Abstract
Using concrete examples of clusters of chemical compounds of various types (intermetallic clusters, metal chains with ligands, polyhedral metal clusters with ligands), it is shown how nanomaterials are formed from individual clusters by multiplying their geometric structure by other geometric elements of different dimensions. The considered examples correspond to nanomaterials with a structure of limited complexity. However, the mathematical apparatus developed on the basis of the geometry of high-dimensional polytopes allows, in principle, to describe and study and design nanomaterials of this type of any complexity and any dimension. In particular, nanomaterials with the simultaneous use of elements with different metric characteristics can be attributed to such nanomaterials.
TopPolytopic Prismahedron From Mackay Clusters
Chapter 2 examined clusters of intermetallic compounds. It is proved that the Mackay cluster, consisting of two icosahedrons with a common center (Figure 1 in Chapter 2) has dimension 4. In determining polytopic prismahedrons from clusters, we will use the following theorem (Zhizhin, 2019a), preserving the accepted notation,
Theorem 1. (Zhizhin, 2015)
If there are convex polytopes of dimensions n and m, respectively denoted Pn and Qm (or simply P and Q), then their product Pn×Qm, (or simply ×, when it is clear what polytopes are multiplied) has a face 
with numbers
,
(1)j=k, j=k if 0≤
k<
m;
j=m, if
m≤
k≤
n+m; n≥
m.
Here the symbol f indicates the number of faces, the superscript of f and F indicates the dimension of faces, the lower index indicates belonging of a face to the respective polytope.
Consider the polytopic prismahedron formed by the product of the Mackay cluster in a one -dimensional segment. The Mackay cluster (P) with two icosahedrons (Chapter 2) has the following numbers of elements of different dimensions
The one - dimensional segment Q1 has 
. The product of polytopes P=P4 and Q1 according to (1) it is determined by the equalities:
Key Terms in this Chapter
Polytopic Prismahedron: Is the product of a polytope to another polytope, or in a particular case to a one-dimensional segment.
Dimension of the Space: The member of independent parameters needed to describe the change in position of an object in space.
Polytope: Polyhedron in the space of higher dimension.
Nanocluster: A nanometric set of connected atoms, stable either in isolation state or in building unit of condensed matter.