Higher dimensional Apollonian packings,revisited

The Apollonian circle and sphere packings are well known objects that have attracted the attention of mathematicians throughout the ages. The historically natural generalization of the procedure for generating the packing breaks down in higher dimensions, as it leads to overlapping hyperspheres. There is, however, an alternative interpretation that allows one to extend the concept to higher dimensions and in a unique way. For relatively small dimensions (2 through at least 8), those packings can be thought of as ample cones for classes of K3 surfaces. We describe the packings in some detail for dimensions 4 (with plenty of pictures), 5, and 6.