Polytopes Dual to Polytopic Prismahedrons

Polytopes Dual to Polytopic Prismahedrons

DOI: 10.4018/978-1-5225-4108-0.ch006


The polytopes are dual to polytypic prismahedrons. In particular, polytopes dual to the product of two canons. It is shown that these polytopes form a new class of polytopes with different values of the incidence of elements of low-dimensional polytopes to polytopes of higher dimension entering the polytope. If the polygons in their product have equal sides, then the dual polytope to the product consists of tetrahedrons, and the degree of incidence of the edge of the dual polytope is determined by the number of sides of the polygon. The existence of a previously unknown polytope consisting of one hundred tetrahedrons is established. Its election is constructed, all its constituent tetrahedrons are listed.
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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; 978-1-5225-4108-0.ch006.m01 - the value of the incidence of the element with dimension 978-1-5225-4108-0.ch006.m02 in relation to the elements of dimension978-1-5225-4108-0.ch006.m03 For regular polytopes because of their uniform values of the incidence 978-1-5225-4108-0.ch006.m04 are constant for the whole polytope in all dimension range from 0 to n (n-dimension of the polytope). Obviously, that 978-1-5225-4108-0.ch006.m05 = 1 for any 978-1-5225-4108-0.ch006.m06.

In a polygon we have 978-1-5225-4108-0.ch006.m07

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).

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