Abstract
The geometry of polytopes, which are products of other polytopes, is investigated. It is proved that for the existence of a product of polytopes, as a polytope, it is necessary that the Euler-Poincaré equation be fulfilled for its factors. Then the Euler-Poincaré equation must be satisfied for the product of these polytopes. It is proved that the products of polytopes for any factors are incorrect polytopes. Thus, as previously established by the author, the possibility of constructing an n-dimensional space using the product of polytopes is carried out by incorrect polytopes. It is shown that the incorrectness of the product of polytopes, as a polytope, leads to the formation of a continuous closed boundary surface in a polytope with dimension one less than the dimension of the product of polytopes. This, in turn, leads to the fundamental possibility of creating multi-shell systems from unrelated products of polytopes with a steady increase in dimension as we go deeper into the system.
TopThe Structure Of Polytopes Which Are Products Of Polytopes
There is the structure of product of polytopes having different structures of their factors is determined.
Theorem 6.1. If there are convex polytopes of dimensions n and m, respectively denoted Pn and Qm (or simply P and Q), then their product Pn×Qm, (or simply ×, when it is clear what polytopes are multiplied) has a face
with numbers
,
(6.1)j=k, if 0£
k<
m;
j=m, if
m£
k£
n+m;
n3m.
Here the symbol f indicates the number of faces, the lower index of f and F indicates the dimension of faces, the superscript indicates belonging of a face to the respective polytope.
Proof. To find the product of polytopes Pn and Qm one fined the product of geometric elements of different dimensions of one of the polytopes by geometric elements of different dimensions of another polytope. According to the definition of polytopes product (Ziegler, 1995), the product of vertices (elements of zero - dimension) of one of the polytopes by the vertices of another polytope is a set of vertices, the number of which is equal to the product of the number of vertices of one of the polytopes by the number of vertices of another polytope:
.
The number of elements dimension 1 of polytopes two product is sum from two parts: a product of the number of elements dimension 1 of one polytope by the number of elements dimension 0 of another polytope and a product of the number of elements dimension 0 of one polytope by the number of elements dimension 1 of another polytope:
.
Key Terms in this Chapter
5-Simplex-Prism: The product of 4-simplex by one dimension segment.
5-Cross-Prism: The product of 4-cross-polytope by one dimension segment.
Octahedral Prism: The product of octahedral by one dimension segment.
Dodeca-Polytopic Prismahedron: The product of dodecahedron by the triangle.
Icosahedral Prism: The product of icosahedral by one dimension segment.
N*3-Angular Prismahedron: The product of n-angle by the triangle.
Tetrahedral Prismahedron: The product of tetrahedral by the triangle.
Tetrahedral Prism: The product of tetrahedral by one dimension segment.
Octahedral Prismahedron: The product of octahedral by the triangle.
6-Complex-Polytopic Prismahedrons: The product of tetrahedron by the tetrahedron.
6-Simplex-Polytopic Prismahedron: The product of 4-simplex by the triangle.
Dodecahedral Prism: The product of dodecahedral by one dimension segment.
Icosa-Polytopic Prismahedron: The product of icosahedron by the triangle.
Hierarchical and Translation Filling Spaces: At the same time hierarchical and translation filling.
Triangular Prismahedron: The product of triangle by the triangle.
6-Complex-Tetrahedral Prismahedrons and Octahedral Prismahedrons: The product octahedral by the tetrahedral.
Cube-Polytopic Prismahedron: The product of cube by the triangle.
Polytopic Prismahedron: The product of polytope by one dimension segment.