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Jamming is an emergent phenomenon wherein the local stability of individualparticles percolates to form a globally rigid structure. However, the onset ofrigidity does not imply that every particle becomes rigid, and indeed someremain locally unstable. These particles, if they become unmoored from theirneighbors, are called \textit{rattlers}, and their identification is criticalto understanding the rigid backbone of a packing, as these particles cannotbear stress. The accurate identification of rattlers, however, can be atime-consuming process, and the currently accepted method lacks a simplegeometric interpretation. In this manuscript, we propose two simplerclassifications of rattlers based on the convex hull of contacting neighborsand the maximum inscribed sphere of the radical Voronoi cell, each of whichprovides geometric insight into the source of their instability. Furthermore,the convex hull formulation can be generalized to explore stability inhyperstatic soft sphere packings, spring networks, non-spherical packings, andmean-field non-central-force potentials.

Local stability of spheres via the convex hull and the radical Voronoi diagram