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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.
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 ...
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.
Diagrams of cubic close packing (left) and hexagonal close packing (right). Imagine filling a large container with small equal-sized spheres: Say a porcelain gallon jug with identical marbles. The "density" of the arrangement is equal to the total volume of all the marbles, divided by the volume of the jug.
This yields the greatest possible packing density and the lowest energy state. — — Below is a candidate caption for use in Close-packing article, added 16:33, 26 February 2007 (and revised 20:15, 26 February 2007) — — Shown above is what the science of sphere packing calls a closest-packed arrangement.
An a × b rectangle can be packed with 1 × n strips if and only if n divides a or n divides b. [15] [16] de Bruijn's theorem: A box can be packed with a harmonic brick a × a b × a b c if the box has dimensions a p × a b q × a b c r for some natural numbers p, q, r (i.e., the box is a multiple of the brick.) [15]
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.
In mathematics, the theory of finite sphere packing concerns the question of how a finite number of equally-sized spheres can be most efficiently packed. The question of packing finitely many spheres has only been investigated in detail in recent decades, with much of the groundwork being laid by László Fejes Tóth.