Abstract
The paper focuses on two groups of geometrical problems closely related to formations of crystal/quasi-crystal structures, and fullerenes. In section one we discuss a minimum radius of local identity that guarantee that a Delone set is a point regular set, and prove the local theorem for crystals. New results related to locally rigid packings are discussed in section two. One of the goals of the paper is to establish some internal (mathematically), and external (applications to material science), connections between research agendas of various studies in material sciences, and two classes of geometric problem related to Delone sets and packings.
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