On the Regularity Radius of Delone Sets in $${\mathbb {R}}^3$$

Abstract

Delone sets are discrete point sets X in R d characterized by parameters (r, R), where (usually) 2r is the smallest inter-point distance of X, and R is the radius of a largest "empty ball" that can be inserted into the interstices of X. The regularity radius ρd is defined as the smallest positive number ρ such that each Delone set with congruent clusters of radius ρ is a regular system, that is, a point orbit under a crystallographic group. We discuss two conjectures on the growth behavior of the regularity radius. Our "Weak Conjecture" states that ρd = O(d 2 log d)R as d → ∞, independent of r. This is verified in the paper for two important subfamilies of Delone sets: those with full-dimensional clusters of radius 2r and those with full-dimensional sets of d-reachable points. We also offer support for the plausibility of a "Strong Conjecture", stating that ρd = O(d log d)R as d → ∞, independent of r.

Bounds for the Regularity Radius of Delone Sets

“…The next theorem is a reformulation of [16, Prop. 2.1] (see also [19]) and provides an important tool for bounding the regularity radius ρd in terms of the dimension d and the radius R of the largest empty ball. It gives a necessary condition for the regularity of a Delone set X in terms of the total number of prime factors of the order of its 2R-cluster groups.…”

“…For d = 2, 3 it is known that ρ2 = 4R [14,15] and 6R ≤ ρ3 ≤ 10R [6,19]. Moreover, for d = 3, if N (2R) = 1, then the order of any axis in S x (2R) does not exceed 6 [24].…”

VoroNoodles: Topological Interlocking with Helical Layered 2‐Honeycombs

“…The standard mathematical model of an ideal crystal also involves a specific type of Delone sets, called symmetric (or crystal) sets, [56,57] which are invariant with respect to a crystallgraphic group. [22,56] The important property of symmetric Delone sets is that the Voronoi decomposition [58] of symmetric Delone sets in 3D can be used to obtain 3-honeycombs and space-filling congruent polyhedra.…”

“…This property is useful since we can formally ensure each layer to be a symmetric Delone set. [25,57] This is based on Dolbilin's result [56,57] that demonstrates the following; if n-number of symmetric Delone sets represent the same crystallographic (in 2D wallpaper) group, their union also represents the same wallpaper group with a crystallographic orbit of n-number points. Using this result, it is possible to be arbitrarily close to any given higher-dimensional shape, such as a planar curve, which can result from 2D crystals.…”

Substitution tilings with transcendental inflation factor

“…Delone sets arise in many topics related to mathematical crystallography, to distribution of points in Euclidean spaces, and in applications as well-spaced point sets. We refer to [6,7,11,14,15] and references therein for more detailed discussions of these sets and related properties. We want to emphasize their importance as point sets that can be used to model atomic structure of crystals or ordered matter.…”