The Local Theory for Regular Systems in the Context of t-Bonded Sets

Authors

Mikhail M. Bouniaev, The University of Texas Rio Grande Valley
Nikolay Dolbilin

Document Type

Article

Publication Date

5-2018

Abstract

The main goal of the local theory for crystals developed in the last quarter of the 20th Century by a geometry group of Delone (Delaunay) at the Steklov Mathematical Institute is to find and prove the correct statements rigorously explaining why the crystalline structure follows from the pair-wise identity of local arrangements around each atom. Originally, the local theory for regular and multiregular systems was developed with the assumption that all point sets under consideration are (r,R)" role="presentation">(r,R) -systems or, in other words, Delone sets of type (r,R)" role="presentation">(r,R) in d-dimensional Euclidean space. In this paper, we will review the recent results of the local theory for a wider class of point sets compared with the Delone sets. We call them t-bonded sets. This theory, in particular, might provide new insight into the case for which the atomic structure of matter is a Delone set of a “microporous” character, i.e., a set that contains relatively large cavities free from points of the set.

Comments

Original published version available at https://doi.org/10.3390/sym10050159

Recommended Citation

Bouniaev, Mikhail, and Nikolay Dolbilin. 2018. “The Local Theory for Regular Systems in the Context of T-Bonded Sets.” Symmetry 10 (5): 159. https://doi.org/10.3390/sym10050159.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Publication Title

Symmetry

DOI

10.3390/sym10050159