Polytopic Prismahedrons: Fundamental Regions of the n-Dimension Nanostructures

Polytopic Prismahedrons: Fundamental Regions of the n-Dimension Nanostructures

DOI: 10.4018/978-1-5225-4108-0.ch005

Abstract

The structure of polytopes—polytopic prismahedrons, which are products of polytopes of lower dimensionality—is investigated. The products of polytopes do not belong to the well-studied class of simplicial polytopes, and therefore their investigations are of independent interest. Analytical dependencies characterizing the structure of the product of polytopes are obtained as a function of the structures of polytope factors. Images of a number of specific polytopic prismahedrons are obtained, tables of structures of polytopic prismahedrons are compiled, depending on the types of polytopes of the factors. Polytopic prismahedrons can be considered as a result of the chemical interaction of molecules, which, from among which there is a polytope of a certain dimension.
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The Structure Of Polytopes As A Function Of Factors Structure

While investigating diffraction patterns of quasi-crystals the golden hyper-rhombohedron with dimension 4 was built (Chapter 4). As it was noted, it may be isolated in the quasi-crystals diffraction patterns as a fundamental domain. It is formed as a product of the golden rhombohedron by a one-dimensional segment. Pontryagin mentioned the structures resulting from the product of a polyhedron by a one-dimensional segment the cylinder ones (Pontryagin, 1976). In Chapter 4 it was noted that polytopes of dimension greater than 4 can be distinguished on the grid of vertices of a hyper-rhombohedron. We can say that the product of a polytope by a segment is a prism (Robertson, 1984) with a base in the form of a polytope. To distinguish it from a usual three-dimensional prism, we call it polytopic prism. Ziegler noted that the product of polytopes is not a simplex even if the factors are simplexes, so the polytopes are of considerable interest (Ziegler, 1995). Especially taking into account that a multi-dimensional world has its own peculiarities having no analogues in the three-dimensional world (Ziegler, 1995), contrary to some opposite statements (Panina, 2006). In this regard, the developed theory of simplicial polytopes (Fomenko, 1992; Pontryagin, 1976; Alexandrov, 1975) for the analysis of polytopes product becomes inapplicable, especially in the case of high-dimensional factors.

The product of two polytopes is the result of the product of one of them by one-dimensional edges of another polytope. Thus, the product of polytopes is a complex of polytopic prisms. Let’s call this complex a polytopic prismahedron. The interest to the study of polytopic prismahedrons is connected, in the first place, with the novelty of this field and, secondly, with the fact that polytopic prismahedrons, due to their construction, can be, as we’ll see later, “bricks” to fill the spaces of higher dimension face in face. The definition of polytopes product (Ziegler, 1995) does not give the possibility to specify the structure of the product as a function of the factors structures. There is the structure of product of polytopes having different structures of their factors is determined.

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