In two dimensions, there are two periodic circle packings for identical circles: square lattice and hexagonal lattice. In 1940, Fejes Tóth proved that the hexagonal lattice is the densest of all possible plane packings (Conway and Sloane 1993, pp. 8-9). The analog of face-centered cubic packing is the densest lattice packing in four and five dimensions. In eight dimensions, the densest lattice packing is made up of two copies of face-centered cubic. In six and seven dimensions, the densest lattice packings are cross sections of the eight-dimensional case. In 24 dimensions, the densest packing appears to be the Leech lattice. For high dimensions (~1000-D), the densest known packings are nonlattice.
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