Generalization of space-filling bearings to arbitrary loop size

Abstract

A model for space-filling packing of discs rolling sliplessly on each other is presented. This model is an extension of the model presented by Herrmann et al to a basic loop size of an arbitrary even number of discs. This model might have application in geophysics, dense granular flows and turbulence. We show the existence and uniqueness of such constructions, analytically, for loop size up to eight, and numerically for bigger basic loop sizes, and make precise the algorithm to construct them. A classification of the solution is also given and extensive studies of the fractal dimension are performed, showing that it varies considerably with the base loop size.