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

Nikolay N Klevanskiy,Sergey I Tkachev,Ludmila A Voloshchuk,Rouslan B Nourgaziev,Vladimir S Mavzovin

doi:10.3390/app11115128

Nikolay N Klevanskiy, Sergey I Tkachev + Show 3 more

Open Access

https://doi.org/10.3390/app11115128

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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.

Highlights

The problem with the regular placing of geometric objects on a plane according to classification [1,2] belongs to the class 2/V/I/C common problem of “Cutting and Packing”
By selecting one of the directions of regular packing (Figure 3), we indicate different rows of objects of the selected direction in color (Figure 4a)
Cognitive theory encompasses mental activiof finding heuristic judgments is related the use of cognitive ties, One suchmethod as the observation of various phenomena in antoenvironment; patternvisualizarecognition

Summary

Introduction

The problem with the regular placing of geometric objects on a plane according to classification [1,2] belongs to the class 2/V/I/C common problem of “Cutting and Packing”(C & P). The problem with the regular placing of geometric objects on a plane according to classification [1,2] belongs to the class 2/V/I/C common problem of “Cutting and Packing”. Russian research in the field of C & P dates back to the work of V.L. Kantorovich. Theoretical and practical studies were carried out by such scientists as V.F. Zalgaller, V.L. Rvachev, Yu.G. Stoyan, E.A. Stoyan, E.A Moukhacheva and their followers [2,3,4]

Methods

Discussion

Conclusion