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It would be interesting to look at the "believed optimal" schemes for packing spheres in each dimension and analyze how the packing density changes.

MathWorld has a really good article [0] with density numbers from R2 through R8 if you're interested.

Thanks! You might also be interested in this thing, sort of a slideshow, that I posted up above: http://www.ams.org/programs/students/wwtbam/cohn-slides-2014...