What is Incidence in Polytopes

For irregular n-simplex, n-cross-polytope, n-cube (n-prismahedron) analytic expressions are obtained for calculating the number of faces of different dimensions included in these polytopes. It is shown that the expressions obtained for each of these types of polytopes lead to the Euler-Poincaré equation, regardless of its general topological conclusion. The fulfillment of the Euler-Poincaré equation is the main condition for the existence of polytopes of higher dimension. It is proved that from the obtained analytical expressions for the numbers of faces of different dimensions, the necessary condition for the existence of polytopes follows, which determines the incidence coefficients of low-dimensional faces with respect to high-dimensional faces. It was found that a cross-polytope of any dimension is the result of the rotation of a simplex around the helicoid axis.