Central Configurations

2D — Maria Comparison

3D

Equal Masses

Mass Ratio 1:80

N=100, varying β

N=200

What This Measures

The “bond angle” here is a packing statistic, not a chemical bond. It answers: when 100 points settle to minimum shape complexity under the gravitational potential, how are they locally arranged? The nearest-neighbour angle is determined by how particles pack in 3D under the constraint of the pre-shape sphere. We mark water’s 104.5° and the ideal tetrahedral angle 109.5° as reference lines — not because we expect to recover water from gravity, but because both are signatures of the same underlying geometry: optimal packing of repulsive points in three dimensions.

Equal Masses — 107.9°

All 500 starts converge to the same minimum — the landscape has a single basin. The mean bond angle is 107.9°, close to the ideal tetrahedral angle (109.5°). This is expected: 100 equal-mass particles minimising ∑ 1/r_ij on a sphere is essentially the Thomson packing problem, whose local structure is approximately tetrahedral. The proximity to water’s 104.5° is a consequence of shared packing geometry, not a claim about chemistry.

1:10 — 98.1°

Still a single basin. Introducing mass asymmetry shifts the packing: heavy particles claim more “space” in the potential (their m_im_j/r terms dominate), compressing the angles of lighter particles’ local neighbourhoods. Two light-particle clusters emerge.

1:80 — 86.6°

The landscape bifurcates — 3 distinct minima from 500 starts. The dominant minimum (434/500 starts) has 7 light-particle clusters and a mean angle of 86.6°. Two rarer configurations exist at slightly higher complexity. The angle compression continues: heavy particles force light particles into tighter local arrangements.

1:2000 — 71.4°

Two distinct minima. The dominant (447/500 starts) has a mean angle of 71.4° — well below tetrahedral, reflecting the extreme mass asymmetry. At this ratio (comparable to proton/electron), heavy particles form a rigid scaffold and light particles are confined to tight clusters in the interstices, compressing local angles far below the equal-mass packing value.

Angle Trend with Mass Ratio

The trend is monotonic: 108° → 98° → 87° → 71°. The equal-mass angle reflects Thomson-like packing on the shape sphere. As mass ratio increases, the heavy-particle scaffold increasingly constrains light-particle geometry, compressing bond angles. The question for further work is whether specific mass ratios select for angles that match known molecular geometries — and if so, whether that is coincidence (shared packing geometry) or something deeper (the shape potential encoding chemical structure through ratios alone).

The Main Result: Landscape Bifurcation

For mass ratios below 1:50, all 500 starts converge to the same minimum — the shape potential has a single basin. At 1:50 and above, two distinct minima appear. The landscape topology changes with mass ratio.

This is not a numerical artifact. The two minima have measurably different V_S values, different local structure, and different bond-angle distributions. The lower-V_S minimum attracts the vast majority of starts (434–447 out of 500); the higher one is a rarer basin.

What the Two Minima Look Like

For each bifurcated family, we show the two distinct configurations side by side. Left is the lower-V_S (dominant) minimum, right is the higher-V_S (rare) one. Both are PCA projections of 3D positions. Red = heavy particles, blue = light.

The key observation: the heavy-particle scaffold (red dots) is similar in both minima. What differs is how the light particles (blue dots) arrange themselves within that scaffold. This is the “fossil” idea — the scaffold is a structural trace that persists across the two basins. It is NOT a time sequence. It is two static configurations that share partial structure.

Measuring Structural Similarity

To quantify how much the two minima share, we compute three similarity scores between them:

• Shape persistence (Procrustes) — align the two configs optimally, measure residual distance. Higher = more similar overall shape.
• Angle similarity — compare the distribution of nearest-neighbour bond angles. Higher = same local packing geometry.
• Radial similarity — compare how particles distribute by distance from centre. Higher = same density profile.

Angle similarity is consistently high (0.85–0.90): the two minima have nearly identical local packing. Shape persistence is lower (0.50–0.63): the global arrangement differs. This is what we’d expect if the scaffold persists but the light-particle clusters rearrange.

Bond Angle and Structure vs Mass Ratio

Left: the mean nearest-neighbour bond angle at the dominant minimum, measured across all particles. It decreases monotonically: 108° (equal) to 71° (1:2000). At equal masses this is Thomson-like sphere packing (near the tetrahedral angle 109.5°). At high mass ratios, the heavy-particle scaffold compresses light-particle local geometry. Blue bars = single-basin families; red = bifurcated (2 minima).

Right: a composite structural score measuring how “non-trivial” each configuration is. It combines: asphericity (deviation from spherical shape), nearest-neighbour distance variation (uneven spacing = internal landmarks), cluster count (distinct sub-groups within the config), and low radial entropy (concentrated rather than uniform density). The score is not calibrated to any external standard — it ranks configurations relative to each other. Higher mass ratios produce more internally structured configurations.

Initialisation Independence

A key concern (raised by Maria): for large mass ratios, do light particles simply freeze at their starting positions? We tested three deliberate initialisation strategies at 1:2000 and 1:80, 100 starts each:

• Random normal — our default, no structured placement
• Light outside — light particles start at radius ~2, heavy near centre
• Light inside — light particles start near centre, heavy at radius ~2

All three strategies converge to the same dominant minimum with 85–92% of starts. The dominant V_S values agree to 4+ decimal places. Mass preconditioning eliminates initialisation dependence.

Limitations

At N=100 with 500 gradient-descent starts, we find at most 2 distinct minima per mass-ratio family. This is not enough for rich “age tracks” or “growth sequences.” To test whether longer sequences of partially-persistent configurations exist, we would need:

• Larger N (more complex landscape with more basins)
• Saddle-point methods (GAD/HiSD) to find transition states between basins
• More mass species (not just 2-species 50/50 split)
• Systematic continuation methods that trace solution branches

Face-like or household-object forms do not emerge from pure N-body at N=100. The strongest recurring motifs are mass-shell segregation and heavy-particle scaffolds.