Regular Two-Dimensional Packing of Congruent Objects: Cognitive Analysis of Honeycomb Constructions

Abstract

A new approach to investigate the two-dimensional, regular packing of arbitrary geometric objects (GOs),using cognitive visualization, is presented. GOs correspond to congruent non-convex polygons with their associated coordinate system. The origins of these coordinate systems are accepted by object poles. The approach considered is based on cognitive processes that are forms of heuristic judgments. According to the first heuristic judgment, regular packing of congruent GOs on the plane have a honeycomb structure, that is, each GO contacts six neighboring GO, the poles of which are vertices of the pole hexagon in the honeycomb construction of packing. Based on the visualization of the honeycomb constructions a second heuristic judgment is obtained, according to which inside the hexagon of the poles, there are fragments of three GOs. The consequence is a third heuristic judgment on the plane covering density with regular packings of congruent GOs. With the help of cognitive visualization, it is established that inside the hexagon of poles there are fragments of exactly three objects. The fourth heuristic judgment is related to the proposal of a triple lattice packing for regular packing of congruent GOs.

Investigation of the Possibilities of Optimizing the Model of Potential Containers to Increase the Speed of Placement of Orthogonal Polyhedra

“…The most common method of forming the placement of objects of complex shape is the method based on the construction of a hodograph vector function of dense placement [12,16]. To obtain a locally optimal solution to the problems of packing some basic geometric shapes (for example, rectangles, circles, ellipses, parallelepipeds, spheres, etc.…”