Nanostructures as Tillings of Higher Dimension Spaces

Introduction

In Chapter 1, when studying intermetallic nanostructures, it was found that the space of intermetallic nanostructures has the highest dimension and the fundamental region of these nanostructures is the golden hyper-rombohedron whose dimension is greater than 3 (Zhizhin, 2014c; Zhizhin, 2015; Zhizhin & Diudea, 2016; Zhizhin, Khalaj, & Diudea, 2016). The golden hyper-rombohedron of intermetallic compounds forms a partition of higher - dimensional space. It is a type of nanocluster. Other types of intermetallic clusters were considered in Chapter 2. Various types of metal clusters with ligands were discussed in Chapters 3 - 6. All these clusters have the highest dimension. However, not all types of clusters of chemical compounds can serve as a fundamental area of ​​partitions of spaces of higher dimension, i.e. fill the space of the highest dimension by broadcasting, adjoining each other along whole faces. Only polytopes of the highest dimension, which are polytopic prismahedrons, possess such properties (Zhizhin, 2019). In particular, the golden hyper-rombohedron is a polytopic prismahedron. In the previous chapter, the process of formation of a polytopic prismahedron from any cluster of chemical compounds was considered. To do this, you need to multiply the original cluster as a convex closed polytope by a geometric element with a dimension other than zero. Continuing to multiply the cluster by various geometric elements, you can get clusters of larger size and so fill the space. However, at each step of increasing the size of the cluster, its dimension also increases. This is not a partition of space. At each step, multiplication changes the appearance of the cluster. This chapter discusses the partitioning of higher - dimensional spaces by polytopic prismahedrons, which can be obtained from clusters of chemical compounds, with the formation of nanomaterials.

The problem of completing space by polyhedrons is one of the fundamental problems of mathematics, which has long attracted the attention of scientists. In 1900, D. Hilbert formulated 23 mathematical problems that require solution (Hilbert, 1901). One of these problems (eighteenth) was devoted to this question. It was formulated as follows: “Construction of space from congruent polyhedrons”. This problem is especially complicated in the case of n -dimensional spaces (Delone, 1969), and up to the end it has not been solved to this day under these conditions. In 1961, Delone proved that if we require that polytopes in n - dimensional space be adjacent along entire (n - 1) faces, then for any n there are only finitely many topologically different partitions of space into polytopes. These partitions are called normal. Delone and Sandakova (1961) obtained a finite algorithm, in principle (according to the authors), which allows one to find all such partitions for a given n. In this case, the polytopes themselves (stereohedrons by the Delone terminus) of these partitions for a fixed n can be only of a finite number of topologically different types. However, unfortunately, the authors did not obtain concrete examples of normal partitions of higher - dimensional space into polytopes and the specific types of corresponding polytopes. The question of the existence of fundamental domains in n - dimensional space is quite nontrivial. The main definition of the fundamental domain it was formulated by Delone (1969). The fundamental domain of a group of motions is a set of points of space such that, firstly, all its points are not equivalent to each other with respect to the group of motions, and secondly, any point of space is equivalent to some point of this region relative to the group of motions. What kind of these fundamental areas for a space with a dimension greater than three remains is uncertain to this day. Here we must also keep in mind the existence of polytopes, whose congruential copies can fill a gapless space, but each of them is not a fundamental area (Reingard, 1928).