Boundary Complexes and Interior Points of the Polytopes

Boundary Complexes and Interior Points of the Polytopes

Copyright: © 2019 |Pages: 28
DOI: 10.4018/978-1-5225-6968-8.ch002

Abstract

This chapter describes how the structure of a polytope of dimension n consisting of points of the boundary complex including a set of faces from zero to n - 1 and a set of interior points that are not belonging to the boundary complex is considered. The value is equal to the number of elements of the boundary complex, which the given element belongs, having dimension one greater than the given element of the boundary complex is denoted coefficient incidence of the given element. It is proven that the coefficient incidence of an element of dimension i of the boundary complex of an n - cube and n - simplex is equal to the difference of the dimension of the cube or simplex n and the dimension of this element i. The incidence coefficient of elements of n – cross - polytopes is substantially higher than this difference.
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Introduction

A finite family C of convex polytopes will be called a complex under the condition

  • 1.

    Each face of an element of the family C is an element of the family C;

  • 2.

    The intersection of any two elements of C is a face of both.

Let P is an n – polytope, i.e. a convex polytope of dimension n. We denote by B (P) the boundary complex P, that is, a complex consisting of all faces of P with dimension n - 1 or less (Grunbaum, 1967). The points not belonging to B (P) will be called interior points of P. The set of interior points of the polytope P will be denoted by V. Thus, we have equality 978-1-5225-6968-8.ch002.m01. Since all the faces of a polytope from vertices (which in chemical compounds are images of atoms, molecules or functional groups) to facets enter the boundary complex, it is of interest to establish relationships between faces of different dimensions in the boundary complex B (P).

For each element 978-1-5225-6968-8.ch002.m02 of the boundary complex B (P) can to introduce of the notation its dimension 978-1-5225-6968-8.ch002.m03

There is denoted 978-1-5225-6968-8.ch002.m04 the incidence value of the element of the boundary complex 978-1-5225-6968-8.ch002.m05 to the elements of the boundary complex having dimension 978-1-5225-6968-8.ch002.m06 This value is equal to the number of elements of the boundary complex which the given element belongs having dimension one greater than the given element of the boundary complex .

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The Structure Of The Boundary Complex Of A N – Cube

If polytope P is point (vertex), so evidence that n = 0, 978-1-5225-6968-8.ch002.m07 If polytope P is segment with vertices 978-1-5225-6968-8.ch002.m08 so the boundary complex B (P) is points 978-1-5225-6968-8.ch002.m09. In this case are978-1-5225-6968-8.ch002.m10 Can to introduce the origin of coordinates O in the middle of the segment [978-1-5225-6968-8.ch002.m11]. Thus, if put of length of the segment equal one, so the boundary elements (points) have coordinates 978-1-5225-6968-8.ch002.m12 The set of interior points in this case is 978-1-5225-6968-8.ch002.m13 If n = 2 so polytope P is square (Figure 1). There elements of the boundary complex are vertices 978-1-5225-6968-8.ch002.m14, 978-1-5225-6968-8.ch002.m15 and edges 978-1-5225-6968-8.ch002.m16, 978-1-5225-6968-8.ch002.m17978-1-5225-6968-8.ch002.m18978-1-5225-6968-8.ch002.m19. The dimensions of the vertices equal zero 978-1-5225-6968-8.ch002.m20 The dimensions of the edges equal one 978-1-5225-6968-8.ch002.m21 The incidence value of the elements of the boundary complex equal corresponding 978-1-5225-6968-8.ch002.m22 If to introduce the origin of coordinates O in centrum of square and put of length of edges equal one so vertices of square have coordinates (x, y):

Key Terms in this Chapter

The Boundary Complex of Polytope With Dimension n: The set of faces of a polytope with dimension from zero to n - 1.

The Set of Interior Points of the Polytope: The set of points belonging to a polytope, but not belonging to its boundary complex.

Cluster: A chemical compound that is intermediate between a molecule and a volumetric body.

The Coefficient Incidence of an Element of the Boundary Complex: The number of elements of the boundary complex to which the given element of the boundary complex belongs, having a dimension one greater than this element.

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