Two cities with the same average population density. In one, a neat grid of blocks; in the other, a maze of narrow alleys and unexpected squares. On a global map they look like twins, but life inside flows quite differently. This is roughly how two central configurations (critical points of variety in shape space) of 5000 equal-mass particles, studied in a new paper, are arranged. Their global clustering measure, variety, differs by just 1.686%: 0.5857 and 0.5956. This quantity, introduced in the approach of Barbour, Koslowski, and Mercati, serves as an indicator of the gravitational arrow of time in the Newtonian N-body problem with zero energy, momentum, and angular momentum. It grows in both directions from the Janus point — the moment of minimal system size. But whether such a description is complete remained an open question.

To look inside these "cities," the authors introduced a scale-invariant measure — the anisotropy of the six nearest neighbors, A6. For each particle, they constructed the tensor of second moments of neighbor displacements, and A6 was defined as the ratio of the difference to the sum of its eigenvalues: 0 for an isotropic environment (like points on a uniform grid), 1 for a collinear one (like beads on a string). The mean A6 for the configuration with lower variety was 0.1331—an almost isotropic background. For the configuration with higher variety, it was 0.3419, a difference of 156.8%. The medians differed even more: 0.1133 versus 0.3226.

In the first configuration, no particle had A6 above 0.5—everything like a city with smooth transitions. In the second, 18.1% of particles did — entire neighborhoods of elongated structures. The probability distributions of A6 have a Kolmogorov–Smirnov distance of 0.557, confirming their strong difference. High A6 values concentrate along filaments and curved arcs, like narrow alleys threading through the second city. The sign of the difference persisted for all tested neighborhood sizes (k from 4 to 12), although the magnitude varied from 8.8% to 161.8%. A robustness test against very close pairs (replacing connected components with centroids) reduced the mean A6 to 0.3169, but the difference from the first configuration remained at 138.1%.

The result shows that the global scalar quantity variety does not uniquely determine the local geometric organization of a system. Two "cities" with nearly identical average density can be fundamentally different in layout: one a uniform scatter, the other a labyrinth of filaments. This does not diminish the role of variety as an arrow indicator, but points to the need for additional, local relational observables for a more complete description of structural evolution. The gravitational arrow may be a multidimensional phenomenon, where global clustering is only the first level in a hierarchy of descriptors.

So we return to the image of two cities. On a map they are almost indistinguishable, but inside, one is a quiet suburb, the other a tangle of tense highways. To understand their fate, knowing the average density is not enough: you need to see the geometry of the streets.