Abstract
This chapter first establishes the existence of integral equality in relation to the issue of the transmission of information by elements of lower and higher dimensions in the polytopes of higher dimension that describe natural objects. This integral equality is called the law of conservation of incidents. There is the incidence interpreted as the transfer of information from one material body to another. The fulfillment of the law of conservation of incidents for the n-simplex of the n-cube and the n-cross-polytope is proved in general terms. 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, 2019 a) the number of elements with dimension d is
.
(1)For example, if d = 0, so
(2)This is the number of vertices in n - simplexes. If d = 1, so
(3) Key Terms in this Chapter
N-Simplex: The convex polytope of dimension n in which each vertex is joined by edges with all remain vertices of polytope.
Dimension of the Space: The member of independent parameters needed to describe the change in position of an object in space.
Polytope: Polyhedron in the space of higher dimension.
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.
Nanocluster: A nanometric set of connected atoms, stable either in isolation state or in building unit of condensed matter.
N-Cube: The convex polytope of dimension n in which each vertex incident to n edges.
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.
N-Cross-Polytope: The convex polytope of dimension n in which opposite related of centrum edges does not have connection of edge.