Abstract
Polytopes of the highest dimension corresponding to sugar molecules with five and six carbon atoms have been constructed. Based on the study of the geometry of these polytopes of dimensions 12 and 15, simplified three-dimensional models of sugar with five and six carbon atoms were obtained, and the chains of these molecules were analyzed. The geometry of protein molecules in various conformations was investigated, taking into account their highest dimension. It was found that Pauling's quasi-flat models do not reflect the spatial structures of the polypeptide chains of protein molecules. An opinion was expressed that the highest dimension of biomolecules is a consequence of the conditions for the emergence of life from the LUCA protocells more than 4.5 million years ago.
TopThe Structure Of Water
The water molecule has a peculiar shape. Oxygen in the second energy level p orbital has one quantum cell with a pair of electrons with opposite spins and two quantum cells with one electron in each of them. These two electrons combine with two electrons of the hydrogen atoms to form a covalent chemical bond. In addition, the oxygen atom has one more pair of electrons at the second energy level in the s orbital. The hydrogen atoms, which donated their two electrons to form a bond, remain sufficiently distant from the oxygen atom and have a positive charge +. Unshared electron pairs of the oxygen atom of the outer energy level naturally have a negative charge -. Conventionally, the shape of a water molecule can be represented as shown in Figure 1.
Figure 1. The shape of a water molecule
It is obvious that positively charged protons of other water molecules can join the orbitals with negative charges of the second energy level of the oxygen atom. The result is a tetrahedral structure with a center (Figure 2).
Figure 2. The tetrahedral structure with a center of molecules water with joined two protons
The tetrahedron with a center on Figure 2 has 5 vertices a0÷a4, f0=5;10 edges
;10 two - dimensional faces
;5 three-dimensional faces
To determine the dimension of a tetrahedron with a center, it is sufficient to use the Euler-Poincaré equation (3.3) in Chapter 3
(10.1)There n is dimension of a polytope P, f i is the number of elements with dimension i in the polytope P.
In this case f0=5, f1=10, f2=10, f3=5.
Key Terms in this Chapter
Hybridization of Electronic Orbitals: This is the interaction of the electronic orbitals of atoms entering the molecule, leading under certain conditions to the formation of higher dimensionality of molecules.
N–Simplex: The convex polytope of dimension n in which each vertex is joined by edges with all remain vertices of polytope.
Functional (Topological) Dimension of a Molecule: The dimension of a convex polytope, as a model of a molecule, at the vertices of which not only individual atoms but also functional groups of the molecule can be located.
Geometrical Image of a Chemical Compound: The geometrical image of a chemical compound (molecule) is a convex polytope, at the vertices of which atoms (or functional groups) are located. The edges of the polytope connecting the vertices correspond to the chemical bonds of the compound. The part of edges only carry a geometric function. They are necessary to give the molecule the image of a convex geometric figure. The dimension of the polytope is determined by the Euler-Poincare equation.
Tetrahedral Coordination of Electron Pairs: The location of the electronic pairs of the outer and the pre-outer electron layer at the vertices of the tetrahedron.
N–Cross-Polytope: The convex polytope of dimension n in which opposite related of centrum edges not have connection of edge.
Transitional Elements: Chemical elements in which electrons fill d - and f -orbitals of an atom ( d - and f -elements).