Central Configurations: N = 100 – 5000

Gravitational molecules at minimum shape complexity

Aaron Lax — March 2026

These are central configurations — critical points of the shape potential VS = √I × W on the pre-shape sphere, computed via gradient descent with 100 random restarts per configuration. Each plot shows the minimum-complexity configuration for N particles with a 50/50 mass split. Heavy particles (red) and light particles (blue) are sized by mass. For 3D configurations, positions are projected onto the plane of greatest variance.

All Results

Select a grouping below to see the full plots, or use the tabs above to jump directly.

Highlights

Gravitational molecules emerge from pure geometry. Click any image to explore the full set.

N=5000 equal
N = 5,000 — equal masses — uniform fill
N=1000 1:2000
N = 1,000 — 1:2000 — mass shell structure
Oblate
Oblate — Jupiter/Saturn-like flattening (λ=2)
V_EM
VEM — magnetic vorticity breaks symmetry (β=1)

Mass Ratio Progression — N = 100, 2D

As mass ratio increases, light particles (blue) segregate from heavy (red). Click to see all scales.

Equal
1:2
1:10
1:80
1:2000

Scaling — Equal Masses, N = 100 to 5,000

Structure persists from 100 to 5,000 particles in 3D.

N=100
N=500
N=1,000
N=2,000
N=5,000

Scaling — Mass Ratio 1:2000, N = 100 to 5,000

Molecular segregation strengthens with particle count. Light particles pushed to outer shell at large N.

N=100
N=500
N=1,000
N=2,000
N=5,000

All Configurations

N = 100

2D (Maria comparison) + 3D — 5 mass ratios each

N = 500, 3D

Five hundred particles — 5 mass ratios

N = 1,000, 3D

One thousand particles — 5 mass ratios

N = 2,000, 3D

Two thousand particles — 5 mass ratios

N = 5,000, 3D

Five thousand particles — 5 mass ratios

Oblate Shapes

Jupiter/Saturn-like — breaking spherical symmetry

VEM — Electromagnetic

Maria’s 3-body triangle vorticity term

Shape Complexity Values

Config VS
N=100, 2D, equal52,231
N=100, 2D, 1:2142,784
N=100, 2D, 1:103,555,587
N=100, 2D, 1:80514,763,640
N=100, 2D, 1:20001,557,299,569,754
N=100, 3D, equal43,325
N=100, 3D, 1:2118,835
N=100, 3D, 1:102,988,269
N=100, 3D, 1:80434,340,799
N=100, 3D, 1:20001,314,787,171,612
N=500, 3D, equal2,537,139
N=500, 3D, 1:26,980,961
N=500, 3D, 1:10178,325,230
N=500, 3D, 1:8026,132,347,551
N=500, 3D, 1:200079,211,871,566,381
N=1000, 3D, equal14,479,237
N=1000, 3D, 1:239,862,518
N=1000, 3D, 1:101,021,235,975
N=1000, 3D, 1:80149,882,119,501
N=1000, 3D, 1:2000454,432,163,200,919
N=2000, 3D, equal82,361,363
N=2000, 3D, 1:2226,827,504
N=2000, 3D, 1:105,821,639,500
N=2000, 3D, 1:80855,223,804,181
N=2000, 3D, 1:20002,593,391,805,253,071
N=5000, 3D, equal817,408,304
N=5000, 3D, 1:22,251,798,693
N=5000, 3D, 1:1057,874,707,862
N=5000, 3D, 1:808,508,278,627,932
N=5000, 3D, 1:200025,803,776,176,303,990

Method

Configurations are computed by minimising VS = √Icm × W on the pre-shape sphere {∑qi = 0, ∑|qi|² = 1}, where Icm = ∑ mi |qi − qcm|² is the moment of inertia about the centre of mass and W = ∑i<j mi mj / rij is the Newton potential. Gradient descent uses mass preconditioning (dividing the gradient by particle mass) to equalise convergence rates for light and heavy particles. Optimisation uses cosine-annealed learning rate over 8,000–18,000 iterations with 40–100 random restarts, selecting the lowest VS. Computed on NVIDIA H100 GPUs via Modal using JAX with float64 precision.

N = 100 Configurations

100 particles with 50/50 mass split at five mass ratios. The 2D results reproduce Maria’s methodology and can be compared directly to her figures. The 3D results use the same parameters, projected to 2D via PCA for display.

