Penrose tiling

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

Background

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

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

All angles are therefore multiples of 36 °. Both tiles are related to the golden section, in the thick diamond the long diagonal has length, is the length of the short diagonal of the thin rhombus. The area ratio of the two diamonds is also, as is the number ratio of the total tiles used in the tiling.

During assembly of the tiles must be noted that this may not be joined together as required. The application of bulges and indentations on the tiles (like puzzle pieces ) can ensure that only correct assembly, alternatively, color pattern, which may only be combined appropriately. For aesthetic reasons, the tiling is usually depicted with straight edges. The often wrongly called parallelogram, which prohibits that two tiles are combined so that they together form a parallelogram is, at least not enough to prevent a periodic tiling.

If one considers this rule, we obtain many ( even infinitely many ) different tilings of the plane, ie surpluses " without holes", which can be continued infinitely. The pictures show two examples, which moreover have a five -fold rotational symmetry and five mirror symmetries. There is in these samples but no translational symmetry, i.e., the patterns are aperiodic. However, one can show that every finite cut of such a pattern infinitely often finds (and even in any other consisting of the same tiles Penrose tiling ). One can therefore based on a finite cutout never determine which pattern is present.

The fact that it is possible to tile the plane with an aperiodic tiling, was first in 1966 (not 1964) by Robert Berger proved shortly afterwards was able to give a concrete example with 20426 different tiles. As a result, smaller sets of tiles were always given for such aperiodic tiling, Penrose was finally able to reduce the number of tiles on two. Besides the aforementioned rhombic tiles, there are another pair of tiles, which provides an aperiodic tiling, called " dragons" and "arrow". For all Penroseparkettierungen with dragon arrows and the distance between two same partial pattern is less than ( conjecture of Penrose and Ammann, so far unproven ), the diameter of the sub- pattern. This means that the same pattern in each part tiling not only infinitely often contain, but also " close together ".

Whether a single tile shape exists, can only be realized with the aperiodic tilings, is unknown.

Aperiodic tilings were first considered only as an interesting mathematical structure, but in the meantime materials were found, in which the atoms are arranged as in Penrose tiles. These materials may not form periodic crystals, quasicrystals, however, because of the repetitive pattern of the "almost".

Islamic precursor

On a trip through Uzbekistan in 2007 were Peter Lu of Harvard University, who works in the field of quasicrystals, on a building tile ornaments on, which reminded him of Penrose tilings. With the sighting of many photographs, he came in Darb -e Imam shrine in Isfahan, Iran, on works from the 15th century, which seem to anticipate the results of Penrose.

This Kachelornamentik has clearly shown its beginnings in the sense of not repeating infinite tilings as early as the 12th century ( as Makovicky showed on Gonbad -e- Kabud in Maragha 1992 ), a set of five easy to constructing basic shapes, the so-called girih - tiles, was used. Unlike, for example, for Celtic knot, where the construction of the pattern is understandable, are for the methods for structural pattern generation in this case, but no clues. From the 15th century the remarks was further supplemented by the property of self-similarity, as they are known among other things for fractals.

At present there are still no known finds of templates that represent the aforementioned basic forms. On one hand, they would have been well able to recognize difficult than those in previous years of archaeological research, on the other hand there is also the possibility that these were not durable enough or may have been destroyed even after the work. The use of such a system has at least, that the application has understood the same and dominated and used specifically for the ornamentation work. Whether this is an indication of a deeper, mathematical understanding of the parties in the area of ​​structures and patterns that is currently open.