Regular packing of congruent polygons on the rectangular sheet

This paper deals with the problem of searching for the fragment of dense lattice (dense double lattice) packing of congruent polygons on the rectangular sheet which is optimal by the dense factor. In doing so we consider only those packings for which the polygons form rows (columns) parallel to one of the sheet sides. The problem arises in many branches of industry in particular in footwear industry. The mathematical model of the problem is built. Simplification of the mathematical model, reducing of the problem dimensionality and its decomposition on a number of independent subproblems is done on the basis of analysis of the properties of the mathematical model. The approximate method to solve the problem is suggested. The example of solving of the real problem is given.