Gravitational molecules at minimum shape complexity
Critical points of VS = √Icm × W on the pre-shape sphere. Heavy particles red, light particles blue. 3D positions projected via PCA. N = 100 to 10,000.
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Shell structure strengthens with N.





2D + 3D, 5 mass ratios





5 mass ratios





5 mass ratios





5 mass ratios





5 mass ratios





500 starts, angle distributions, recurring motifs





Jupiter/Saturn-like flattening





Maria’s 3-body triangle vorticity





| Config | Equal | 1:2 | 1:10 | 1:80 | 1:2000 |
|---|---|---|---|---|---|
| N=100, 2D | 52,231 | 142,784 | 3.6×106 | 5.1×108 | 1.6×1012 |
| N=100, 3D | 43,325 | 118,835 | 3.0×106 | 4.3×108 | 1.3×1012 |
| N=500 | 2.5×106 | 7.0×106 | 1.8×108 | 2.6×1010 | 7.9×1013 |
| N=1,000 | 1.4×107 | 4.0×107 | 1.0×109 | 1.5×1011 | 4.5×1014 |
| N=2,000 | 8.2×107 | 2.3×108 | 5.8×109 | 8.6×1011 | 2.6×1015 |
| N=5,000 | 8.2×108 | 2.3×109 | 5.8×1010 | 8.5×1012 | 2.6×1016 |
| N=10,000 | 4.6×109 | — | — | — | 1.5×1017 |
2D results reproduce Maria’s methodology. 3D results projected via PCA.










Molecular segregation persists at scale.





Mass-dependent shell structure clearly visible.





Shell structure increasingly pronounced.





Five thousand particles. Clear mass-dependent spatial organisation.





z-variance penalty breaks spherical symmetry → Jupiter/Saturn-like shapes. N=200, 3D, aligned to principal axes.








3-body oriented triangle terms: VEM = √I × (α Welec + β Wmag). The magnetic term couples triangle areas to a global vorticity axis — oblate shapes from pure geometry.








500 random starts per mass ratio, keeping ALL local minima. Bond angles, asphericity, and cluster counts classify each configuration.
All 500 starts converge to one minimum. Bond angle peaks at water’s 104.5° line.


Still one minimum. Two light-particle clusters.


3 distinct minima. Dominant (434/500) has 7 light clusters.


2 distinct minima. Dominant (447/500) with pronounced segregation.


| Mass Ratio | Distinct Minima | Dominant Angle | Light Clusters |
|---|---|---|---|
| Equal | 1 | 107.9° | — |
| 1:10 | 1 | 98.1° | 2 |
| 1:80 | 3 | 86.6° | 7 |
| 1:2000 | 2 | 71.4° | — |
Bond angle decreases monotonically: 108° → 98° → 87° → 71°.