Nanostructures as Tillings of Higher Dimension Spaces

Abstract

It is proved that clusters in the form of the polytopic prismahedrons have the necessary properties for partitioning the n-dimensional spaces of a face into a face, that is, they satisfy the conditions for solving the eighteenth Hilbert problem of the construction of n-dimensional spaces from congruent figures. Moreover, they create extended nanomaterials, in principle, of any size. General principles and an analytical method for constructing n-dimensional spaces with the help of polytopic prismahedrons are developed. On the example of specific types of the polytopic prismahedrons (tetrahedral prism, triangular prismahedron), the possibility of such constructions is analytically proved. It was found that neighboring polytopic prismahedrons in these constructions can have common geometric elements of any dimension less than n or do not have common elements.