CUTTING SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY, (2018)DOI: 10.1142/S0218348X18500135
Abstract
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.