AmmannBeenker

Ammann–Beenker tiling, sometimes called the AmmannBeenker tiling or octagonal tiling, is a nonperiodic tiling of the plane with eightfold rotational symmetry. It is a canonical example in the study of quasicrystals and aperiodic order.

The standard tiling uses a pair of rhombic prototiles with equal edge lengths. The two tile types are decorated or marked to enforce local matching rules, which guarantee a nonperiodic arrangement while preserving eightfold symmetry at large scales.

Construction methods include inflation/deflation rules and projection techniques. The inflation factor for Ammann–Beenker tilings is 1 plus the square root of two (1+√2). One can also realize the tiling by the cut-and-project method from the four-dimensional integer lattice Z4, projecting onto a two-dimensional plane chosen to exhibit octagonal symmetry; an octagonal acceptance window selects which projected points are kept.

Key properties include aperiodicity, finite local complexity, and a hierarchical structure that reflects the inflation rules. Local patches recur in a nonrepeating, aperiodic fashion, and the tiling displays eightfold rotational symmetry in its global organization, though not in every finite patch.

Relation to broader mathematics and science is strong: the Ammann–Beenker tiling serves as a mathematical model for quasicrystalline order and as a testbed for algorithms in tiling generation, tiling theory, and symbolic dynamics.

See also: Penrose tiling; quasicrystal; projection method; Robinson triangles; inflation tilings.