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Here there is a choice between separating the spheres into regions of close-packed equal spheres, or combining the multiple sizes of spheres into a compound or interstitial packing. When many sizes of spheres (or a distribution) are available, the problem quickly becomes intractable, but some studies of binary hard spheres (two sizes) are ...
The same packing density can also be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction. The Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular.
An arrangement in which the midpoint of all the spheres lie on a single straight line is called a sausage packing, as the convex hull has a sausage-like shape.An approximate example in real life is the packing of tennis balls in a tube, though the ends must be rounded for the tube to coincide with the actual convex hull.
A compact binary circle packing with the most similarly sized circles possible. [7] It is also the densest possible packing of discs with this size ratio (ratio of 0.6375559772 with packing fraction (area density) of 0.910683). [8] There are also a range of problems which permit the sizes of the circles to be non-uniform.
Random close packing (RCP) of spheres is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are packed randomly. For example, when a solid container is filled with grain, shaking the container will reduce the volume taken up by the objects, thus allowing more grain to be added to the container.
Sphere packing in a sphere is a three-dimensional packing problem with the objective of packing a given number of equal spheres inside a unit sphere. It is the three-dimensional equivalent of the circle packing in a circle problem in two dimensions.
Kepler wrote, in what today is known as the Kepler conjecture, that no other arrangement of spheres can exceed its packing density of 74%. [1] Mathematically, there is an infinite quantity of closest-packed arrangements (assuming an infinite-size volume in which to arrange spheres).
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, people are given: