Properties of the Ammann–Beenker Tiling and its Square Periodic Approximants

Abstract

Our understanding of physical properties of quasicrystals owes a great deal to studies of tight-binding models constructed on quasiperiodic tilings. Among the large number of possible quasiperiodic structures, two dimensional tilings are of particular importance – in their own right, but also for information regarding properties of three dimensional systems. We provide here a users manual for those wishing to construct and study physical properties of the 8-fold Ammann–Beenker quasicrystal, a good starting point for investigations of two dimensional quasiperiodic systems. This tiling has a relatively straightforward construction. Thus, geometrical properties such as the type and number of local environments can be readily found by simple analytical computations. Transformations of sites under discrete scale changes – called inflations and deflations – are easier to establish compared to the celebrated Penrose tiling, for example. We have aimed to describe the methodology with a minimum of technicalities but in sufficient detail so as to enable non-specialists to generate quasiperiodic tilings and periodic approximants, with or without disorder. The discussion of properties includes some relations not previously published, and examples with figures.