Polytopes of Higher Dimension in the Nature

Polytopes of Higher Dimension in the Nature

Copyright: © 2019 |Pages: 45
DOI: 10.4018/978-1-5225-6968-8.ch001

Abstract

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.
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Introduction

Visual image of any object on the retina of the human eye is two - dimensional. For this reason, any object is perceived initially as a two - dimensional one. Similarly, we touch only the surface of objects, that is, the palpable image of objects has again two dimensions. The idea of three -dimensionality of an object forms only as a result of the comparison of mismatched images in the right and left eye, this difference being observed in motion. Comparison is a result of thinking and judgment appears as a result of it.

Therefore, even the three - dimensional conception of objects is an abstract idea and, therefore, it is difficult for many people. The way of abstraction leads us to the concept of space, i.e. to a conceptual space (space studied by geometry - geometrical space, (Poincare, 1985)). This conceptual space arose on the basis the perception of space, i.e. it based on the perceptual space (the space of ideas (Poincare, 1985)). A perceptual space itself is a reflection of the real material space. Perceptual space is an image of the real space, which exists regardless of whether we perceive it or not, and on what the perception we get from it (Coleman, 1970). However, given that perceptual space is the result of mental activity, the image of the real space is ambiguous. It may take into account the different human views about surrounding world, as well as either properties of the world that seem the most important to a person in the given concrete circumstances. It is also the reason for co - existence of different geometries. It should be borne in mind that the geometrical axioms are neither a priori synthetic judgments nor experimental facts. They are conventional provisions (agreement): choosing between all possible agreements, we are guided by experimental facts, but the choice is free and limited only by the need to avoid any contradiction “(Poincare, 1985).

Historically two main concepts of the space formed within which there were a lot of modifications of these concepts and related geometries. According to the first of these concepts associated with the names of Aristotle and Leibniz, the real space is a property of position of material objects. It is inextricably linked with the matter. The development of these ideas led to the well- known philosophical theses that there is no space without matter, just as there is no matter without space, and that space is a form of existence of the matter.

According to the second of these concepts, associated with the names of Democritus and Newton, space is the receptacle of all the material objects having no influence on space (Einstain, 1930). Exactly this idea of space was determinative for many centuries and serve as philosophical foundation of Euclidean geometry (Euclid, 2012). According to this view geometric space is continuous, endless, three - dimensional, uniform (all points of space are identical to each other), isotropic (all lines passing through a point are identical to each other).

Ideas of Leibniz in his time did not have any support, because they lead to ambiguity of Euclidean geometry. N.I. Lobachevsky’s discovery of non-Euclidean noncontradictory geometry according to which through the point outside straight line more than one parallel lines can be drawn on the plane initiated rapid development of geometry. It is important that N. I. Lobachevsky associated geometry with physics and space with matter (Lobachevsky, 1945). The works of Lobachevsky at non - Euclidean geometry it were continued by B. Riemann. In his famous lecture “On the hypotheses that lie at the foundations of geometry” (Riemann, 1854) he introduces the concept of n - extended manifold. It is essential that the n – dimensional extension it is determined by B. Riemann without introduction of infinite spaces. Moreover, the infinity of space is obviously contrary to the views of B. Riemann on n - dimensional extension. Definition it B. Riemann initially considers a finite region, and as a sign of manifold n - dimension manifold the position in this manifold is characterized by a change in n - dimensional (“simple”) of extended values. B. Riemann showed that a manifold of (n + 1) - dimensions is a manifold of 1 - st dimension, consisting of elements (pixels) n - dimensions. Three - dimensional space can be considered as a special case of n - dimensions manifold.

Key Terms in this Chapter

Dimension of the Space: The number of independent parameters needed to describe the change in position of an object in space.

Simplex: A convex polytope, any two vertices of which are joined by an edge.

Stereochemistry of Higher Dimension: Representation of the structure of chemical compounds in the space of higher dimensions.

Deoxyribonucleic Acid: A biopolymer, the monomer of which is the nucleotide.

Zero Surface: The surface at the points of which the derivative of one of the phase coordinates with respect to the independent variable is zero.

Phase Portrait: Representation of solutions of a system of differential equations by trajectories in the space of the corresponding phase coordinates.

Polytope: A polyhedron in the space of higher dimension.

Nucleotide: A phosphoric acid residue attached to sugar deoxyribose, to which one of the four nitrogen bases is attached also.

The Order Parameter: A vector characterizing the phase transition of the second kind. Its dimension coincides with the dimension of irreducible representations of the symmetry group of high-symmetry phase.

Golden Hyper – Rhombohedron: A rhombohedron in a space of higher dimensions, the metric relations between the geometric elements of which are determined by the golden section.

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