Fixed-Parameter Extrapolation and Aperiodic Order: Open Problems

Abstract

For a number of years we have been investigating the geometric and algebraic properties of a family of discrete sets of points in Euclidean space generated by a simple binary operation: pairwise affine combination by a xed parameter, which we call xed-parameter extrapolation. By varying the parameter and the set of initial points, a large variety of point sets emerge. To our surprise, many of these sets display aperiodic order and share properties of so-called \quasicrystals" or \quasilattices." Such sets display some ordered crystal-like properties (e.g., generation by a regular set of local rules, such as a nite set of tiles, and possessing a kind of repetitivity), but are \aperiodic" in the sense that they have no translational symmetry. The most famous of such systems are Penrose's aperiodic tilings of the plane [16]. Mathematically, a widely accepted way of capturing the idea of aperiodic order is via the notion of Meyer sets, which we de ne later. Our goal is to classify the sets generated by xed-parameter extrapolation in terms of a number of these properties, including but not limited to aperiodicity, uniform discreteness, and relative density. We also seek to determine exactly which parameter values lead to which types of sets.