The development path on G/Γ for the input K(t) = (1, β + α sin 2πωt) with mean μ = (1, β). The trajectory winds around both cycles of the base torus when β ≠ 0. The arches fold into the central fibre; H₂ is independent of β.
G/Γ — fundamental domain [0,1)³
Drag to rotate. Each period is a different color. The z-coordinate wraps with the Heisenberg twist: when x resets, z shifts by +y/2.
Universal cover G — the arches
The same path before reduction. Arches grow without bound; on G/Γ they fold into the central fibre.
Base torus T² — horizontal projection
The projection G/Γ → T² forgets the central fibre. When ω = 1 the path traces the same closed loop every period. When ω is incommensurate, each period traces a different curve and the path fills the torus.