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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]
In geometry, sphere packing in a cube is a three-dimensional sphere packing problem with the objective of packing spheres inside a cube. It is the three-dimensional equivalent of the circle packing in a square problem in two dimensions. The problem consists of determining the optimal packing of a given number of spheres inside the cube.
The hexagonal packing of circles on a 2-dimensional Euclidean plane. 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.
[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 ...
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 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.
A line can be drawn from the top corner of a cube diagonally to the bottom corner on the same side of the cube, which is equal to 4r. Using geometry, and the side length, a can be related to r as: =. Knowing this and the formula for the volume of a sphere, it becomes possible to calculate the APF as follows:
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 ...