The density of a (point) lattice sphere packing in n dimensions is the volume of the sphere in Rn divided by the volume of a fundamental region of the (point) lattice. We will give examples of packings where the centers of the spheres are points on the Zn;An , and Dn lattices, calculate their densities, center densities, and covering radii, and state the densest known lattice packings in 1 through 4 dimensions. We will then explain how the Z4 and D4 packings have enough room for a second copy of the respective packing to be placed next to the rst one without any sphere intersections, resulting in lattice packings with twice the densities of the originals. In addition, we will mention Rogers's upper bound and Minkowski's lower bound regarding sphere packing densities, and also prove Mordell's inequality.
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