Local Groups in Delone Sets

TL;DR

This paper proves that in any 3D Delone set, points with local symmetry groups of order up to 6 form a Delone set, advancing understanding of local symmetries and their global implications in crystallography.

Contribution

It provides the first rigorous proof that the subset of points with local symmetry axes of order ≤6 in a Delone set is itself a Delone set, and introduces conjectures on the prevalence of crystallographic axes.

Findings

Subset of points with local symmetry axes of order ≤6 is a Delone set.

Main theorem implies boundedness of local groups in certain Delone sets.

Supports crystallographic restrictions on global symmetries.

Abstract

In the paper, we prove that in an arbitrary Delone set $X$ in $3D$ space, the subset $X_6$ of all points from $X$ at which local groups have axes of the order not greater than 6 is also a Delone set. Here, under the local group at point $x\in X$ is meant the symmetry group $S_x(2R)$ of the cluster $C_x(2R)$ of $x$ with radius $2R$, where $R$ (according to Delone's theory of the 'empty sphere') is the radius of the largest 'empty' ball, that is, the largest ball free of points of $X$. The main result seems to be the first rigorously proved statement on absolutely generic Delone sets which implies substantial statements for Delone sets with strong crystallographic restrictions. For instance, an important observation of Shtogrin on the boundedness of local groups in Delone sets with equivalent $2R$-clusters immediately follows from the main theorem. In the paper, the 'crystalline kernel conjecture' (Conjecture 1) and its two weaker versions (Conjectures 2 and 3) are suggested. According to Conjecture 1, in a quite arbitrary Delone set, points with locally crystallographic axes (of order 2,3,4, or 6) only inevitably constitute an essential part of the set. These conjectures significantly generalize the famous statement of Crystallography on the impossibility of (global) 5-fold symmetry in a 3D lattice.