Abstract
Using three-dimensional visualization of nucleic acid molecules, obtained in the previous chapter, an analysis of the geometry of nucleic acid molecules in the space of higher dimension is carried out. It is shown that phosphoric acid residues and five-carbon sugar molecules in a double-stranded nucleic acid form polytopes of higher dimension with anti-parallel edges. These polytopes are of type n-cross-polytope (n = 5 for phosphoric acid residues, n = 13 for sugar molecules). It was found that these n-cross-polytopes located in right- and left-twisted spirals are enantiomorphic. It has been found that in cross-polytopes constructed of two sugar molecules there are 12 coordinate planes, each of which may contain a bond of nitrogenous bases (one of the 12 known ones). The formation of codons (triplets) corresponds to the separation in space of the highest dimension of nucleic acids of three-dimensional regions. This also occurs in the ribosomes upon contact with transport and adapter RNA during protein synthesis.
TopPolytopes With Antiparallel Edges
In single - stranded and double - stranded nucleic acids (RNA, DNA), the constituents of acids (residues of phosphoric acid and sugar molecules) interact with each other (Watson & Crick, 1953a, b; Spirin & Gavrilova, 1971; Frank – Kamenetskiy, 1986, 1988). Phosphoric acid residues are connected by divalent metal ions, mainly magnesium ions, due to the interaction of negative charges of phosphoric acid residues with positive charges ions (Spirin & Gavrilova, 1971). This interaction is essential for the stability of nucleic acid structures, especially in the ribosomes. Sugar molecules interact with each other due to the hydrogen bond between the nitrogenous bases attached to the sugar molecules. Being geometric forms, the constituents of nucleic acids interact with each other to form new geometric forms - new polytopes. Nitrogenous bases are known to be flat structures. However, it is not known how nitrogenous bases are oriented in space, whether their orientation depends on the type of nitrogenous base. Currently there is no information on this. There is also no information on how exactly the metal ions are located, connecting the phosphoric acid residues. It should be remembered here that the adopted three - dimensional model of the components and the nucleic acid molecule itself is only a model for visual perception. As it was shown earlier, the phosphoric acid residue is a polytope of dimension 4, and the sugar molecule has a dimension of 12. When two phosphoric acid residues or two sugar molecules are combined, the dimension of formation in each case increases by one. In this case, as will be shown, the arrangement of flat nitrogenous bases in the space of higher dimension will become clear.
The movement of triangles along a helix leads to the formation of polytopes with antiparallel edges. Consider an arbitrary triangle ABC on the plane. Choose some point O/ on the plane outside the triangle to his left. Let this point be the base of the axis of the helix passing through the triangle. Rotate the ABC triangle 180 degrees to the right, moving it up the helix, parallel to the initial plane. In the projection on the plane, both triangles ABC and the displaced triangle 
will be located as shown in Figure 1.
Figure 1.
Polytopes of dimension 3 with anti – parallel edges
It is easy to see that the edges of the triangle ABC and 
are antiparallel. It can now connect in space the vertices of the triangle ABC with the vertices of the triangle
so that there are no connections of the vertices with the same letters. In a projection on the plane the connection are represented by dotted segments. It can be seen that the connecting segments also break up into pairs of anti - parallel segments. Let us now verify that the image ABCA/B/C/ in Figure 1, along with the dotted segments, is a projection of a three - dimensional convex polytope. One use the Euler – Poincaré equation (Poincaré, 1895) for this aim
(1)
is the number of elements of dimension k in polytope of dimension n.
Key Terms in this Chapter
Simplex: A convex polytope, any two vertices of which are joined by an edge.
Dimension of the Space: The number of independent parameters needed to describe the change in position of an object in space.
Deoxyribonucleic Acid: A biopolymer, the monomer of which is the nucleotide.
Ribosome: The most important non-membrane organelle of a living cell, which serves for protein biosynthesis from amino acids in a given matrix based on genetic information provided by messenger RNA (mRNA).
Polytope: A polyhedron in the space of higher dimension.
Nucleotide: A phosphoric acid residue attached to sugar deoxyribose, to which one of the four nitrogen bases is attached also.
N-Cross-Polytope: A convex polytope of dimension n in which every two vertices opposite to the center of the polytope have no connection by an edges, and the remaining vertices joined by edges.