Scale Invariance, Variety and Central Configurations

Scale invariance has received very little attention in physics. Nevertheless, it provides a natural conceptual foundation for a relational understanding of the universe, where absolute size loses meaning and only dimensionless ratios retain physical significance. We formalize this idea through the $N$-body problem, introducing a scale-invariant function--the variety, $V$--built from the square root of the center-of-mass moment of inertia and the Newtonian potential. Critical points of $V$, known as central configurations, correspond to special particle arrangements that preserve their shape under homothetic collapse or expansion. Numerical exploration of these critical points reveals that even slight deviations from the absolute minimum of $V$, which corresponds to a remarkably uniform configuration, lead to the spontaneous formation of filaments, loops, voids and other patterns reminiscent of the cosmic web. This behavior is a consequence of the intrinsic structure of shape space--the space of configurations modulo translations, rotations and dilatations--in which regions of higher variety act as attractors. Our results suggest that scale-invariant dynamics not only captures the relational nature of physical laws but also naturally generates organized patterns, offering a novel perspective on the formation of cosmic structures and on the emergence of a gravitational arrow of time from scale-invariant, relational dynamics.