Turn the knob smoothly. The answer moves in steps.

One thing spins. Another gives it a small shove once every turn. Ask the only question that matters — how many turns does the follower make per turn of the driver? — and the answer stops being willing to take most values.

What you're seeing

The upper graph answers one question at every knob setting: how many turns does the follower make for each turn of the driver? Turn the nudge off and the graph is a plain diagonal. Every ratio is available, and a small move of the knob makes a small move in the answer.

Turn the nudge on and the line grows flat steps. Inside a step, moving the knob does nothing at all — the follower has been captured by a whole-number ratio and stays there. Between steps the answer climbs, briefly, until the next fraction catches it.

The strip below shows why. It marks where the follower stands at each turn of the driver, newest at the bottom. When the setting is locked, those marks fall in the same few columns forever: the pattern repeats exactly. When it is not, they slide sideways and never come home.

Now push the nudge toward 1. The steps widen until they are touching, and the graph climbs from bottom to top while standing still almost everywhere. That is the devil's staircase, and it is not a drawing artefact — look closer and there are more steps between the steps, at every scale you care to look. The readout says “locked, found by this probe” and shows a ≥ sign for a reason: it wiggles the knob by a fixed amount and counts the settings that do not budge, so any step narrower than that wiggle is invisible to it. The number can only ever be too small. Look with a finer wiggle and it climbs — at the critical line it keeps climbing, which is the measurable shadow of the theorem that up there the locked settings cover everything except a dust of zero width.

Push past 1 and something else happens: a second, blue curve appears. Above that point the follower can settle into more than one rhythm at the same knob setting, and which one you get depends on where it happened to be when you started. The blue line is the same calculation from a different starting position. Where the two lines part company, the question “how many turns per turn?” has no single answer at all.

This is the same kind of locking that keeps one face of the Moon toward us and turns Mercury exactly three times for every two trips around the Sun. Neither started at the perfect ratio; each was near enough that a once-a-turn tug could pull it in and hold it — and what makes those ratios stable is that they occupy a whole band of conditions rather than one exact setting, which is precisely what the flat steps here are. The shape carries over; the cause does not. Real bodies are not driven by a knob a visitor turns — tidal friction moves them into the lock and dissipates the energy that keeps them there, and Mercury’s 3:2 rather than 1:1 depends on its eccentric orbit and lumpy shape. None of that is in this equation. The same family of capture is also why coupled clocks fall into step, why some heart arrhythmias beat in tidy ratios, and why a wobbling washing machine finds one bad speed and digs in.

The rule, exactly. The follower's position θ on its circle advances by θ → θ + Ω − (K/2π) sin(2πθ) where Ω is the knob and K the nudge strength (the standard circle map). The answer plotted is the winding number W = lim (θ_N − θ₀)/N. The frozen step at W = 0 exists exactly when |Ω| ≤ K/2π — an equation, not an estimate. The companion proof (improve/verify/108-devils-staircase.js, 28 checks) measures that step at 0.2546 against a predicted 0.2546 at K = 0.8, confirms W never decreases and satisfies W(1−Ω) = 1−W(Ω), and checks the page's own 2000-iteration columns against a 60 000-iteration computation to 2×10⁻⁴. Four negative controls: at K = 0 the widest step shrinks to the detector's own tolerance, so the steps are not a finite-sample artefact; the width constant is shown not to be K/π or K/4π; locking is tested by wiggling the knob rather than by looking for nearby fractions; and replacing the nudge with a constant of the same size destroys every step and merely slides the diagonal sideways — so what makes the staircase is a nudge that knows where the follower is. And the claim has a domain: the map is invertible only for K ≤ 1, so everything above is a statement about that range. Past it the map folds, more than one rhythm can be stable at the same Ω, and W stops being a function of the knob — measured at 3 of 201 settings differing by up to 2.9×10⁻² between two starting phases at K = 1.2, against 1.3×10⁻⁵ (pure finite-average noise) at and below K = 1. That is why the slider runs past 1 and why a second curve appears when it does. Likewise the “locked” percentage is a probe floor, marked ≥: it counts steps wider than the wiggle used to find them, and at K = 1 theory says the true figure is 100% with the leftovers a measure-zero dust.