Ammann–Beenker tiling

In geometry, an Ammann–Beenker tiling is a nonperiodic tiling generated by an aperiodic set of prototiles named after Robert Ammann, who first discovered the tilings in the 1970s, and after F. P. M. Beenker who discovered them independently and showed how to obtain them by the cut-and-project method.Because all tilings obtained with the tiles are non-periodic, Ammann–Beenker tilings are considered aperiodic tilings. They are one of the five sets of tilings discovered by Ammann and described in "Tilings and Patterns" [ B. Grünbaum and G.C. Shephard, "Tilings and Patterns", Freemann, NY 1986] .

The Ammann–Beenker tilings have many properties similar to the more famous Penrose tilings, most notably:
*They are nonperiodic, which means that they lack any translational symmetry.
*Any finite region in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. Thus, the infinite tilings all look similar to one another, if one looks only at finite patches.
*They are quasicrystalline: implemented as a physical structure an Ammann–Beenker tiling will produce Bragg diffraction; the diffractogram reveals both the underlying eightfold symmetry and the long-range order. This order reflects the fact that the tilings are organized, not through translational symmetry, but rather through a process sometimes called "deflation" or "inflation."

Various methods to construct the tilings have been proposed: matching rules, substitutions, cut and project schemes [ Beenker FPM, Algebric theory of non periodic tilings of the plane by two simple building blocks: a square and a rhombus, TH Report 82-WSK-04 (1982), Technische Hogeschool, Eindhoven ] and coverings [F. Ga¨hler, in Proceedings of the 6th International Conference on Quasicrystals, edited by S. Takeuchi and T. Fujiwara, WorldScientific, Singapore, 1998, p. 95.] [ [http://www.itap.physik.uni-stuttgart.de/~gaehler/papers/octcluster.pdf S. Ben Abraham and F. Gahler, Phys. Rev. B60(1999)860] ] . In 1987 Wang, Chen and Kuo announced the discovery of a quasicrystal with octagonal symmetry [ Wang N., Chen H. and Kuo K., Phys Rev Lett. 59(1987) 1010 ] .

Description of the tiles

The most common choice of tileset to produce the Ammann–Beenker tilings includes a rhombus with 45 and 135 degree angles (these rhombi are shown in white in the diagram at the top of the page) and a square (shown in red in the diagram above). The square may alternatively be divided into a pair of isosceles right triangles. (This is also done in the above diagram.) The matching rules or substitution relations for the square/triangle do not respect all of its symmetries, however.

Pell and silver ratio features

The Ammann–Beenker tilings are closely related to the silver ratio ($1\; +\; sqrt\; 2$) and the Pell numbers.

*the substitution scheme $R\; o\; RrR;\; r\; o\; R$ introduces the ratio as a scaling factor: its matrix is the Pell substitution matrix, and the series of words produced by the substitution have the property that the number of $r$s and $R$s are equal to successive Pell numbers.
*the eigenvalues of the substitution matrix are $1+sqrt\; 2$ and $1-sqrt\; 2$.
* One set of Conway worms, formed by the short and long diagonals of the rhombs, forms the above strings, with r as the short diagonal and R as the long diagonal. Therefore the Ammann bars also form Pell ordered grids. [cite journal|author = Socolar, J E S | title = Simple octagonal and dodecagonal quasicrystals | journal = Physics Review B | year = 1989 | volume = 39 | pages = 10519–10551 | MR0998533]

References and notes

External links

* [http://tilings.math.uni-bielefeld.de/substitution_rules/ammann_beenker Tilings Encyclopedia entry] .
* [http://www.quadibloc.com/math/oct01.htm Ammann–Beenker tiles on John Savard's website] .