> >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

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