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 V_S = √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 buttons above to jump directly.

N = 100, 2D

Reproducing Maria’s results — 5 mass ratios

N = 100, 3D

Same counts, three dimensions — 5 mass ratios

N = 500, 3D

Five hundred particles — 5 mass ratios

N = 1000, 3D

One thousand particles — 5 mass ratios

N = 2000, 3D

Two thousand particles — 5 mass ratios

N = 5000, 3D

Five thousand particles — 5 mass ratios

Shape Complexity Values

Config	V_S
N=100, 2D, equal	52,231
N=100, 2D, 1:2	142,784
N=100, 2D, 1:10	3,555,587
N=100, 2D, 1:80	514,763,640
N=100, 2D, 1:2000	1,557,299,569,754
N=100, 3D, equal	43,325
N=100, 3D, 1:2	118,835
N=100, 3D, 1:10	2,988,269
N=100, 3D, 1:80	434,340,799
N=100, 3D, 1:2000	1,314,787,171,612
N=500, 3D, equal	2,537,139
N=500, 3D, 1:2	6,980,961
N=500, 3D, 1:10	178,325,230
N=500, 3D, 1:80	26,132,347,551
N=500, 3D, 1:2000	79,211,871,566,381
N=1000, 3D, equal	14,479,237
N=1000, 3D, 1:2	39,862,518
N=1000, 3D, 1:10	1,021,235,975
N=1000, 3D, 1:80	149,882,119,501
N=1000, 3D, 1:2000	454,432,163,200,919
N=2000, 3D, equal	82,361,363
N=2000, 3D, 1:2	226,827,504
N=2000, 3D, 1:10	5,821,639,500
N=2000, 3D, 1:80	855,223,804,181
N=2000, 3D, 1:2000	2,593,391,805,253,071
N=5000, 3D, equal	817,408,304
N=5000, 3D, 1:2	2,251,798,693
N=5000, 3D, 1:10	57,874,707,862
N=5000, 3D, 1:80	8,508,278,627,932
N=5000, 3D, 1:2000	25,803,776,176,303,990

Method

Configurations are computed by minimising V_S = √I_cm × W on the pre-shape sphere {∑q_i = 0, ∑|q_i|² = 1}, where I_cm = ∑ m_i |q_i − q_cm|² is the moment of inertia about the centre of mass and W = ∑_i<j m_i m_j / r_ij 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 V_S. 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	V_S
2D, equal	52,231
2D, 1:2	142,784
2D, 1:10	3,555,587
2D, 1:80	514,763,640
2D, 1:2000	1,557,299,569,754
3D, equal	43,325
3D, 1:2	118,835
3D, 1:10	2,988,269
3D, 1:80	434,340,799
3D, 1:2000	1,314,787,171,612

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	V_S
3D, equal	2,537,139
3D, 1:2	6,980,961
3D, 1:10	178,325,230
3D, 1:80	26,132,347,551
3D, 1:2000	79,211,871,566,381

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	V_S
3D, equal	14,479,237
3D, 1:2	39,862,518
3D, 1:10	1,021,235,975
3D, 1:80	149,882,119,501
3D, 1:2000	454,432,163,200,919

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.