Gauss was the first to prove anything about the Kepler conjecture. He showed that if all of the centers of the balls of a packing are aligned along the points of a lattice, then it can do no better than the face-centered cubic packing.

Gauss's name confers an undeserved prestige to this elementary result. The proof takes only a few lines and requires no calculations. In the best case, it will certainly be true that two balls will touch each other. Once two balls touch, the lattice constraint forces the balls to touch along long parallel strings of balls, like a thick row of marshmallows on a roasting stick. In the best case, it will also certainly be true that two of the long parallel beaded strings will touch. The lattice constraint forces the balls to be laid out in identical parallel plates. The centers of four balls in the plate form a parallelogram, as shown in Figure 3. The parallel plates should be set one on the other so that the plates are as close as possible. A ball $D$ of the next layer is set in the pocket between three balls $A,B,C$ in the layer below, so that it touches all three. The triangle $\mathrm{ABD}$ formed by the centers is equilateral.

We now change our point of view. We view all of the balls as arranged in planes parallel to $\mathrm{ABD}$. In each of those layers, the centers of the balls repeat the pattern of the equilateral triangle, $\mathrm{ABD}$. The balls of one layer should be nestled in the pockets of the layer before, so that each ball rests on three below it. The lattice this describes is the face-centered cubic.