Cutting self-similar space-filling sphere packings

TL;DR

This paper investigates how cutting self-similar space-filling sphere packings with a plane affects their fractal properties, revealing general rules and special cases with unique fractal dimensions.

Contribution

It proves that random cuts reduce fractal dimension by one and identifies special cuts that are also generated by inversive geometry, expanding understanding of fractal packings.

Findings

Random hyperplane cuts decrease fractal dimension by one.

Special cuts can be constructed via inversive geometry.

Numerous special cuts suggest infinitely many topologies with distinct fractal dimensions.

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.