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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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."