The first stage consisted of studying identical dual spaces, their properties and parameters trying to develop insights into their nature and order.
The tunnel-like periodic spaces, represented by networks, are defined within the Euclidean three-dimensional space, and are composed of periodic cells. This property indicates the relation between the network-pairs and the surface separating them and the symmetry groups acting on this space.
The smallest repetitive cell that is derived from a periodical space using the symmetry operations of the symmetry group that acts on this space is called “Elementary Periodic Region” (E.P.R.). The E.P.R. contains complete representation of all the phenomena taking place within the “periodic complex” and particularly, representation of the whole periodic space, its symmetry group, the two complementary (identical) subspaces, the self dual lattice-pairs characterizing them, as well as the surface partition in between.
At this stage of the research, the topological properties of smooth 2-manifolds in general, and of smooth 2-mainfolds which divide the space between two dual networks in particular, were explored.