Ammann–Beenker tiling

In geometry, an Ammann–Beenker tiling is a nonperiodic tiling which can be generated either by an aperiodic set of prototiles as done by Robert Ammann in the 1970s, or by the cut-and-project method as done independently by F. P. M. Beenker. They are one of the five sets of tilings discovered by Ammann and described in Tilings and patterns.

The Ammann–Beenker tilings have many properties similar to the more famous Penrose tilings:

• They are nonperiodic, which means that they lack any translational symmetry.
• Their non-periodicity is implied by their hierarchical structure: the tilings are substitution tilings arising from substitution rules for growing larger and larger patches. This substitution structure also implies that:
• Any finite region (patch) 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."
• All of this infinite global structure is forced through local matching rules on a pair of tiles, among the very simplest aperiodic sets of tiles ever found, Ammann's A5 set.

Various methods to describe the tilings have been proposed: matching rules, substitutions, cut and project schemes and coverings. In 1987 Wang, Chen and Kuo announced the discovery of a quasicrystal with octagonal symmetry.