Dennis Clarke wrote:

> > >The problem is how to place the spheres in the box to get the maximum
> > >numbers of spheres that can fit in the box. I think that this is not a
> > >trivial problem, but is there a complete analytic solution at all (using
> > >with, height, depth and radius as parameters)?
> > >
>
> A crystaline matrix that has the body-centered cubic lattice construction is the
> most efficient way to pack spheres.  Mother nature has proven this in many
> materials that we run into all the time.  There are arguments for the FCC ( face
> centered cubic lattice ) but those arguments are typically based on materials
> that are under stress and demonstrate any of a number of dislocations in order
> to achieve plastic deformation.  If you have a cubic box with three axi a,b,c of
> equal length and angles alpha, beta, and gamma ( between each pair of axi ) that
> are also equal then the lattice structure of Si or NaCl (salt) would be best.
> If you get into a triclinic system where all axi and angles are different then
> the problem gets ugly very fast and even mother nature has a fit.  The best
> material that would get close would be Al2SiO5 but again, like salt, we are
> dealing with spheres of different sizes.  Kepler suggested this problem about
> 400 years ago and suggested a solution that was correct, without the benefit of
> x-ray crystalography or modern numerical methods.  His solution was based on the
> simple observation that a man selling grape-fruit in the market will stack in a
> close packed haxagonal plane and then add another layer on top of that one
> slightly shifted to allow the next layer to sit in the crevass between three
> other touching grape-fruit on the lower layer.  Why?  Because it works.  The
> grape-fruit don't tumble over the edge and this seems to be a good arrangement.
> There is plenty of lost volume in this packing but such is life.  If the sphere
> radius is much smaller than the box dimensions then we have a solution.
>
> Dennis Clarke

I'm no math expert (high school sophomore) but I think this is right. Also, the most
efficient packing would depend on the ratio of the box size to the sphere size. For
example, if you have a 2x2x2 box and spheres of diameter one, obviously you could
pack at most eight spheres using a cube-type pattern. However, if the apheres are
much smaller, I believe the best arrangement is the hexagonal planes. The best way
to pack them is the way you would build a tetrahedron out of spheres, hexagonal
planes with each sphere resting on the three below it. There is a third way,
hexagonal planes where each sphere rests on two below it, so as to create hexagonal
tilings along the x-z plane and the x-y plane. I think this also depends on the
box-sphere ratio, but I'm pretty sure the tetrahedral stacking is best for
infinitessimal spheres.

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