  Any space-filling packing of spheres can be cut by a plane to obtain a
space-filling packing of disks. Here, we deal with space-filling packings
generated using inversive geometry leading to exactly self-similar fractal
packings. First, we prove that cutting along a random hyperplane leads in
general to a packing with a fractal dimension of the one of the uncut packing
minus one. Second, we find special cuts which can be constructed themselves by
inversive geometry. Such special cuts have specific fractal dimensions, which
we demonstrate by cutting a three- and a four-dimensional packing. The increase
in the number of found special cuts with respect to a cutoff parameter suggests
the existence of infinitely many topologies with distinct fractal dimensions.
