Matching Rules and Quasiperiodicity: the Octagonal Tilings

Abstract

This lecture discusses one of the most important question raised by the discovery of quasicrystals: the onset of quasiperiodic order. In fact, one of the main problems about quasicrystals is to understand the simple possibility of a non periodic long range order, since no two atoms have exactly the same environment up to infinity. One possible solution to this problem is to consider that the order stems from privileged local configurations and is able to propagate throughout the structure. This point of view deals with the existence of local constraints which would enforce the quasiperiodic order: these are the so-called “local rules”, or “matching rules” in tiling language.