Self-similar space-filling sphere packings in three and four dimensions

TL;DR

This paper introduces a method using inversive geometry to generate self-similar, space-filling sphere packings in two, three, and four dimensions, discovering numerous new topologies and analyzing their properties.

Contribution

It generalizes a construction method to higher dimensions and identifies new sphere packing topologies, including bearings, with detailed characterization.

Findings

Discovered 29 three-dimensional and 13 four-dimensional packings.

Identified 10 three-dimensional and 5 four-dimensional bearings.

Estimated fractal dimensions and analyzed contact networks of the packings.

Abstract

Inversive geometry can be used to generate exactly self-similar space-filling sphere packings. We present a construction method in two dimensions and generalize it to search for packings in higher dimensions. We newly discover 29 three-dimensional and 13 four-dimensional topologies of which 10 and 5, respectively, are bearings. To distinguish and characterize the packing topologies, we numerically estimate their fractal dimensions and we analyze their contact networks.