A class of two-dimensional regular graphs called circulants and its special case of the double-loop networks are considered. Such graphs provide a practical interest as reliable interconnection networks for the multimodule supercomputer systems. A solution to the problem of determining the best double-loop networks with the minimum of a diameter and mean distance for the structures of computer systems is considered. A new method of geometrical representation (visualization) of the optimal circulants at the plane and connected with it complete design of transformations and movements generating the optimal graphs are obtained. Some new classes of analytical representation of optimal circulants are proposed. New results on solution of a problem of existence for the optimal two-dimensional loop networks are presented.