Abstract
The structure of polytopes of higher dimension (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, and tables of structures of polytopic prismahedrons are compiled depending on the types of polytopes of the factors. The geometric properties of the boundary complexes of polytopic prismahedrons are investigated.
TopThe Structure Of Product Of Polytopes As Function Of Factors Structures
There is the structure of product of polytopes having different structures of their factors is determined.
Theorem 5.1. (Zhizhin, 2015). 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
,
(5.1)j=k, if 0≤
k<
m;
j=
m, if
m≤
k≤
n+m;
n≥
m Key Terms in this Chapter
Tetrahedral Prismahedron: The product of tetrahedral by the triangle.
Octahedral Prism: The product of octahedral by one dimension segment.
Poly-Incident Polytopes: The product of icosahedron by the triangle.
Triangular Prismahedron: The product of triangle by the triangle.
Poly-Incident Polytopes: The product of n-angle by the triangle.
5-Simplex-Prism: The product of 4-simplex by one dimension segment.
Poly-Incident Polytopes: The product of dodecahedron by the triangle.
Hierarchical and Translation Filling Spaces: At the same time hierarchical and translation filling.
Poly-Incident Polytopes: The product of cube by the triangle.
Dodecahedral Prism: The product of dodecahedral by one dimension segment.
Poly-Incident Polytopes: The product octahedral by the tetrahedral.
Poly-Incident Polytopes: The product of 4-simplex by the triangle.
Poly-Incident Polytopes: The product of tetrahedron by the tetrahedron.
Tetrahedral Prism: The product of tetrahedral by one dimension segment.
Polytopic Prismahedron: The product of polytope by one dimension segment.
Poly-Incident Polytopes: The product of 4-cross-polytope by one dimension segment.
Octahedral Prismahedron: The product of octahedral by the triangle.
Icosahedral Prism: The product of icosahedral by one dimension segment.