Abstract
For the first time it was established that for any convex polytope of higher dimension there is an integral equality in the transfer of information from low-dimensional elements to higher-dimensional elements and vice versa. This integral equality is called the law of conservation of incidents. In previous works of the author, this law was established for some polytopes of a particular kind. There is the incidence interpreted as the transfer of information from one material body to another. It is shown that the law of conservation of incidents is valid for both regular bodies and irregular bodies, which can be clusters of chemical compounds. The incident conservation law can serve as a mathematical basis for the recently discovered epigenetic principle of the transmission of hereditary information without changing the sequence of genes in DNA and RNA molecules.
TopIncident Conservation Law For N - Simplex
In n - simplex (Zhizhin, 2013, 2014 c, 2019 a) the number of elements with dimension d is
.
(3.1)For example, if d = 0, so
(3.2) This is the number of vertices in n - simplexes. If d = 1, so
(3.3) Key Terms in this Chapter
N-Cube: The convex polytope of dimension n in which each vertex incident to n edges.
N-Simplex: The convex polytope of dimension n in which each vertex is joined by edges with all remain vertices of polytope.
Nanocluster: A nanometric set of connected atoms, stable either in isolation state or in building unit of condensed matter.
Incidence Coefficients of Elements of Higher Dimension With Respect to Elements of Lower Dimension: The number of elements of a given lower dimension that are included in a particular element of a higher dimension.
Incidence Coefficients of Elements of Lower Dimension With Respect to Elements of Higher Dimension: The number of elements of a certain higher dimension to which the given element of a lower dimension belongs.
Polytope: Polyhedron in the space of higher dimension.
Dimension of the Space: The member of independent parameters needed to describe the change in position of an object in space.
N-Cross-Polytope: The convex polytope of dimension n in which opposite related of centrum edges do not have connection of edge.