The Mathematics of Maximizing Parallelepiped Box with Spherical Objects

This study investigated a special case of packing problems involving identical and fixed sized spherical objects and rectangular parallelepiped box. It determined the possible patterns of piling spherical objects into a parallelepiped box so that the box attains its maximum content. The study explored on the different ways of piling and identified those that yield content greater than that in ordinary piling. The effects of the dimensions of a box on whether or not it is possible to obtain a new pile that tends to increase the population density of the box is determined. The results show that there are smooth and behaved piling patterns that tend to increase the population density of the box from its default piling pattern. It is found that if the box can contain at least 5 spheres along its length, at least 3 spheres along its width and at least 4 spheres along its height, then it is possible to modify the default pile pattern so that additional spheres can be fitted into the box. The mathematics of maximizing the content of the box, or by increasing its population density, is given in theorems and corollaries. Also, the proofs of the theorems are supplied to establish their mathematical viability. The results also show that it is possible to generate mathematical models that establish deterministic algorithms of maximizing content, or increasing population density of the parallelepiped box. The mathematical models developed are recommended for use in calculating the maximal content of a given parallelepiped container. Finally, it is recommended that further investigations on the same topic be conducted as the present study is in no way exhaustive.