rattling neighbors. When p is close to unity, the fraction of rattlers is approximately 2 . 6% and 1 . 2% for two and three dimensions, respectively. As p moves away from unity, the fraction of rattlers decreases quickly and practically vanishes for large p (e.g., p > 2 . 75). This behavior results from the increasing protuberance of the particle shape, which makes it more difficult to form isotropic cages and also requires more average contacts per particle to achieve jamming. Note that rattlers are excluded when reporting average contact numbers in the following discussion. 

## B. Packing Density 

The rapid increase of the density is mainly due to the broken rotational symmetry of the particles. In particular, the cubic-like (square-like) and octahedral-like particles are more efficient to cover the space than spheres (circular disks), i.e., near the jamming point the particles can rotate to accommodate the neighbors by orienting the “far corners” to fill the available gaps and thus cover more space. For small values of p , the increase in φ is also attributed to the expected increase in the number of contacting neighbors per particle, which means locally more particles can be packed in a given volume. The manner in which rotational symmetry is broken in superball packings is distinctly different from that in ellipsoid packings. For example, the asphericity γ [27, 28], defined as the ratio of the radii of circumsphere and insphere of a nonspherical particle, is always bounded and close to unity for all values of p for superballs, while it can increase without limit as the largest aspect ratio α grows for ellipsoids. For very elongated or flake-like ellipsoids with large aspect ratios, the effect of a very anisotropic exclusion volume becomes dominant and causes the density of random ellipsoid packings to decrease. By contrast, the shape of superballs becomes more efficient in filling space as the deformation parameter deviates more from unity and thus results in a monotonically increasing density. The nonanalyticity of φ at p = 1 is also associated with the broken symmetry of superdisks and superballs. This nonanalytical behavior has also been observed in the optimal packings of these particles realized by various Bravais lattices [25, 26]. This stands in contrast to the densest known ellipsoid packings, which are periodic packings with a two-particle basis possessing a smooth initial increase of φ$_{max}$ as the aspect ratio moves away from unity [21]. 

