Higher dimensional Apollonian packings, revisited (Q1663773)

The Apollonian packing is a packing of nonoverlapping disks of unequal size that fill a larger disk except from a fractal set of measure zero. The process is inductive, and begins with mutually tangent disks of equal size at a simplex position. At every step, disks of maximal size are placed in the remaining gaps, and this process goes on indefinitely. This inductive process works smoothly in 3 dimensions. In higher dimensions, however, there might appear overlapping hyperspheres at one of the steps. The author proposes a process that produces an Apollonian packing in higher dimensions, inspired by results in arithmetic geometry (specifically, ample cones of \(K3\) surfaces). This process generalizes the Apollonian packings in 2 and 3 dimensions, and for small dimensions up to 6 satisfies certain properties that are expected to hold for such a packing (for example, there is an ``initial'' configuration of hypersheres in the simplex position, and two hyperspheres can only intersect tangentially). The author works in the hyperbolic geometry setting; in particular, he considers intersections between hyperplanes in a lattice arrangement and a hyperboloid (the hyperbolic sphere of radius i). The author then proves that the projection of the above configuration in the compactification of the hyperboloid produces a configuration of hyperspheres in a Euclidean space, which satisfies the desired properties of an Apollonian packing in dimensions 4, 5, and 6.