2D — Reproducing Maria’s Results

These match Maria’s methodology: 100 particles in 2D with 50/50 mass split. Compare directly to her figures.

N=100, 2D, equal masses
Equal masses (m = 1)
N=100, 2D, mass ratio 1:2
Mass ratio 1:2
N=100, 2D, mass ratio 1:10
Mass ratio 1:10
N=100, 2D, mass ratio 1:80
Mass ratio 1:80
N=100, 2D, mass ratio 1:2000
Mass ratio 1:2000

3D

Same particle counts and mass ratios, now in three dimensions. Projected to 2D via PCA for display.

N=100, 3D, equal masses
Equal masses (m = 1)
N=100, 3D, mass ratio 1:2
Mass ratio 1:2
N=100, 3D, mass ratio 1:10
Mass ratio 1:10
N=100, 3D, mass ratio 1:80
Mass ratio 1:80
N=100, 3D, mass ratio 1:2000
Mass ratio 1:2000

Shape Complexity — N = 100

Config VS
2D, equal52,231
2D, 1:2142,784
2D, 1:103,555,587
2D, 1:80514,763,640
2D, 1:20001,557,299,569,754
3D, equal43,325
3D, 1:2118,835
3D, 1:102,988,269
3D, 1:80434,340,799
3D, 1:20001,314,787,171,612

N = 500 Configurations

Five hundred particles in 3D. The molecular segregation at high mass ratios persists at scale — heavy particles form a uniform scaffold, light particles cluster in the interstices.

3D

Projected to 2D via PCA for display. Five mass ratios from equal to 1:2000.

N=500, 3D, equal masses
Equal masses (m = 1)
N=500, 3D, mass ratio 1:2
Mass ratio 1:2
N=500, 3D, mass ratio 1:10
Mass ratio 1:10
N=500, 3D, mass ratio 1:80
Mass ratio 1:80
N=500, 3D, mass ratio 1:2000
Mass ratio 1:2000

Shape Complexity — N = 500

Config VS
3D, equal2,537,139
3D, 1:26,980,961
3D, 1:10178,325,230
3D, 1:8026,132,347,551
3D, 1:200079,211,871,566,381

N = 1000 Configurations

One thousand particles in 3D. At this scale the mass-dependent shell structure is clearly visible: light particles are pushed to the outer boundary of the configuration.

3D

Projected to 2D via PCA for display. Five mass ratios from equal to 1:2000.

N=1000, 3D, equal masses
Equal masses (m = 1)
N=1000, 3D, mass ratio 1:2
Mass ratio 1:2
N=1000, 3D, mass ratio 1:10
Mass ratio 1:10
N=1000, 3D, mass ratio 1:80
Mass ratio 1:80
N=1000, 3D, mass ratio 1:2000
Mass ratio 1:2000

Shape Complexity — N = 1000

Config VS
3D, equal14,479,237
3D, 1:239,862,518
3D, 1:101,021,235,975
3D, 1:80149,882,119,501
3D, 1:2000454,432,163,200,919

N = 2000 Configurations

Two thousand particles in 3D. At this scale the shell structure becomes increasingly pronounced — heavy particles form a regular interior lattice while light particles occupy the outer layers.

3D

Projected to 2D via PCA for display. Five mass ratios from equal to 1:2000.

N=2000, 3D, equal masses
Equal masses (m = 1)
N=2000, 3D, mass ratio 1:2
Mass ratio 1:2
N=2000, 3D, mass ratio 1:10
Mass ratio 1:10
N=2000, 3D, mass ratio 1:80
Mass ratio 1:80
N=2000, 3D, mass ratio 1:2000
Mass ratio 1:2000

Shape Complexity — N = 2000

Config VS
3D, equal82,361,363
3D, 1:2226,827,504
3D, 1:105,821,639,500
3D, 1:80855,223,804,181
3D, 1:20002,593,391,805,253,071

N = 5000 Configurations

Five thousand particles in 3D. The largest configurations computed, showing clear mass-dependent spatial organisation at extreme mass ratios.

3D

Projected to 2D via PCA for display. Five mass ratios from equal to 1:2000.

