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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]
Sphere packing in a sphere is a three-dimensional packing problem with the objective of packing a given ... Packing density Optimality ... Toggle the table of ...
An example uniform structure and its corresponding rolled-out contact network. The identical vicinity of each sphere defines a uniform structure. A uniform structure is identified by each sphere having the same number of contacting neighbours. [22] [1] This gives each sphere an identical neighbourhood. In the example image on the side each ...
[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 ...
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.
For example, in the case =, it is known that the optimal packing is not a tetrahedral packing like the classical packing of cannon balls, but is likely some kind of octahedral shape. [ 1 ] The sudden transition in optimal packing shape is jokingly known by some mathematicians as the sausage catastrophe (Wills, 1985). [ 4 ]
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.
The optimal packing density or packing constant associated with a supply collection is the supremum of upper densities obtained by packings that are subcollections of the supply collection. If the supply collection consists of convex bodies of bounded diameter, there exists a packing whose packing density is equal to the packing constant, and ...