Points of view.

One mathematical object — the complex wave packet f(t) = e^{−γ(t−t₀)²} · e^iωt. Seen from three sides it becomes three different things: a spiral in the complex plane, a dying sine wave, and a dying cosine wave. Drag the time slider and watch the same point tell three stories at once.

The object itself — the helix (Re f, Im f, t) in 3D. The amber bead is “now”.

Point of view ① — the Re–Im plane: an inward spiral

Point of view ② — Im f = e^{−γ(t−t₀)²} sin(ωt)

Point of view ③ — Re f = e^{−γ(t−t₀)²} cos(ωt)

t = 0.00 T · f(t) = …

Twist ω 1.00× Spread σ 1.40 T Speed 1.00×

The Gaussian envelope e^{−γ(t−t₀)²} squeezes the endless oscillation e^iωt into a packet localized near t₀ — the same shape (a Morlet wavelet) used to hunt for brief oscillations hidden in signals, from seismic tremors to brain waves. Turn the Twist up and the spiral winds tighter; widen the Spread and the packet relaxes toward a pure tone.