Abstract
The transitions from regular and semi-regular polytopes of various dimensions to dual polytopes are considered in detail. It is shown that any deviation of polytopes from the correct forms leads, in their dual modifications, to new classes of polytopes. Polytopes dual to the products of polytopes form the class of polyincidental polytopes. Their images are given, and their composition is precisely indicated. For example, the polytope dual to the product of a decagon and a decagon contains 100 tetrahedra. All of them are listed by vertices in the dual polytope. Regularly faceted polytopes with a large number of different types of gonohedrons (polytypichedron and polytypictops) for dual transitions form a new class of polytopes with faces of irregular shape and, accordingly, a new type of gonohedron. This can be important in the analysis of chemical structures, often characterized by deviations from regular geometric shapes.
TopIntroduction
From the previous chapters it follows that there are different classes of polytopes of the highest dimension: regular, semi-regular, regular - faceted, quasi-regular with edges of different incidence, polytopic prismahedrons, polytypictopes. Polytopes of each of these classes can have dual polytopes. For a general definition of the dual polytope to some polytope of higher dimension Pn, we introduce the concept of a tower of a polytope Φ. A tower Φ of a polytope Pn is a set 
of faces of Pn such that, i. e. their sequence is subject to strict inclusion
.
(8.1)Thus, the dimensions of the faces in the tower Φ satisfy the inequalities:
.
(8.2) If
P =
n, the tower Φ is said to be maximal.
Let there be a polytope P. Then they say that a polytope P * is dual to a polytope P if there is a one-to-one correspondence between the set of faces of the polytope P and the set of faces of the polytope P *, which reverses the inclusion relation (Grünbaum, 1967). Let us apply this definition to some three-dimensional polytope. In it f0 is the number of vertices, f1 is the number of edges, f2 is the number of flat faces. These numbers satisfy the Euler equation (Euler, 1736):
f0 –
f1 +
f2 = 2.
According to the definition of a dual polytope for it, the faces that have dimension zero in the original polytope become faces that have dimension 2 in the dual polytope. This can be done by associating a vertex in the dual polytope with each planar face in the original polytope (for example, choosing the center of the planar face of the original polytope beyond the vertex of the dual polytope). The resulting vertices are connected by edges that form flat faces of the dual polyhedron in an amount equal to the number of vertices in the original polyhedron. In this case, the number of edges in the dual polytope must be equal to the number of edges in the original polytope so that the Euler equation for the dual polyhedron is satisfied:
.
For the well-known regular polyhedrons this is indeed the case. For example, an octahedron inscribed in a cube (Figure 1), a tetrahedron inscribed in a tetrahedron (Figure 2).
Figure 1. The dual polyhedrons an octahedron and a cube
Figure 2. The dual two the tetrahedrons
The same situation is observed when the dodecahedron is inscribed in the icosahedron or when the icosahedron is inscribed in the dodecahedron (Steinhaus, 1950; Coxeter, 1963). Thus, the tetrahedron is dual to itself, the octahedron is dual to the cube, the icosahedron is dual to the dodecahedron. Dual polytopes are related to each other by the inclusion relation and together characterize the structure of the polytope. Therefore, it is of interest to construct and study dual polytopes included in the already defined classes of polytopes. This chapter is devoted to this analysis.
Key Terms in this Chapter
Tetrahedral Prism: The product of tetrahedral by one dimension segment.
Tetrahedral Prismahedron: The product of tetrahedral by the triangle.
Gonohedron: A polyhedral corner whose faces are regular polygons.
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.
Convex Regularly Faceted Polyhedrons: Convex three-dimensional polytopes whose faces are regular polygons.
Incidence in Polytopes: Incidence in polytopes define the number of elements of higher dimension the given element of lower dimension belongs.
Triangular Prismahedron: The product of triangle by the triangle.
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.
N*3-Angular Prismahedron: The product of n-angle by the triangle.
Polytypichedron (Max): The regular-faceted polyhedron with max number of different types of genohedrons.
Polytypichedrons: The regular-faceted polyhedrons, in which three or more of different types of genohedrons can be present.
Semiregular Polyhedrons: Convex polyhedrons with faces of regular polygons of two or more types, if all gonohedrons of the polyhedron have the same composition and arrangement of faces.
Polytopic Prismahedron: The product of polytope by one dimension segment.
Regular Polyhedron: Convex polyhedron with faces of regular polygons of one type.
Polytypictops: Polytopes of the highest dimension, which include regular-faceted three-dimensional polyhedra with a large number of types of vertices (gonohedrons).
Mechanical Operations on a Polyhedron: Conceivable transformations of a polyhedron, including cutting a polyhedron (cutting off vertices), deforming a section of a cut to obtain a regular polygon in a section, rotating parts of a polyhedron after cutting relative to each other.