Geometric Models for Quasicrystals II. Local Rules Under Isometries

Abstract

The atomic structures of quasicrystalline materials exhibit long range order under translations. It is believed that such materials have atomic structures which approximately obey local rules restricting the location of nearby atoms. These local constraints are typically invariant under rotations, and it is of interest to establish conditions under which such local rules can nevertheless enforce order under translations in any structure that satisfies them. A set of local rules $ \cal L $ in $ {\frak R}^n $ is a finite collection of discrete sets {Y i } containing 0, each of which is contained in the ball of radius ρ around 0 in $ {\frak R}^n $ . A set X satisfies the local rules $ \cal L $ under isometries if the ρ -neighborhood of each $ {\bf x} \in X $ is isometric to an element of $ \cal L $ . This paper gives sufficient conditions on a set of local rules $ \cal L $ such that if X satisfies $ \cal L $ under isometries, then X has a weak long-range order under translations, in the sense that X is a Delone set of finite type. A set X is a Delone set of finite type if it is a Delone set whose interpoint distance set X-X is a discrete closed set. We show for each minimal Delone set of finite type X that there exists a set of local rules $ \cal L $ such that X satisfies $ \cal L $ under isometries and all other Y that satisfy $ \cal L $ under isometries are Delone sets of finite type. A set of perfect local rules (under isometries or under translations, respectively) is a set of local rules $ \cal L $ such that all structures X that satisfy $ \cal L $ are in the same local isomorphism class (under isometries or under translations, respectively). If a Delone set of finite type has a set of perfect local rules under translations, then it has a set of perfect local rules under isometries, and conversely.