Lattice Packings of Circles into Squares.

Points of a regular lattice serve as centers of congruent circles, packed without overlapping in the plane. The packings are designed to fit a tiling of the plane by squares, each square containing the same number N and arrangement of circle centers (a circle may protrude outside its square but then another circle, centered outside the square, will intrude across the opposite side). Packings without the symmetry of a square tiling can have density rho sub 0 = .9096... but, in the problem considered here, only somewhat smaller densities are possible, especially if N is not large. Special values of N, related to rational to sqrt 3, allow lattice packings with densities abnormally close to rho sub 0. These densities are within 0(1/N) of rho sub 0 for large N. Lattice packings are not the densest possible.