The infinitary structure of an n-cube, global k-ary trees, and natural numbers are considered as a single genetic structure. A number of geometric characteristics of the shortest paths in an n-cube are specified and the properties of prime number symmetry among the natural numbers are studied on the basis of this structure.
Keywords: n-cube, symbolic matrix, global k-ary tree, k-tuples of natural numbers, difference tabloid, symmetry of prime numbers, incompatibility relation.