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In an effort to understand sphere packing in 24 dimensions, I figured that I should start with 1 and 2 first. The proof of 1 dimension is obvious, since we can achieve 100% density, and you cannot do better than that. It seems intuitive that the best circle packing in the plane is described by one in which their centers lie on points of the hexagonal lattice. However, Wikipedia and other sites on the Internet do not seem to offer a proof of this fact, and I do not have access to journals in which this is proved. Sadly, I have looked at the problem for a bit but cannot find a proof that does not go by exhaustion - ie examining the density for all of the representative members of the family of 2D lattices. Surely this cannot be the "best" way to prove such a seemingly simple concept, right?

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@WillJagy I am aware of Kepler's Conjecture but I am just looking for a proof of optimal-ness in 2D rather than 3D. I get that 3D is non trivial in comparison, but in my experience it has often been accepted as "fact" that the hex lattice describes the best packing in 2D simply by intuition. – tacos_tacos_tacos Nov 15 '12 at 2:53
@WillJagy it is elementary for you but not for me! Otherwise I wouldn't ask the question – tacos_tacos_tacos Nov 15 '12 at 2:58
And to further expand, I get that you could just check out all the representative lattices and see that the packing is optimal for the hex lattice. But I was thinking there would have to be much better, less exhaustive ways of doing so. – tacos_tacos_tacos Nov 15 '12 at 3:00
I'm just quoting Wikipedia. What books do you have on lattices? – Will Jagy Nov 15 '12 at 3:00
Just lecture notes. I am just looking for any available accessible proof that the 2D lattice is optimal that does not have to examine every individual case of lattice. What do you suggest? – tacos_tacos_tacos Nov 15 '12 at 3:04
up vote 2 down vote accepted

Perhaps you can get something out of Hai-Chau Chang, Lih-Chung Wang, A simple proof of Thue's theorem on circle packing. The abstract goes, "A simple proof of Thue theorem on Circle Packing is given. The proof is only based on density analysis of Delaunay triangulation for the set of points that are centers of circles in a saturated circle configuration."

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