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Background: If you pack $n^2$ circles in a square, for 1, 4, 9, 16, 25, and 36 circles, the densest packing is with the circles stacked in an $n\times n$ grid. But for 49 circles the densest packing becomes irregular - for large $n^2$ it approaches a hexagonal packing (see http://hydra.nat.uni-magdeburg.de/packing/csq/csq.html#overview).

Question So my question is when (I am assuming it does at some point), if packing $n^3$ spheres in a cube, does regular cubic packing become less efficient than some irregular approximation of fcc or hexagonal packing?

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  • $\begingroup$ I've made a couple of dimensionality fixes (assuming that you were talking about packing $n^2$ circles, not spheres); please make sure I've got the correct fixes. Also, nice question! $\endgroup$
    – Steven Stadnicki
    Dec 1, 2013 at 23:37
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    $\begingroup$ randomwalk.de/sphere/incube/index_6.htm suggests that at least for $n^3=27$, the cubic lattice is still densest. That same data seems to suggest that the cubic packing isn't best for $n^3=64$, but it's hard to tell for certain. $\endgroup$
    – Steven Stadnicki
    Dec 1, 2013 at 23:40
  • $\begingroup$ Thanks @Steven Stadniki - that was what I am looking for. $\endgroup$
    – Penguino
    Dec 1, 2013 at 23:47

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