About New Internal Symmetries of the Integer Lattice (ZN)

Kornyushkin A

doi:10.4172/2090-0902.1000178

Abstract

The numerous internal symmetries are found in N-dimensional integer lattices (ZN). The relation of these symmetries with the new mathematical category named the Masks (or Neighborhoods) is shown. A set of definitions for the Correct Masks and Perfect Masks is presented; the identity between the Correct and Perfect Masks is hypothesized. The relationship between the Perfection of the Mask and the new category named “Mathematical String” is shown. The Correctness of the several Masks in ZN (N=1,2) is proven and a simple method to find the Correctness for all other N is outlined. The hypothesis of high population density of Perfect Masks in integer lattices ZN with large N is stated.

Highlights

A new set of internal symmetries for N-dimensional integer lattices ZN are inseparably linked to a new mathematical category named the Mask (Neighborhood)
We remind that the number of symmetries is greater than the number of central-symmetric Masks in the N-dimensional integer lattice ZN
The author developed the approach to finding the Correct Masks for lowdimensional ZN

Summary

Introduction

A new set of internal symmetries for N-dimensional integer lattices ZN are inseparably linked to a new mathematical category named the Mask (Neighborhood). From point 5) it follows that the Automaton must be able to make at least the first step, the Tables should contain all the 2n+1 states formed from the letters of the same type “x”. The second row (Figure 6B) shows two examples of Correct Masks for n=5 composed by the addition of points (x=1, y=1) or (x=2, y=0).

Results

Conclusion