1. Delone sets in ℝ3: Regularity Conditions
- Author
-
N. P. Dolbilin
- Subjects
Statistics and Probability ,Discrete mathematics ,Euclidean space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Delone set ,01 natural sciences ,Identity (music) ,010305 fluids & plasmas ,Set (abstract data type) ,0103 physical sciences ,Homogeneous space ,Mathematics::Metric Geometry ,0101 mathematics ,Symmetry (geometry) ,Orbit (control theory) ,Link (knot theory) ,Mathematics - Abstract
A regular system is a Delone set in Euclidean space with a transitive group of symmetries or, in other words, the orbit of a crystallographic group. The local theory for regular systems, created by the geometric school of B. N. Delone, was aimed, in particular, to rigorously establish the “local-global-order” link, i.e., the link between the arrangement of a set around each of its points and symmetry/regularity of the set as a whole. The main result of this paper is a proof of the so-called 10R-theorem. This theorem asserts that identity of neighborhoods within a radius 10R of all points of a Delone set (in other words, an (r, R)-system) in 3D Euclidean space implies regularity of this set. The result was obtained and announced long ago independently by M. Shtogrin and the author of this paper. However, a detailed proof remains unpublished for many years. In this paper, we give a proof of the 10R-theorem. In the proof, we use some recent results of the author, which simplify the proof.
- Published
- 2020