Polytopes Dual to Polytopic Prismahedrons

The Incidence In Polytope

In Chapter 4 it was established that one of the types of semi-regular polytopes, as deviations from the conditions for the correctness of polytopes that occur in the structures of chemical compounds, are poly-incident polytopes. In each of these polytopes there are simultaneously edges with different incidence values of elements of higher-dimensional polytopes. We consider the question of the incidence of elements of polytopes in a more general form, i.e. let us consider the incidence of elements of polytopes of different dimensions to each other.

The incidence in polytopes indicates to what number of elements of higher dimension the given element of lower dimension belongs. Let’s denote e(d) - an element of dimension d; - the value of the incidence of the element with dimension in relation to the elements of dimension 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 .

In a polygon we have

In a polyhedron we have

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

In a simplex:

In a hypercube:

In a 4-cross-polytope:

In semi-regular polytopes relations of incidence keep their form the same as in regular polytopes (Zhizhin, 2014). But there different figures in one polytope may serve as elements e(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. If we build a polyhedron dual to this prism, we’ll receive an irregular polyhedron. Let’s connect in a prism the centers of flat faces incident to one edge. We’ll get a double pyramid (Figure 1).

Figure 1.

Triangle prism and double pyramid dual to it

In a double pyramid a two vertices are incident to three edges, and another two vertices are incident to four edges. The same picture will be if we take a pentagonal prism.

Two vertices of double pentagonal pyramid are incident to 5 edges, and 5 of the remaining vertices are incident to 4 edges (Figure 2).

Figure 2.

Pentagonal prism and a double pyramid dual to it

Further it will be shown that the polytopes dual to polytopes products have not only vertices with different values of incidence to the edges, but the edges with different values of incidence to three-dimensional figures. This new type of polytopes will be called poly-incident polytopes (see Chapter 4).