Hidden symmetries in jammed systems

TL;DR

This paper uncovers hidden geometric symmetries in jammed systems through iterative transformations, revealing invariant structures and scale-invariant behaviors in their Voronoi tessellations.

Contribution

It introduces and analyzes two iterative processes, MIS inversion and coarsening, to reveal geometric symmetries and invariant properties in jammed packings.

Findings

MIS inversion leads to fixed points with uniform inscribed spheres

Coarsening reveals length-scale invariant geometric order

Hidden symmetries are associated with fixed points in transformations

Abstract

There are deep, but hidden, geometric structures within jammed systems, associated with hidden symmetries. These can be revealed by repeated transformations under which these structures lead to fixed points. These geometric structures can be found in the Voronoi tesselation of space defined by the packing. In this paper we examine two iterative processes: maximum inscribed sphere (MIS) inversion and a real-space coarsening scheme. Under repeated iterations of the MIS inversion process we find invariant systems in which every particle is equal to the maximum inscribed sphere within its Voronoi cell. Using a real-space coarsening scheme we reveal behavior in geometric order parameters which is length-scale invariant.