Исследование асимптотической последовательности высот правильных симплексов при увеличении размерности евклидова пространства

Abstract

The problem of spherical packing and the problem of the distribution of points on a sphere are well-known unsolved mathematical problems. In particular, the problem of the distribution of points on a sphere is included in the well-known list of unsolved mathematical problems of Steve Smale. Accordingly, a significant interest for the study is both the direct search for solutions to problems and research in areas related to known problems. In particular, the problem of spherical dense packing and coating packing is directly related to the study of regular, irregular and layered lattices, problems of noise-resistant coding, synthesis and analysis of broadband signals, mathematical research of multidimensional spaces, crystallography, methods for solving differential equations and calculating n-dimensional integrals, and many other particular problems and entire fields of knowledge. The paper presents the results of studies of some properties of n-dimensional regular simplex in Euclidean space. The vectors forming the n-dimensional regular simplex define the fundamental basis of the hexagonal lattice. The subject of the study is some properties and regularities of a vector object forming a regular simplex in n-dimensional Euclidean space, the novelty of the results lies in a formulated and proven theorem, several formulas that allow recursive and non-recursive calculations of some characteristics of n-dimensional regular simplex and spherical hexagonal packages. The object of the study is an n-dimensional regular simplex. Main results a theorem on the asymptotic character of the sequence of heights of regular simplex with increasing dimension of space is formulated and proved, the consequences of the theorem and remarks supplementing the theorem are formulated. The conclusion of the article contains additions and numerical examples with comments for some research results. The practical significance lies in the proposed mathematical tools that contribute to the development of the theory of spherical packing and the study of the properties of geometric objects in n-dimensional Euclidean space.