About New Internal Symmetries of the Integer Lattice ($\mathrm{Z^N}$)

Abstract

The numerous internal symmetries are found in N-dimensional integer lattices ($\mathrm{Z^N}$). 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 $\mathrm{Z^N}$ (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 $\mathrm{Z^N}$ with large N is stated.