Packing problems

Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real life packaging, storage and transportation issues. Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap. Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real life packaging, storage and transportation issues. Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap. In a bin packing problem, you are given: Usually the packing must be without overlaps between goods and other goods or the container walls. In some variants, the aim is to find the configuration that packs a single container with the maximal density. More commonly, the aim is to pack all the objects into as few containers as possible. In some variants the overlapping (of objects with each other and/or with the boundary of the container) is allowed but should be minimized. Many of these problems, when the container size is increased in all directions, become equivalent to the problem of packing objects as densely as possible in infinite Euclidean space. This problem is relevant to a number of scientific disciplines, and has received significant attention. The Kepler conjecture postulated an optimal solution for packing spheres hundreds of years before it was proven correct by Thomas Callister Hales. Many other shapes have received attention, including ellipsoids, Platonic and Archimedean solids including tetrahedra, tripods (unions of cubes along three positive axis-parallel rays), and unequal-sphere dimers. These problems are mathematically distinct from the ideas in the circle packing theorem. The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere. The counterparts of a circle in other dimensions can never be packed with complete efficiency in dimensions larger than one (in a one-dimensional universe, the circle analogue is just two points). That is, there will always be unused space if you are only packing circles. The most efficient way of packing circles, hexagonal packing, produces approximately 91% efficiency. In three dimensions, close-packed structures offer the best lattice packing of spheres, and is believed to be the optimal of all packings. With 'simple' sphere packings in three dimensions ('simple' being carefully defined) there are nine possible defineable packings. The 8-dimensional E8 lattice and 24-dimensional Leech lattice have also been proven to be optimal in their respective real dimensional space. Cubes can easily be arranged to fill three-dimensional space completely, the most natural packing being the cubic honeycomb. No other Platonic solid can tile space on its own, but some preliminary results are known. Tetrahedra can achieve a packing of at least 85%. One of the best packings of regular dodecahedra is based on the aforementioned face-centered cubic (FCC) lattice. Tetrahedra and octahedra together can fill all of space in an arrangement known as the tetrahedral-octahedral honeycomb.