Higher Dimensional Polytopes That Are Products of Lower Dimensional Polytopes

The geometry of polytopes, which are products of other polytopes, is investigated. It is proved that for the existence of a product of polytopes, as a polytope, it is necessary that the Euler-Poincaré equation be fulfilled for its factors. Then the Euler-Poincaré equation must be satisfied for the product of these polytopes. It is proved that the products of polytopes for any factors are incorrect polytopes. Thus, as previously established by the author, the possibility of constructing an n-dimensional space using the product of polytopes is carried out by incorrect polytopes. It is shown that the incorrectness of the product of polytopes, as a polytope, leads to the formation of a continuous closed boundary surface in a polytope with dimension one less than the dimension of the product of polytopes. This, in turn, leads to the fundamental possibility of creating multi-shell systems from unrelated products of polytopes with a steady increase in dimension as we go deeper into the system.