Hexagonal closest packing of spheres in a 3-dimensional space was (I believe) only recently proven to be the tightest packing possible
(remarkably, the tightest packing of spheres in 24-dimensional space has been known for quite some time).
Once you've packed the universe with spheres in this way, 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 in the animation at right
(each dodecahedron has half a face removed so you can see the inner sphere). When the ensemble rotates to the right
they separate a bit so you can better see the arrangement.