Focus the 3D view, then use arrow keys to orbit, plus and minus to zoom, and Home to reset. On touch screens, enable Rotate before dragging.
THE CALCULATION HAS TO HOLD UP
Numerical agreement is a consistency check, not a rigorous bound on true error. Rendered paths interpolate saved samples; marker sizes are illustrative.
Particular systems: yes. We can calculate trajectories, and special configurations have exact formulas. Even convergent infinite-series solutions exist. What we lack is a generally useful closed-form recipe for arbitrary starting conditions.
The experiments here show three different achievements: an exact symmetric solution, a numerical search for a known periodic orbit, and finite-time prediction checked against more accurate calculations. None is a new general theorem.
BEHIND THE ORBITS
Each body accelerates toward every other body. Add all those pulls, update the motion, then evaluate the new configuration.
The verified presets use SciPy's adaptive eighth-order DOP853 integrator. The triangle and octagon were checked against their exact formulas. Both sensitive trajectories were recalculated at tighter tolerances. The eight-future ensemble adds distinct deterministic initial states; it is not a probability distribution.
Your experiments use an adaptive Dormand–Prince 5(4) solver in a background worker, followed by a second run with ten times tighter tolerance and a smaller maximum step. Point-mass singularities or excessive work end the calculation with an explanation.
RMS position difference measures the distance between corresponding bodies in two calculations. Return error measures the final-minus-initial position and velocity state in the figure-eight search. Energy change checks conservation. Good energy conservation alone does not prove an accurate trajectory, and rerun agreement is not a rigorous error bound.
The preset motion is planar. Orbit view lets you inspect it with a 3D camera. Time sculpture adds elapsed time as height; it is a diagram of history, not physical vertical motion. Outlined bodies indicate a comparison state. Gravity arrows show directions, with equal display length. Saved-path interpolation is approximate. The search inset magnifies position miss only; its total objective includes velocity error as well. Its displayed scale changes as the miss shrinks.
For equal masses on a regular polygon of radius R, the tangential force components cancel. This angular speed supplies exactly the required inward acceleration. Stability to disturbances is a separate question.