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. 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.

Polytopes Dual to Polytopic Prismahedrons

The polytopes are dual to polytypic prismahedrons. In particular, polytopes dual to the product of two canons. It is shown that these polytopes form a new class of polytopes with different values of the incidence of elements of low-dimensional polytopes to polytopes of higher dimension entering the polytope. If the polygons in their product have equal sides, then the dual polytope to the product consists of tetrahedrons, and the degree of incidence of the edge of the dual polytope is determined by the number of sides of the polygon. The existence of a previously unknown polytope consisting of one hundred tetrahedrons is established. Its election is constructed, all its constituent tetrahedrons are listed.

Stereohedrons and Partition of n-Dimensional Space

The process of hierarchical filling of space by p-dimensional regular polytopes is considered under the condition of large-scale discrete increase in the size of polytopes and preservation of their shape (scaling process). It is shown that the polytopic prismahedrons are a concrete realization of the stereohedrons. The polytopic prismahedrons have the necessary properties for translational filling of spaces of higher dimension without slits face to face. Moreover, it is proved that the polytopic prismahedrons forming such fillings can have common elements of any dimension included in the polytope. On the basis of the research carried out in spaces of higher dimension, a new paradigm for describing a discrete world has been put forward.

The Structure and Higher Dimension of Molecules s- and p-Elements

There are considered chemical compounds in which s- and p-elements participate, i.e., elements in which electrons fill with the s- and p-orbitals of atoms. Many of these elements, showing increased chemical activity, play an important role in the vital activity of living organisms and are included in drugs for the treatment of living organisms. The structures of these compounds have been determined and classified, and the molecules of these compounds have been shown to have both rule of higher dimensionality (4, 5, 6, and more). This can be of significant importance for nanomedicine.

The Structure and Higher Dimension of Molecules d- and f-Elements

The chapter deals with the chemical compounds formed by the transition elements of the periodic system elements, i.e. d- and f-elements. All these elements are metals and many of them have valuable physical and chemical properties. In the transition elements, the electrons are filled the d- and f-orbital atoms. The filling of the energy levels of the orbitals should occur as the electron energy increases in accordance with the rules of Pauli and Hund. However, many of the transient elements fill electronic orbitals in violation of these rules. This chapter shows that these anomalies can be described by analytic relationships and they lead to an increase in the chemical and physical activity of the elements. It is shown that the molecules of most compounds with the participation of transition elements are of higher dimensionality, which must be taken into account when analyzing their properties.

Convex Semi-Regular Polytopes

The geometry of polytopes of higher dimension having deviations from the conditions for the correctness of the geometric figure is considered. These deviations reflect the shapes of the molecules of the chemical compounds studied in Chapters 1-3. From the validity conditions in all cases the condition of topological equivalence of the vertices of the polytope is preserved. All these polytopes are called semi-regular. We study the hierarchical filling of spaces with polytopes of higher dimension, different from the well-known filling of spaces with spheres of constant diameter. The considered fillings characterize the distribution of atoms in nanostructures, in which the growth centers are distributed throughout the volume of the structure.

The Structure, Topological, and Functional Dimension of Carbohydrates, Proteins, Nucleic Acids, ATP

New structures of biomolecules have been constructed: carbohydrates, proteins, nucleic acids. It is shown that glucose molecules and ribose molecules have dimensions of 15 and 12, respectively. The enantiomorphic forms of biomolecules in space of higher dimension make it possible to explain the experimentally observed facts of branching of chains of biomolecules in one of the enantiomorphic forms and the absence of chain branching in another enantiomorphic form. The enantiomorphic forms of the tartaric acid molecule in a space of higher dimension reveal the cause of the reversal in different directions of the polarization plane of light in two opposite forms.