Poly - Incident and Dual Polytopes

Poly - Incident and Dual Polytopes

Copyright: © 2019 |Pages: 26
DOI: 10.4018/978-1-5225-6968-8.ch005

Abstract

The incidence of elements of low dimension in convex regular polytopes of dimension n with respect to elements of higher dimension up to elements of dimension n - 1 is investigated. It is shown that polytopes are dual to polytopic prismahedrons form a new class of polytopes with simultaneously different values of the incidence of elements of low dimension to elements of higher dimension entering in the polytope. This new type of polytopes is called poly – incident polytopes. The existence of a previously unknown polytope consisting of one hundred tetrahedrons is established. All its constituent tetrahedrons are listed. The concept of a polytope with a factor structure whose vertices consist of polytopes of large dimension is introduced.
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Introduction

In 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 978-1-5225-6968-8.ch005.m01 - an element of dimension i; 978-1-5225-6968-8.ch005.m02 - 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 978-1-5225-6968-8.ch005.m03 are constant for the whole polytope in all dimension range from 0 to n (n - dimension of the polytope). Obviously, that978-1-5225-6968-8.ch005.m04 = 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 978-1-5225-6968-8.ch005.m10 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.

Figure 1.

The triangle prism and double pyramid dual to it

978-1-5225-6968-8.ch005.f01

Key Terms in this Chapter

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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