The Law of Conservation of Incidents in the Space of Nanoworld

Introduction

In previous works of the author (Zhizhin, 2012; 2013; 2014a, 2014b, 2014c; 2015; 2016; 2018; 2019a, 2019b, 2019c, 2019d, 2019e; Zhizhin, & Diudea, 2016; Zhizhin, Khalaj, & Diudea, 2016), it was convincingly shown that various objects of the nanoworld have a higher dimension. In particular, many molecules of chemical compounds geometrically have the form of the polytopes of the higher dimension. Moreover, many of them are constructions with the main part in the form of an n - simplex or n - cross - polytope. Strictly speaking, any polytope can be reduced to an n - simplex by adding a sufficient number of edges to it without changing the number of vertices. As it turned out, an important place among polytopes describing chemical compounds are occupied by the cross - polytopes of higher dimension. It is enough to say that the connection of helices in DNA molecules with nitrogenous bases is carried out by the cross - polytope of dimension 13 (Zhizhin, 2019d, 2019e). In nanoclusters of chemical compounds, multi - shell structures of the highest dimension are widespread (Zhizhin, 2019c). In the analysis of the continuously distributed nanoworld, it was found (Zhizhin, 2015, 2019a) that the fundamental areas of their structures are the polytopic prismahedrons of the higher dimension, and not the stereohedrons that dreamed Delone B.N. (Delone, 1961; Delone, & Sandakova, 1961). A polytopic prismahedron is a multidimensional prism with bases in the form of polytopes of higher dimension (Zhizhin, 2019a). The simplest polytopic prismahedron is the golden hyper - rhombohedron of dimension 4, which was found in the analysis of intermetallic diffraction patterns (Zhizhin, 2013; Shevchenko, Zhizhin, Mackay, 2013; Zhizhin, 2014c). It is interesting to note that our children's ideas about elementary bricks (cubes and parallelepipeds) filling the space, based on the assumption of the three - dimensionality space, have no continuation in the space of the nanoworld. A logical continuation of our children`s ideas in the space of higher dimension could be a cube of the higher dimension, but the n - cube is not a fundamental area in the space of the nanoworld, since the study of numerous chemical compounds did not reveal elementary cells in the form of an n – cube (Zhizhin, 2018).

This work is devoted to the presentation and proof of the law found in the space of the nanoworld, characterized by the higher dimension of the components. It is connected with the concept of incidence in polytopes of the higher dimension.