Finding large databases of sphere packings
I have found a partial answer to finding a large database of sphere packings, thanks to the comment from permeakra!  

So far I have been obtaining .cif files from the large databases of crystal structures online such as, http://www.crystallography.net/.  Once I have the .cif files I use openbabel to convert those files to .xyz files.  From here I have a table that describes a crystal's structure by the x,y, and z coordinates of each particular element for example,

\begin{array}{c|c c c}
\ce{Mg}      &amp;   1.01054     &amp;   1.01054   &amp;     1.01054\\\hline
\ce{Al}      &amp;   1.01054     &amp; 1.01054     &amp;   1.01054\\\hline
\ce{Al}      &amp;   4.04217     &amp;   4.04217   &amp;     4.04217\\\hline
\ce{Mg}      &amp;   4.04217     &amp;   4.04217    &amp;    4.04217\\\hline
\ce{O}       &amp;   2.11390     &amp;   2.11390    &amp; 2.11390
\end{array}

But my problem from here is that babel has failed to return the ionic radius which I suppose can be retrieved online via wikipedia or something.  I posted another question that is more specific to .cif files which expands on this problem which can be found here, Extracting &quot;Crystal Radii&quot; from a CIF file.\

Limitations to this approach include that I can only find 3D sphere packings and it's a rather long and clunky procedure for obtaining lots of sphere packings.