On the origin of crystallinity: a lower bound for the regularity radius of Delone sets

The mathematical conditions for the origin of long-range order or crystallinity in ideal crystals are one of the very fundamental problems of modern crystallography. It is widely believed that the (global) regularity of crystals is a consequence of `local order', in particular the repetition of local fragments, but the exact mathematical theory of this phenomenon is poorly known. In particular, most mathematical models for quasicrystals, for example Penrose tiling, have repetitive local fragments, but are not (globally) regular. The universal abstract models of any atomic arrangements are Delone sets, which are uniformly distributed discrete point sets in Euclidean d space. An ideal crystal is a regular or multi-regular system, that is, a Delone set, which is the orbit of a single point or finitely many points under a crystallographic group of isometries. The local theory of regular or multi-regular systems aims at finding sufficient local conditions for a Delone set X to be a regular or multi-regular system. One of the main goals is to estimate the regularity radius \hat{\rho}_d for Delone sets X in terms of the radius R of the largest `empty ball' for X. The celebrated `local criterion for regular systems' provides an upper bound for \hat{\rho_d} for any d. Better upper bounds are known for d ≤ 3. The present article establishes the lower bound \hat{\rho_d}\geq 2dR for all d, which is linear in d. The best previously known lower bound had been \hat{\rho}_d\geq 4R for d ≥ 2. The proof of the new lower bound is accomplished through explicit constructions of Delone sets with mutually equivalent (2dR - ℇ)-clusters, which are not regular systems. The two- and three-dimensional constructions are illustrated by examples. In addition to its fundamental importance, the obtained result is also relevant for the understanding of geometrical conditions of the formation of ordered and disordered arrangements in polytypic materials.