Parallelohedrons of Higher Dimension and Partition of N-Dimensional Spaces

For more than 100 years in science, many researchers, when trying to solve Hilbert's 18th problem of constructing n-dimensional space, used the principles of the Delaunay geometric theory. In this book, as a result of a careful analysis of the work in this direction, it is shown that the principles of the Delaunay theory are erroneous. They do not take into account the features of figures of higher dimensionality, do not agree with modern advances in the physics of the structure of matter, and lead to erroneous results. A new approach to solving the 18th Hilbert problem is proposed, based on modern knowledge in the field of the structure of matter and the geometric properties of figures of higher dimension. The basis of the new approach to solving the 18th Hilbert problem is the theory developed by the author on polytopic prismahedrons.

Joint Normal Partitions and Hierarchical Filling of N-Dimensional Spaces

The structures arising in spaces of various dimensions with simultaneous normal partitioning of spaces and their hierarchical fillings are considered. The conditions for the appearance of translational symmetry in these structures are investigated. It is shown that simultaneous hierarchical filling and normal tiling in three-dimensional spaces do not lead to the formation of translational symmetry. Such consistent transformations lead to many elements of translational symmetry in spaces of higher dimension. The higher the dimension of space, the more complex the emerging structure and the more symmetry the elements.

Parallelohedrons and Stereohedrons of Delaunay

The works of Delaunay and the followers of his ideas about the geometry of n-dimensional parallelohedrons and stereohedrons are considered. It is proved that these representations do not take into account the conditions for the existence of polytopes of higher dimension and the properties characteristic of figures of higher dimension. They are an attempt to extend the properties of three-dimensional figures to figures of higher dimension. A direct verification of the parallelohedrons from the Delaunay classification taking into account the Shtogrin parallelohedron showed that these figures do not satisfy the Euler-Poincaré equation and therefore the assertion that they are parallelohedrons with dimension 4 is erroneous.

The Products of Molecules and Clusters on Polytopes of Higher Dimension

The molecules and clusters given in the previous chapters are multiplied by polytopes of different dimensions and types: from a one-dimensional segment to polytopes of arbitrary dimension n. The structure of their products is analytically determined. Due to the presence in the products of polytopes of several families of parallel edges, these polytopes can become the basis of parallelohedrons, dividing the space of higher dimension face to face. The products of polytopes can simulate continuous areas of living matter of finite sizes.

The Law of Conservation of Incidents in the Space of Nanoworld

For the first time it was established that for any convex polytope of higher dimension there is an integral equality in the transfer of information from low-dimensional elements to higher-dimensional elements and vice versa. This integral equality is called the law of conservation of incidents. In previous works of the author, this law was established for some polytopes of a particular kind. There is the incidence interpreted as the transfer of information from one material body to another. It is shown that the law of conservation of incidents is valid for both regular bodies and irregular bodies, which can be clusters of chemical compounds. The incident conservation law can serve as a mathematical basis for the recently discovered epigenetic principle of the transmission of hereditary information without changing the sequence of genes in DNA and RNA molecules.

Clusters of Chemical Compounds as Polytopes of the Highest Dimension

General analytical expressions are obtained for calculating the dimension of multi-shell clusters with a common center of shells in those cases when there is a metal atom in the center of the shells and when it is not. The shells can be in the shape of any body of Plato. It has been established that the gamma-copper cluster has the form of a cross-polytope of high dimension. The forms of clusters with ligands of the core of which is a chain of metal atoms or a metal polyhedron are geometrically investigated. It is shown that if the skeleton is a chain of metal atoms, then the cluster is polytope composed of two polytopes of higher dimension adjacent to each other along a flat section containing a chain. If the skeleton is a metal polyhedron, then a cluster of higher dimension has several ligand shells.

Hierarchical Filling of N-Dimensional Spaces

The hierarchical filling of the n-dimensional space with geometric figures is studied, accompanied by a process of discrete similar changes in their sizes, that is, 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 Geometric Structure of the Product of Polytopes

The structure of polytopes of higher dimension (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, and 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 prismahedrons are investigated.

Higher Dimensions in the Theory of Heredity

On the basis Mendel's experiments, a mathematical model is constructed that describes the results of these experiments in a wide range of parameters. There is shown that in the mathematical model of Mendel's experiments, based on real patterns of plant development, there are equilibrium positions between the dominant and recessive forms. This equilibrium position is stable and located in the multidimensional space of system phenotypes. This newly discovered behavior of the dominant and recessive forms in the vicinity of the equilibrium position (true) differs significantly from the logistic equilibrium position in the Hardy-Weinberg principle, built without taking into account the real patterns in the plant population. The geometry of the neighborhood of the compound of two nucleic acid helices with nitrogen bases was investigated. It is proved that this neighborhood is a cross-polytope of dimension 13 (polytope of hereditary information), in the coordinate planes of which there are complementary hydrogen bonds of nitrogenous bases.

On the Conditions for the Existence of Higher-Dimensional Polytopes

For irregular n-simplex, n-cross-polytope, n-cube (n-prismahedron) analytic expressions are obtained for calculating the number of faces of different dimensions included in these polytopes. It is shown that the expressions obtained for each of these types of polytopes lead to the Euler-Poincaré equation, regardless of its general topological conclusion. The fulfillment of the Euler-Poincaré equation is the main condition for the existence of polytopes of higher dimension. It is proved that from the obtained analytical expressions for the numbers of faces of different dimensions, the necessary condition for the existence of polytopes follows, which determines the incidence coefficients of low-dimensional faces with respect to high-dimensional faces. It was found that a cross-polytope of any dimension is the result of the rotation of a simplex around the helicoid axis.