Development path in the Heisenberg group
The horizontal input KA(t) = (1, α sin 2πt) drifts along X₁ while oscillating in X₂. The development lifts off the horizontal plane into the Z direction, forming arches that touch z = 0 at every half-integer and peak at z = nα/(2π) at every integer n.
Drag to rotate. The development has coordinates:
x(t) = t, y(t) = α/(2π)(1 − cos 2πt), z(t) = αt/(4π)(1 + cos 2πt) − α sin(2πt)/(4π²).
At integer times z(n) = nα/(2π) — the running area total. At half-integers z = 0 — the area unwinds completely.
The persistent oscillation of H₂[T]/T in the period array animation is this arch structure viewed from above.
development path (x, y, z)
horizontal projection (x, y, 0)
drop lines at integer t (z = nα/(2π))
drop lines at half-integer t (z = 0)