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The upper bound for the density of a strictly jammed sphere packing with any set of radii is 1 – an example of such a packing of spheres is the Apollonian sphere packing. The lower bound for such a sphere packing is 0 – an example is the Dionysian sphere packing. [27]
[1] [2] Highest density is known only for 1, 2, 3, 8, and 24 dimensions. [3] Many crystal structures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling the spaces between them. The cubic and hexagonal arrangements are very close to one another in energy, and it may be difficult to ...
Packing density Optimality Arrangement Diagram Exact form Approximate 1 1.0000 1 Trivially optimal. Point: 2 0.5000 0.25 ... Sphere packing in a sphere.
However, the optimal sphere packing question in dimensions other than 1, 2, 3, 8, and 24 is still open. Ulam's packing conjecture It is unknown whether there is a convex solid whose optimal packing density is lower than that of the sphere.
The higher the packing density, the less empty space there is in the packing and thus the smaller the volume of the hull (in comparison to other packings with the same number and size of spheres). To pack the spheres efficiently, it might be asked which packing has the highest possible density.
Sphere packing in a cylinder is a three-dimensional packing problem with the objective of packing a given number of identical spheres inside a cylinder of specified diameter and length. For cylinders with diameters on the same order of magnitude as the spheres, such packings result in what are called columnar structures .