On the Structure of Parallelohedrons of Higher Dimension: Hilbert's 18th Problem

Introduction

In 1885, the book of Efgraf Fedorov “Began the doctrine of figures” (Fedorov, 1885) went out of print. It classifies two - and three - dimensional figures that can fill the Euclidean plane and three - dimensional space adjacent to each other along whole edges and, accordingly, flat faces without gaps. These figures were called parallelogons and, correspondingly, parallelohedra, due to the fact that their sides are pairwise parallel. It gives the concept of stereohedrons as equal figures that make up a parallelohedron together. In 1889 and in 1891 the following works of Fedorov, “Symmetry of finite figures” and “Symmetry of regular systems of figures” (Fedorov, 1889, 1891), went out of print. In these works, Fedorov deduced 230 spatial groups, i.e. 230 unique geometric laws of arrangement of elementary particles of crystalline structures in three - dimensional Euclidean space and 17 two - dimensional groups of arrangement of elementary particles in the Euclidean plane. In 1891, Schönflis's book “Crystal Systems and the Structure of Crystals” was also published (Schönflis, 1891). Her author has repeatedly quoted Fedorov, pointing to his primacy in a number of questions on theoretical crystallography. Fedorov’s extensive correspondence with Schönflis (Shafranovsky, 1963) has been preserved, from which the leading role of Fedorov in the derivation of 230 spatial symmetry groups follows. The studies were reflected in a report by David Hilbert (Hilbert, 1901) at the International Congress of Mathematicians in Paris, in which a number of mathematical problems were formulated that determined the development of mathematics for the next century. One of these problems (the eighteenth) raised the question of constructing spaces of higher dimension from congruent polyhedrons. The report expressed the desire to investigate the question of constructing spaces of higher dimension from congruent figures (Aleksandrov, 1969). The answer to this request was a series of researchers' work in subsequent years. However, many of them tried to consider spaces of higher dimension and figures of higher dimension by mechanically extending to a space of higher dimension ideas about three - dimensional Euclidean space without taking into account the essential properties and features of spaces of higher dimension (Minkowski, 1911; Delaunay, 1929, 1937, 1961; Delaunay, & Sandakova,1961; Voronoi, 1952; Alexandrov, 1934, 1954; Venkov, 1954, 1959). Although back in 1880, the work of the American mathematician Stringham appeared on figures in n - dimensional space (Stringham, 1880). This work was apparently unknown to these researchers, as well as the work of Stott from Amsterdam (Stott, 1900, 1910). Much later, the works of Coxeter and Grünbaum, widely known at present, on figures of higher dimensions appeared (Coxeter, 1963; Grünbaum, 1967). From the analysis of geometric figures of higher dimensions, it would be necessary to begin the solution of the question of constructing spaces of higher dimensions with the help of these figures. That is exactly what the ingenious crystallographer Fedorov did when he studied the question of constructing three - dimensional space with the help of three - dimensional figures. However, this did not happen. As a result, mechanically extending Fedorov’s teaching on parallelohedra in three - dimensional spaces to the four - dimensional space, Delaunay in 1929 systematized, as he believed, four-dimensional parallelohedons (Delaunay, 1929). Moreover, no evidence was given for any of these 51 parallelohedrons that they have dimension 4. Later, a student of Delaunay Shtogrin, as part of the teachings of Delaunay, discovered the existence of another figure, additionally included in the classification of Delaunay. It was believed that this figure is also a parallelogon with dimension 4 (Shtogrin, 1973). In this chapter, it will be shown that, as a result of neglecting the necessary conditions for the existence of polytopes of higher dimension, none of the Delaunay classification figures, as well as the figure proposed by Shtogrin, are polytopes of higher dimension, i.e. due to this, they do not have, as it was supposed, a dimension equal to 4.