Local groups in Delone sets in the Euclidean space

A Delone (Delaunay) set is a uniformly discrete and relatively dense set of points located in space, and is a natural mathematical model of the set of atomic positions of any solid, whether it is crystalline, quasi-crystalline or amorphous. A Delone set has two positive parameters: $r$ is the packing radius and R is the covering radius. The value 2r can be interpreted as the minimum distance between points of the set. The covering radius $R$ is the radius of the biggest ‘empty’ ball, i.e. the radius of the biggest ball containing no points from the set. The central concept of this article is the so-called local group at a point of $X$ which is defined as a group of a cluster (neighborhoods) around the point of radius $2R$. This value $2R$ is notable because it is the minimum size of cluster that provides the finiteness of the cluster group at each point for any set X from the family of all Delone sets with the covering radius $R$. A few conjectures and theorems on the local groups for arbitrary Delone sets in the Euclidean plane and 3D space are discussed. Some of these statements significantly refine and generalize the famous Bravais theorem on the impossibility of fifth-order axes in $2D$ and $3D$ lattices. A complete proof is given that, in a Delone set $X$ in the $3D$ Euclidean space, the subset $\widetilde{X}$ of all points at which the local groups contain rotations of order at most 6 is also a Delone set with a certain covering radius $\widetilde{R}$, where $\widetilde{R} < 3R$ and R$$ is the covering radius for $X$.