In: István Hargittai (ed.): Fivefold Symmetry. World Scientific, Singapore / River Edge NJ 1992, ISBN 981-02-0600-3 , pp. 67-86.
• Peter Cromwell The Search for Quasi-Periodicity in Islamic 5-fold Ornament , Mathematical Intelligencer, Vol. 31, No. 1, 2009.

See also

• Parquet with pentagons

Web links

Commons : Penrose tiling  - collection of images, videos, and audio files

• Free Windows program for creating Penrose tiling . Developed by Stephen Collins (JKS Software) in collaboration with the Universities of York (UK) and Tsuka (Japan).
• Two theories about the formation of quasicrystals with similarities to Penrose tiling at www.sciencenews.org
• NUMERATOR mosque builders were 500 years ahead of western mathematicians at Spiegel Online

supporting documents

1. ^ Grünbaum, Shephard Tilings and Patterns , Freeman 1981, p. 563
2. ↑ Socolar's and Taylor's aperiodic tile
3. ↑ Socolar, Taylor, An aperiodic hexagonal tile, Journal of Combinatorial Theory A, Volume 118, 2011, pp 2207-2231
4. ↑ Activities with golden rhombi .
5. ^ Penrose Tilings and Wieringa Roofs .
6. ↑ The Story of Spikey .
7. ↑ Celtic knot ( Memento of the original from February 27, 2007 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.