What is Hierarchical and Translation Filling Spaces

The structure of polytopes of higher dimension (polytopic prismahedrons), which are products of polytopes of lower dimensionality, is investigated. The products of polytopes do not belong to the well-studied class of simplicial polytopes, and therefore, their investigations are of independent interest. Analytical dependencies characterizing the structure of the product of polytopes are obtained as a function of the structures of polytope factors. Images of a number of specific polytopic prismahedrons are obtained, and tables of structures of polytopic prismahedrons are compiled depending on the types of polytopes of the factors. The geometric properties of the boundary complexes of polytopic prismahedrons are investigated.