Scaling in the Process of Hierarchical Filling of n - Dimensional Space

The hierarchical filling of the n - dimensional space with geometric figures is studied, accompanied by a process of discrete similar changes in their dimensions, i.e. process of scaling. The scaling process in these fillings does not depend on time and is determined only by the geometric characteristics of the figures, which are preserved when their size is changed. Two possible ways of hierarchical filling of space are defined, under which the original figure incrementally increases its size fills the space. Investigations of the hierarchical filling of concrete geometric figures of a plane, three -dimensional space, four - and five - dimensional spaces are carried out. The denominator of geometric progressions characterizing sequences of figures in the process of scaling are determined depending on the shape of the figure and its dimension.

The Detailed Structure of n - Cross - Polytopes and Polytopes With Their Participation

The structure of the n – cross - polytopes for large values of n with an exact enumeration of elements of various dimensions entering their boundary complexes is studied in detail. Examples of chemical compounds with the structure of the n – cross - polytopes are given. It is shown that the polytopic prismahedrons can be polytopes, which simultaneously include the n - simplexes and the n – cross- polytopes as faces. It is established that the list of the n - simplexes and the n – cross - polytopes of different dimensions represented by Gosset as faces of this polytope contradicts the analytic laws of increasing incidence coefficients with increasing n, first established in Chapters 1, 3, 4 of this book. This testifies to the impossibility of the existence of Gosset's polytope.

Boundary Complexes and Interior Points of the Polytopes

This chapter describes how the structure of a polytope of dimension n consisting of points of the boundary complex including a set of faces from zero to n - 1 and a set of interior points that are not belonging to the boundary complex is considered. The value is equal to the number of elements of the boundary complex, which the given element belongs, having dimension one greater than the given element of the boundary complex is denoted coefficient incidence of the given element. It is proven that the coefficient incidence of an element of dimension i of the boundary complex of an n - cube and n - simplex is equal to the difference of the dimension of the cube or simplex n and the dimension of this element i. The incidence coefficient of elements of n – cross - polytopes is substantially higher than this difference.

The Partition of n – Dimensional Space of Polytopic Prismahedrons

It is proved that polytopic prismahedrons have the necessary properties for partitioning the n - dimensional spaces of a face into a face, that is, they satisfy the conditions for solving the eighteenth Hilbert problem of the construction of n - dimensional spaces from congruent figures. General principles and an analytical method for constructing n -dimensional spaces with the help of polytopic prismahedrons are developed. On the example of specific types of the polytopic prismahedrons (tetrahedral prism, triangular prismahedron), the possibility of such constructions is analytically proved. It was found that neighboring polytopic prismahedrons in these constructions can have common geometric elements of any dimension less than n or do not have common elements.

On the Possible Electronic Structure of Atoms in a Space of Higher Dimension

It is shown that the stationary Schrödinger equation describing the distribution of electrons in the vicinity of the atomic nucleus has a solution, in principle, for any dimensionality of the space around the nucleus. As an example, a solution of the Schrödinger equation in a five - dimensional space is obtained. It is shown that the solution of the Schrödinger equation in p - dimensional space has p quantum numbers: the principal quantum number, the orbital quantum number and p - 2 magnetic quantum numbers. Taking into account the spin quantum number, the total number of quantum numbers in p - dimensional space is p + 1. This leads to the possibility of increasing the number of quantum cells of orbitals and, consequently, to the possibility of increasing the valence of the elements.

Poly - Incident and Dual Polytopes

The incidence of elements of low dimension in convex regular polytopes of dimension n with respect to elements of higher dimension up to elements of dimension n - 1 is investigated. It is shown that polytopes are dual to polytopic prismahedrons form a new class of polytopes with simultaneously different values of the incidence of elements of low dimension to elements of higher dimension entering in the polytope. This new type of polytopes is called poly – incident polytopes. The existence of a previously unknown polytope consisting of one hundred tetrahedrons is established. All its constituent tetrahedrons are listed. The concept of a polytope with a factor structure whose vertices consist of polytopes of large dimension is introduced.

The Number of Symmetry Transformation of Convex Regular Polytopes in the n - Space

The number of symmetry transformations of regular polytopes of dimension n (n - cubes, n - simplexes, n - cross polytopes) are considered, using symmetry transformation of their facets. In this chapter, it is investigated how a certain symmetry transformation of the facet leads to the transformation of the symmetry of the polytope, under the condition of continuity of the polytope as a whole. It is established that the number of symmetry transformations of a regular n - polytope is equal to the product of the number of symmetry transformations of the facet of the corresponding polytope by the number of facets in this polytope.

Polytopic Prismahedrons

The structure of polytopes - polytopic prismahedrons, which are products of polytopes of lower dimensionality, is investigated. The products of polytopes do not belong to the well-studied class of simplicial polytopes, and therefore their investigations are of independent interest. Analytical dependencies characterizing the structure of the product of polytopes are obtained as a function of the structures of polytope factors. Images of a number of specific polytopic prismahedrons are obtained, tables of structures of polytopic prismahedrons are compiled, depending on the types of polytopes of the factors. The geometric properties of the boundary complexes of polytopic prismehedrons are investigated. Polytopic prismahedrons can be considered as a result of the chemical interaction of molecules, which, from among which there is a polytope of a certain dimension. The possibility of the multivaluedness of the incidence coefficients of geometric elements of polytopic prisms is revealed. It is shown that polytopic prismahedra, due to their nature, as products of polytopes, ensure the filling of n-dimensional spaces during their translation, creating a structure similar to the structure of quasicrystals.

Polytopes of Higher Dimension in the Nature

Areas of research into the phenomena of nature in which the influence of polytopes of higher dimension is described in this chapter. These include studies of the structures of many chemical compounds whose molecules exhibit the properties of polytopes of higher dimension. This leads to the creation of higher-dimensional stereochemistry. Phase transitions of the second kind are accompanied by a change in the symmetry of the structure of matter, the description of which, in agreement with the experimental data, requires the attraction of spaces of higher dimension. Elementary cells of quasicrystals, having the form of polytopic prismahedrons, are given (polytopes of higher dimension). The structure of DNA as sequence of the higher dimensional polytopes are given.