Dense packing
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Corresponding Wikipedia article: Sphere packing
The dense packing of equal balls is such arrangement of them, when each ball is contacting to at least one neighbor, and its displacement with respect to the neighbors is only possible with breaking a contact to any neighbor. If this arrangement can be modified without breaking contacts between balls, then packing is not dense. For example, the non-dense square arrangement can be deformed to a rhombic arrangement.
The dense packing increases solidity and hardness. The mechanical force, applied to one element, is distributed to a lot of its neighbors, decreasing a tension and a deformation in the structure.
The elementary cell of the dense packing is the equilateral triangle. The spatial dense packing cell is the tetrahedron. The tetrahedron-based packing is known as the hexagonal close packing, because triangles in the same plane produce the regular hexagons.
The extremely dense (without gaps) spherical packing of 13 balls has an icosahedron shape, which consists of 20 tetrahedrons. The higher number of balls cannot provide the extremely dense and uniform arrangement in a spherical surface and volume, because the convex regular polyhedrons with a higher number of the triangular faces don’t exist.
See also
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