DOI: 10.4018/978-1-5225-6968-8.ch005

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TopIn Chapter 4 it was described that the coefficient of incidence of the vertices of the Egyptian pyramid and polytopic prismahedrons with the participation of pyramids in their boundary complexes can have simultaneously several different values in the same polytope. It turns out that polytopic prismahedrons lead to polytopes with different incidence values not only of vertices, but also of other elements of boundary complexes and in the absence of pyramids.

Let’s denote - an element of dimension *i*; - the value of the incidence of the element with dimension *i* in relation to the elements of dimension *j* (*j* > *i*). For regular polytopes because of their uniform values of the incidence are constant for the whole polytope in all dimension range from 0 to *n* (*n* - dimension of the polytope). Obviously, that = 1 for any *i* < *n*.

In a polygon there are:

In a polyhedron there are:

In four - dimensional polytopes relations of the incidence have the following values.

In a 4 - simplex:

In a 4 - cube:

In a 4 – cross - polytope:

In the construction of dual polytopes to polytopes of dimension *n*, the incidence coefficients with respect to facets of polytopes For correct four - dimensional polytopes, these incidence coefficients have, respectively, the values:

In a 4 – simplex:

In a 4 – cube:

In a 4 – cross – polytope:

In semi - regular polytopes relations of incidence keep their form the same as in regular polytopes (Zhizhin, 2014a). But there different figures in one polytope may serve as elements *b*(2), though all vertices of these semi - regular polytopes are superposed by motion. If in a polytope there are vertices which are not superposed by motion, then relations of incidence are variable in a polytope, for example, in a triangle prism. A prism can be considered a semi - regular polytope because it has two triangle faces and three of square faces.

For a deeper study of the geometry of polytopic prismahedron, one apply the analysis of the duality of polytopic prismehedron. The analysis of duality, as an analysis of the unity of opposites, is used in various fields of science: in mathematics, logic, philosophy, physics and chemistry (Bogdanov, 1989; Whitehead, 1990; Feyman, 1968; Hegel; 1998; Bucur and Delanu, 1972).

One will consider the polytope as dual to a given polytope of dimension *n* if the facets of a given polytope are replaced on the vertices of the polytope. Moreover, the vertices of a new polytope are connected by an edge if the corresponding facets have a common face of dimension *n* - 2 in the given polytope. Consider a rectangular prism with triangular bases. Can to introduce the centers of the planar faces of this prism and join the edges of those centers whose faces have a common edge. One will get a double pyramid (Figure 1). It is an irregular polyhedron dual to this prism.

Poly - Incident Polytopes: Polytopes in which elements of lower dimension have different incidence values for elements of higher dimension. Polytopes that are dual to polytopes products are poly - incident polytopes.

Polytope with Factor – Structure: Polytope in which vertices are polytopes with same dimension k , and edges are polytopes with dimension k – 1.

Incidence in Polytopes: Incidence in polytopes define the number of elements of higher dimension the given element of lower dimension belongs.

Duality in Polytopes: If the facets of one polytope correspond to the vertices of another polytope, and the edges of one polytope correspond to common elements of the facets of another polytope, then these polytopes are dual.

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