Abstract
The molecules and clusters given in the previous chapters are multiplied by polytopes of different dimensions and types: from a one-dimensional segment to polytopes of arbitrary dimension n. The structure of their products is analytically determined. Due to the presence in the products of polytopes of several families of parallel edges, these polytopes can become the basis of parallelohedrons, dividing the space of higher dimension face to face. The products of polytopes can simulate continuous areas of living matter of finite sizes.
TopThe Product Of A Tetrahedron With Center On One – Dimensional Segment
Molecules of chemical compounds (many acids, hydrocarbons, oxides) often exhibit tetrahedral coordination (Gillespie, 1972; Gillespie, & Hargittai, 1991; Zhizhin, 2018). In these compounds, atoms are located not only at the vertices of the tetrahedron, but also in the center of the tetrahedron. One example may be chromium oxide Cr O3. In this case, the chromium atom is located in the center of the tetrahedron, and the oxygen atoms are located at the vertices of the tetrahedron (Chapter 1). In accordance with Chapter 5, we denote the tetrahedron with the center by P, and the one - dimensional segment by Q. The number of vertices in the tetrahedron with center is 5, the number of edges is 10, the number of two - dimensional faces is 10, the number of three - dimensional faces is 5 (one large tetrahedron and four tetrahedra inside it). Thus, for a tetrahedron with a center, in accordance with the notation of Chapter 5, we have 
The equation of the Euler - Poincaré (Poincaré, 1895) in this case has view
.
(1) In this equation
fi(
P) is the number of faces with dimension
i in polytope
P with dimension
d.
Substitute obtained numbers fi(P) in the equation of the Euler - Poincaré it is easy to verify that this equation for the polytope (8) is satisfied for d = 4; 5 – 10 + 10 – 5 = 0.
Thus, polytope P is simplex of dimension 4 (4 – simplex). For one – dimensional segment we have 
For the product of tetrahedron with center on a one – dimensional segment in accordance of Theorem 1 of Chapter 5 we have
= 5 ˑ 2 = 10,
.
Consequently, the product of a tetrahedron centered on a one - dimensional segment has 10 vertices, 25 edges, 30 two – dimensional faces, 20 three - dimensional faces, 7 four - dimensional faces. This product itself has dimension 5. This can be easily verified by substituting the obtained values of the numbers of faces of various dimensions into the Euler – Poincare equation 10 – 25 + 30 – 20 + 7 = 2. Thus, the Euler - Poincaré equation (1) is satisfied for dimension d = 5.
TopThe Product Of A Tetrahedron With Center On Triangle
In this case, as well as the previous paragraph for polytope P, we have 
For triangle the numbers of elements with different dimension are 
For the product of tetrahedron with center on a triangle in accordance of Theorem 1 of Chapter 5 we have
= 5 ˑ 3 = 15,
Key Terms in this Chapter
Octahedral Prismahedron: The product of octahedral by the triangle.
5-Simplex-Prism: The product of 4-simplex by one-dimension segment.
Polytopic Prismahedron: The product of polytope by one-dimension segment.
6-Complex-Tetrahedral Prismahedrons and Octahedral Prismahedrons: The product octahedral by the tetrahedral.
Icosahedral Prism: The product of icosahedral by one-dimension segment.
Tetrahedral Prism: The product of tetrahedral by one-dimension segment.
Triangular Prismahedron: The product of triangle by the triangle.
6-Simplex-Polytopic Prismahedron: The product of 4-simplex by the triangle.
N*3-Angular Prismahedron: The product of n-angle by the triangle.
5-Cross-Prism: The product of 4-cross-polytope by one-dimension segment.
Cube-Polytopic Prismahedron: The product of cube by the triangle.
6-Complex-Polytopic Prismahedrons: The product of tetrahedron by the tetrahedron.
Dodeca-Polytopic Prismahedron: The product of dodecahedron by the triangle.
Icosa-Polytopic Prismahedron: The product of icosahedron by the triangle.
Tetrahedral Prismahedron: The product of tetrahedral by the triangle.
Dodecahedral Prism: The product of dodecahedral by one-dimension segment.
Octahedral Prism: The product of octahedral by one-dimension segment.