On the Conditions for the Existence of Higher-Dimensional Polytopes

Introduction

As early as 1880, the American mathematician Stringham published a paper on the correct figures in n - dimensional space (Stringham, 1880). Moreover, a multidimensional figure is considered correct if all the polyhedral angles in the vicinity of all vertices of the figure are correct and compatible with the motion. Polyhedral angles are considered correct if all the edges emanating from the vertex have the same length and the angles between the edges are equal. In this work, for a special type of multidimensional figures, a generalized Euler equation was obtained that relates the numbers of elements of different dimensions in this figure into one equation. For the first time, images of regular four - dimensional figures are shown in projection onto a two - dimensional plane. The conditions for the existence of regular n - dimensional figures are formulated. Figures 1, 2, 3 respectively show the projections of four - dimensional regular figures of three types in the Stringham image, which exist for any finite n: 4 - simplex, 4 - cube, 4 – cross - polytope.

Figure 1.

Simplex

Figure 2.

Cube

Figure 3.

Cross - polytope

It is clearly seen that there are images the three - dimensional figures included in the four -dimensional figures and adjacent to each other along two - dimensional faces, as if pass through each other, being on one side of the two - dimensional face. This is a fundamental feature of multidimensional figures. Five years after the appearance of Stringham's work, Poincare's work (Poincaré, 1885) was published, in which, using topological analysis, a generalized Euler equation was obtained for arbitrary closed convex and non - convex multidimensional figures without determining specific numbers of faces of different dimensions of these figures. Therefore, hereinafter, the generalized Euler equation will be called the Euler - Poincaré equation. In 1967, Grünbaum also obtained the Euler - Poincaré equation without determining specific numbers of faces of various dimensions in convex n - dimensional figures (Grünbaum, 1967). We can say that the main condition for the existence of a multidimensional figure is the fulfillment of the Euler - Poincaré equation for it for some value of dimension n. In the monograph Zhizhin (Zhizhin, 2018), using the Gillespie data on the geometry of molecules (Gillespie, 1972; Gillespie & Hargittai, 1991), the dimensions of the molecules of most elements of the periodic system were determined and it was proved that almost all of them have a higher dimension. Then, the geometry of polytopes of molecules of chemical compounds was studied in detail (Zhizhin, 2019 a). It is noteworthy that among these polytopes, virtually no polytope is correct. Deviations from the correctness of polytopes can consist in the inequality of the angles between the edges in the vicinity of the same vertex, the inequality of the lengths of the edges, the inequality in the number of edges emanating from each vertex. Therefore, in this chapter, when determining the conditions for the existence of polytopes of higher dimension, the condition for the correctness of polytopes is not used. In this regard, the Euler - Poincaré equation is re - formulated with the determination of the number of faces of various dimensions in irregular polytopes of higher dimension. A specific determination of the numbers of elements of different dimensions for different types of polytopes of higher dimension allows us to establish previously unknown other conditions for the existence of polytopes of higher dimension, based on the patterns of incidence of elements of one dimension with respect to elements of another dimension. In the conclusion of the chapter, basing on previous studies of polytopes of higher dimensionality of chemical compounds (Zhizhin, 2018, 2019 a) and biomolecule compounds (Zhizhin, 2019 b), possible types of polytopes of higher dimensionality of chemical compounds are listed.