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Hexagonal closest packing of spheres in 3-space seems likely to be the tightest packing possible. Remarkably
(unless things have changed since last I investigated the matter), this is not proven. No one knows for sure if
there might not be a tighter packing (less empty space) of spheres in 3-dimensional space (remarkably, the
tightest packing of spheres in 24-dimensional space IS known). Anyway, once you've packed the spheres this
way throughout the entire universe, each sphere will have 12 kissing spheres (neighboring spheres with which
it is in contact), and at each point of contact there will be a plane tangent to the two spheres at that point.
So each sphere will have 12 tangent planes defined by its 12 kissing spheres. If we trim these planes where
they intersect we get a polyhedron = regular rhombic dodecahedron. These are the guys you see when you
open the 3-d animation above. Note that when the animation opens you're looking down on what appears to be
stacked parallel planes of spheres (inside the dodecahedra) packed in square configurations - ie., not the
tight triangular configuration you would expect from "closest packing". Only when you rotate things do you
see that from other angles there are parallel planes packed in tight triangles.