Central Configurations of 100-Body Systems

Shape Dynamics — Barbour's Variety Functional

How to Read This Site

1. Start with the minimum. Near V_min, configurations are close to uniform packing.

2. Then move upward in age. Higher age means larger V - V_min, hence more visible structure: clusters, filaments, voids, and local motifs.

3. Compare mass ratios. Equal masses test pure packing. Unequal masses test how the same scale-invariant functional reorganises local structure.

The Variety

Given N particles with positions r_i, define the root-mean-square and mean-harmonic-length of all pairwise separations r_ij = |r_i − r_j|:

ℓ rms = \sqrt(Σ r ij ²) ℓ mhl = 1 / Σ(1/r ij)

The Variety is their ratio:

V = ℓ rms / ℓ mhl

V is purely geometric, scale-invariant, and dimensionless. It measures how far a configuration departs from uniformity. Masses affect only the centre-of-mass weighting, not V itself.

Critical Points

Central configurations are all critical points of V — minima, saddle points, and maxima.

Minima of V are always uniform or hexagonally-packed arrangements. Mathematically interesting, visually unremarkable.

Saddle points produce the visually striking configurations: filaments, clusters, voids, and hierarchical structure. These sit 1–100% above V_min.

Age

Age = (V − V_min) / V_min × 100%

A timeless complexity label. Age = 0 at the minimum (maximum uniformity). Higher age means more structure. For N = 100, V_min ≈ 522,312.

Maria's Electromagnetic Extension

Maria's new functional keeps the scale-invariant logic but adds an orientation-sensitive three-body term. Pairwise separations supply the electric-like part. Oriented triangles supply the magnetic-like part.

Instructional summary:

Pairs control radial structure.
Triangles create an emergent axis through their total oriented area.
The magnetic coefficient β controls how strongly the configuration flattens around that self-generated axis.

Maria's V_EM explained

Baseline, mechanism, and a controlled extension to oblate Jupiter/Saturn-like shapes.

6 figures: Maria's functional alone, then our oblate proxy side by side

This page is meant to read as a mechanism, not just a result: what the pair term does, what the triangle term does, how to read the side views, and what our extension changes.

Interesting Shapes (High Age)

These are saddle-point configurations with age ≥ 5% — shapes with visible structure: filaments, clusters, voids.

Variety, mass ratio 1:2000 — Interesting shapes

40 configurations with age ≥ 5%. One heavy body among 99 light ones.

Range: 5.3% to 97.5% above V_min

Representative high-age configurations (1:2000 mass ratio)

Variety, equal masses — Interesting shapes

34 configurations with age ≥ 5%. All 100 bodies have equal mass.

Range: 5.0% to 107.1% above V_min

Representative high-age configurations (equal masses)

All 500 Configurations

The complete set of optimised configurations from each search. Most land near V_min (uniform packing). The interesting shapes above were drawn from these.

Variety, mass ratio 1:2000 — All 500

Full set. 500 gradient-optimised configurations, sorted by age.

40 with age ≥ 5%, 460 near-minimum

Variety, equal masses — All 500

Full set. 500 gradient-optimised configurations, sorted by age.

34 with age ≥ 5%, 466 near-minimum

Method

Parameter	Value
N (particles)	100
Functional	V = ℓ_rms / ℓ_mhl
Optimiser	L-BFGS via JAX autodiff
Attempts per dataset	500
Initialisation strategies	Clustered, ring, line, cross, random
V_min (N=100)	≈ 522,312

Diverse initialisation is key. From uniform starts, >90% of optimisations land at the global minimum. Structured initialisations (ring, cross, clustered) access saddle-point basins that produce high-age configurations.