N=5000, 3D, equal masses
Equal masses (m = 1)
N=5000, 3D, mass ratio 1:2
Mass ratio 1:2
N=5000, 3D, mass ratio 1:10
Mass ratio 1:10
N=5000, 3D, mass ratio 1:80
Mass ratio 1:80
N=5000, 3D, mass ratio 1:2000
Mass ratio 1:2000

Shape Complexity — N = 5000

Config VS
3D, equal817,408,304
3D, 1:22,251,798,693
3D, 1:1057,874,707,862
3D, 1:808,508,278,627,932
3D, 1:200025,803,776,176,303,990

Oblate Configurations

Adding a z-variance penalty to VS breaks spherical symmetry, producing oblate (Jupiter/Saturn-like) shapes. The parameter λ controls flattening. N = 200 particles in 3D, shown aligned to principal axes.

Equal Masses

oblate lambda=0
λ = 0 — spherical baseline (z/xy = 1.99)
oblate lambda=0.5
λ = 0.5 — mildly oblate (z/xy = 0.92)
oblate lambda=2
λ = 2.0 — strongly oblate (z/xy = 0.39)
oblate lambda=10
λ = 10.0 — pancake (z/xy = 0.006)

Mass Ratio 1:80

1:80 lambda=0
λ = 0 — spherical (z/xy = 2.00)
1:80 lambda=0.5
λ = 0.5 — mildly oblate (z/xy = 0.91)
1:80 lambda=2
λ = 2.0 — strongly oblate (z/xy = 0.40)
1:80 lambda=10
λ = 10.0 — pancake (z/xy = 0.04)

Maria’s VEM — Electromagnetic Shape Potential

Maria’s functional introduces 3-body oriented triangle terms: VEM = √I × (α Welectric + β Wmagnetic). The magnetic term couples each triangle’s oriented area to the global geometric vorticity axis, breaking spherical symmetry from pure geometry — no imposed angular momentum.

N = 100, Varying β

VEM beta=0
β = 0 — gravity only, spherical (aspect = 1.00)
VEM beta=0.1
β = 0.1 — oblate emerges (aspect = 0.70)
VEM beta=0.5
β = 0.5 (aspect = 0.70)
VEM beta=2
β = 2.0 (aspect = 0.90)
VEM beta=10
β = 10.0 (aspect = 0.85)

N = 200

VEM N=200 beta=0
β = 0 — spherical baseline (aspect = 1.00)
VEM N=200 beta=1
β = 1.0 (aspect = 0.79)
VEM N=200 beta=5
β = 5.0 (aspect = 0.85)

Molecule Search

500 random starts per mass ratio, keeping ALL local minima. Each start converges to a potentially different gravitational molecule. Structural descriptors — nearest-neighbour bond angles, asphericity, light-particle cluster count — classify the results.

Equal Masses — 1 unique minimum, bond angle 107.9°

All 500 starts converge to the same configuration. The mean nearest-neighbour angle is 107.9° — strikingly close to water’s tetrahedral bond angle of 104.5°. The angle distribution peaks right at the water line.

Equal mass angle distribution
Angle distribution across 500 starts. The 105° water line (red dashed) falls at the peak.
Equal mass gallery
Gallery of configurations — all are the same minimum viewed from different PCA projections.

Mass Ratio 1:10 — 1 unique minimum, bond angle 98.1°

Still a unique minimum. Angle tightens to 98.1°. Two light-particle clusters emerge.

1:10 angle distribution
Angle distribution, 1:10 mass ratio.
1:10 gallery
Gallery of interesting configurations.

Mass Ratio 1:80 — 3 distinct minima, bond angle 86.6°

The landscape now has multiple basins. The dominant minimum (434 of 500 starts) has 7 light-particle clusters and a mean angle of 86.6°. Two rarer configurations exist.

1:80 angle distribution
Angle distribution, 1:80 mass ratio.
1:80 gallery
Gallery showing the three distinct minima.

Mass Ratio 1:2000 — 2 distinct minima, bond angle 71.4°

Two distinct gravitational molecules. The dominant one (447/500 starts) has a mean angle of 71.4° and pronounced molecular segregation.

1:2000 angle distribution
Angle distribution, 1:2000 mass ratio.
1:2000 gallery
Gallery of gravitational molecules at extreme mass ratio.

Angle Trend with Mass Ratio

Mass RatioDistinct MinimaDominant AngleLight Clusters
Equal1107.9°—
1:10198.1°2
1:80386.6°7
1:2000271.4°—

The dominant bond angle decreases monotonically with mass ratio: 108° → 98° → 87° → 71°. The number of distinct minima increases at intermediate ratios (1:80 has 3), then decreases again at extreme ratios (1:2000 has 2).