A method of optimal lattice packing of congruent oriented polygons in the plane

An approach based on an application of the δ-function apparatus is proposed for constructing a mathematical model of the problem of optimal lattice packing of congruent oriented polygons in the plane. The feasible solution region is contained in the set of pairs of points of the δ-function 0-level surface of the polygon. The region is partitioned into a finite number of subregions corresponding to pairs of linear segments of the δ-function 0-level surface. The frontiers of the subregions are generated by the conditions of non-overlapping of polygons. Only four such conditions imply non-overlapping of each pair of polygons in a lattice packing. The solution can be attained only in a finite number of points on the frontiers of these subregions. Hence, it is possible to use an organized sorting of such points with discarding of subregions whose objective function upper bound is lower than the record.