Local groups in Delone sets: a conjecture and results

Abstract

In the framework of a new approach to the concept of local symmetry in arbitrary Delone sets we obtain new results for such sets without any restrictions. These results have important consequences for lattices and regular systems. A conjecture about the crystal kernel is stated, which generalises significantly the classical theorem on the non-existence of a five-fold symmetry in three-dimensional lattices. The following theorems related to the foundations of geometric crystallography are proved.