# Models for Packing Circular Objects Into Rectangular Air Spaces

Serano L. Oryan

## Abstract

spaces are here presented. The study explores on different packing patterns that tend to increase the population density of a given rectangular space by way of systematic repositioning of the objects and by applications of some trigonometric concepts in determining the effect of repositioning to the vertical distances between the centers of the objects across the contiguous rows. The results showed that if the dimension (rc) of a rectangular space is rc = 8x5, where the unit of measure of the space is the diameter of a circular object, then the default arrangement of the objects can be repositioned so that the content of the space is maximum. The results also showed that a rectangular space attains its maximum content if row, r, is a multiple of [ 7.464 ] and column c ≥ 5. In order to determine whether the population density of a rectangular space can be increased by applying some packing patterns, two mathematical models are developed, through which the exact number of objects that can be accommodated in a space is calculated. This study shows that there are deterministic mathematical models of calculating the maximal number of identical circular objects that can be packed into rectangular spaces. In cases, however, where the rectangular container provides empty space either on the row or column or both with length less than the diameter of one circular object, then adjustment on the models may be made. Hence, it is recommended that such particular cases have to be further explored in future study.

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