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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 ...
Packing identical rectangles in a rectangle: The problem of packing multiple instances of a single rectangle of size (l,w), allowing for 90° rotation, in a bigger rectangle of size (L,W) has some applications such as loading of boxes on pallets and, specifically, woodpulp stowage. For example, it is possible to pack 147 rectangles of size (137 ...
The strictly jammed (mechanically stable even as a finite system) regular sphere packing with the lowest known density is a diluted ("tunneled") fcc crystal with a density of only π √ 2 /9 ≈ 0.49365. [6] The loosest known regular jammed packing has a density of approximately 0.0555. [7]
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 packing constant of a geometric body is the largest average density achieved by packing arrangements of congruent copies of the body. For most bodies the value of the packing constant is unknown. [1] The following is a list of bodies in Euclidean spaces whose packing constant is known. [1]
[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 ...
The atomic packing factor of a unit cell is relevant to the study of materials science, where it explains many properties of materials. For example, metals with a high atomic packing factor will have a higher "workability" (malleability or ductility ), similar to how a road is smoother when the stones are closer together, allowing metal atoms ...
They also reported a glassy, disordered packing at densities exceeding 78%. For a periodic approximant to a quasicrystal with an 82-tetrahedron unit cell, they obtained a packing density as high as 85.03%. [12] In late 2009, a new, much simpler family of packings with a packing fraction of 85.47% was discovered by Kallus, Elser, and Gravel. [13]