Arnold Tongues & the Devil's StaircaseA metronome tries to keep its own beat while a second rhythm keeps nudging it every cycle. Nudge it gently and it drifts freely; nudge it hard enough and it snaps into lock-step, its beat pinned to a clean fraction of the driver's — and it stays pinned even as the driver's tempo wobbles. Each locked zone is an Arnold tongue.Arnold posed this in 1961 to explain how a driven oscillator's frequency gets captured by nearby rational ratios. The locked ranges are single points at zero coupling and swell into wedges as coupling K grows; by K = 1 they cover almost every drive rate, forming a devil's staircase — a curve that keeps climbing yet is flat almost everywhere. Push past K = 1 and the map folds back on itself into chaos. The same lock-in governs heart pacemaker cells and injection-locked lasers.