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When one looks at the way cannon balls and oranges are normally packed by the military and by groceries, it seems intuively clear that there is no way anybody can pack these any tighter. However, it wasn't until recently that Hales proved, using a computer, that this intuition is correct.

Before Hales' proof, it was well known that the kissing number in 3D is 12, i.e., that the maximum number of nonoverlapping spheres that one could fit around one sphere, each touching the one sphere, is 12.

The way cannon balls and oranges are normally packed, every sphere touches 12 other spheres. Therefore, if one could pack spheres any denser than this, one would be able to rearrange these spheres and add another sphere to this space, which would cause at least one of these spheres to be touching 13 spheres. But this is impossible, since the kissing number is 12 in 3D. So I would think that this alone proves that it is impossible to pack spheres denser than the the way cannon balls and oranges are normally packed.

So my question is why doesn't the fact that the kissing number is 12 and the fact that the way cannon balls and oranges are normally packed every sphere touches 12 other spheres imply that the way cannon balls and oranges are normally packed is the most dense packing? Or does it?

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    $\begingroup$ "one would be able to rearrange these spheres and add another sphere to this space, which would cause at least one of these spheres to be touching 13 spheres". Why? I would rather say that if the density could be improved, we could pack the same amount of slightly bigger spheres into the same volume, so we would be able to pack the original spheres in a way that no two of them touch at all. $\endgroup$
    – fedja
    Commented May 19, 2019 at 3:05
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    $\begingroup$ But there's another way to achieve the kissing number 12: the centers of the 1+12 spheres are the 12 vertices of an icosahedron and the center of the icosahedron. $\endgroup$
    – LeechLattice
    Commented May 19, 2019 at 5:45
  • $\begingroup$ Fedja, you are correct, but your observation actually proves my point. You really think it is possible to repack the original spheres so that none of them touch at all? How far would the closest two spheres be from one another? $\endgroup$
    – Craig Feinstein
    Commented May 19, 2019 at 13:32

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Fedja is right that the gap in the argument is why the kissing number should increase if you increase the density. The 12-sphere kissing configuration is not unique, and one can imagine rearranging the spheres so as to keep the same kissing number but increase the density. Worse yet, one might decrease the kissing number while increasing the density. If the density can be increased at all, then one can always separate the spheres slightly, as Fedja points out, but that’s not the only way it can happen. This phenomenon of swapping kissing for density seems to occur naturally in higher dimensions. For example, in twelve dimensions there are packings with average kissing number greater than 770 (see page 140 of Conway and Sloane’s sphere packing book), but the densest packing known is the Coxeter-Todd lattice, which has kissing number 756. Focusing too much on kissing overly constrains the packing, and relaxing the local constraints a little bit gives you more flexibility to arrange the spheres well globally.

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  • $\begingroup$ Thank you, Henry. That is very counterintuitive. My question was based on my intuition that the most dense packing would have as many spheres kiss as possible. $\endgroup$
    – Craig Feinstein
    Commented May 19, 2019 at 19:43
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    $\begingroup$ Yeah, that’s part of why geometric optimization problems like sphere packing can be so subtle. Even very clear and believable intuitions can be difficult to justify rigorously, and sometimes they aren’t even true. $\endgroup$
    – Henry Cohn
    Commented May 19, 2019 at 19:47

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