Integrable dynamics — dense windings vs. closed orbits
0.618
2.5
0.8
800
A geodesic on the flat torus T²=[0,2π)² is a straight line with slope = winding number ω = dθ₂/dθ₁. If ω = p/q is rational, the orbit is closed (period q). If ω is irrational, the orbit is dense — it visits every neighborhood of every point. The Weyl equidistribution theorem guarantees uniform density. The golden ratio φ = (1+√5)/2 ≈ 1.618 gives the "most irrational" winding.