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.