The Three-Body Problem
For 300 years, physicists have called this problem "unsolvable." Three masses, mutual gravitation, no closed-form solution. Newton couldn't solve it. Poincaré proved it chaotic. But chaos isn't a property of reality — it's a symptom of incomplete understanding.
Live Simulation: Chaos vs Geometry
Watch the same initial conditions evolve under two different frameworks. Same masses. Same positions. Different understanding.
Simulation Controls
Classical Mechanics
Newton's laws only — no geometric correction
Geometric Framework [1=-1]
Newton's laws + triaxial closure
The Missing Understanding
"Classical mechanics describes motion through dimension without asking what dimension IS. The three-body problem isn't unsolvable — it was incompletely modeled. Dimension itself has structure: triaxial, self-closing, geometrically necessary."
Three bodies naturally occupy three axes. The fourth component — the hidden observer [Uv] — provides the closure that forces stability. This isn't adding a dimension. It's recognizing the geometry that was always there.
The Measure of Chaos
The Lyapunov exponent (λ) quantifies how fast nearby trajectories diverge. It's the mathematical fingerprint of chaos — or stability.
Classical Mechanics
Positive = Exponential Divergence
Nearby trajectories separate exponentially. A butterfly's wing changes everything.
Prediction becomes impossible beyond ~100 orbits.
Geometric Framework
Zero/Negative = Convergence
Trajectories converge toward stable attractors. Perturbations decay.
The system self-corrects. Prediction horizon: infinite.
Understanding the Framework
The three-body problem reveals a deep truth: our models of reality are incomplete not because reality is complex, but because we haven't understood the geometry it emerges from.
The Classical Problem
Newton solved two-body gravitation exactly: ellipses, parabolas, hyperbolas. Add a third body and everything breaks. Poincaré proved in 1889 that no closed-form solution exists.
The system is deterministic but unpredictable. Given perfect initial conditions, the future is fixed — but we can never measure perfectly, and errors grow exponentially.
The Geometric Solution
The [1=-1] framework doesn't add forces or dimensions. It recognizes that dimension itself has structure. Three bodies naturally map to the triaxial geometry: [+1], [-1], [0].
The coupling constant κ = 2π/180 isn't arbitrary — it's the bridge between discrete (degrees) and continuous (radians) measurement. It's how dimension closes on itself.
Why Three Bodies?
It's not coincidence that three is where classical mechanics fails. Three is the minimum for structure. Two points define a line. Three define a plane. Three axes define space.
The triaxial structure [+1, -1, 0] is the simplest self-balancing geometry. The fourth component [Uv] — the hidden observer — provides closure. This is why particle physics finds three generations, three colors, three forces.
Stable Attractors
With geometric coupling, the three-body system develops stable attractors — configurations it naturally evolves toward regardless of initial conditions.
The famous Lagrange points (L1-L5) are examples. So is the figure-8 orbit discovered in 1993. The geometric framework reveals these aren't special cases — they're the natural resting states of triaxial closure.