Bounce Lab v2
The same two-dot game, now with up to a hundred thousand runners. Every runner n starts at the right dot and crosses the line n times per frame. The carpet shows where all the runners are at every moment — the standing waves they form along the line as they bunch and scatter. The sensor can sit anywhere on the line, at the middle or at any fraction; it writes its waveform in real time as you play, and you can step the clock a notch at a time to watch a coincidence happen.
Runners
Sensor and combination rule
The sensor writes its waveform as the clock runs
The line, now
If the runners were waves — the string's shape, all modes at amplitude 1
The ring — the whole system as one wave: every runner's pitch sounding at once, amplitude 1
The collective wave — the centre of mass of all the runners, swinging as they bunch and scatter
The carpet — where every runner is, at every moment (position across, time down)
The ring. Strike the whole system and every runner rings at its own pitch — runner n at rate n. Added together at one point of the line, that is a single waveform in time: the chord of the system's pitches. For the primes it is the "prime chord," and its shape is spiky at the simple fractions of a frame for the same reason the runners bunch there. The spectrum below it is simply the list of pitches — one line per runner — which is the honest statement of what this system rings at: its own numbers. Press Listen and you hear it: each runner n as a sine at base × n hertz, so the primes up to 50 become a chord of 15 notes. The integers 1 … K, by contrast, ring as a harmonic series — a single buzzy note — because their pitches are evenly spaced. What you will never hear from this system is 14.13, 21.02, 25.01: those are the pitches of a different instrument, and this one rings at the primes themselves. If the runners were waves. Runner n becomes the string's nth mode: n half-wavelengths across the line, amplitude 1, oscillating n times faster than the fundamental. The panel adds all of them at the current instant. At t = 0 every mode is at full stretch and the sum is the "primes as overtones" picture from the overtones lab, laid along the line: spikes at the simple fractions of the way across, with signs set by the Möbius rule (positive at 1/6 and 5/6, negative at 1/3 and 2/3, near zero at 1/4). As t runs, each mode swings at its own rate and the shape reshuffles; at t = 1 the odd modes have flipped and the even ones have not, so the end-of-frame shape is the mirror image of the start. The envelope shows how far each point of the string ever moves during the frame — the standing-wave pattern of the whole set of modes. This is the same mathematics as the runners' game seen through a different lens: the runners' meeting times are the fractions of a frame; the modes' spikes are the fractions of the line. The carpet. Each row is one instant; brightness is how many runners are at each position. At most instants the runners are spread evenly and the row is flat. At the simple fractions of a frame — ½, ⅓, ¼, ⅕ … — they bunch into a few sharp positions and the row lights up: those are the revivals, and the curved bands linking them are the standing-wave pattern of the whole family. With the primes as runners, look at the middle row (t = ½): the odd primes sit exactly at the centre, going in two directions. The sensor. Moving it off the middle changes which fractions "fit": at position a/b the runners that arrive together are the ones whose crossing times share that fraction, so the waveform is a different rhythm at every rational position and nearly silent at an irrational one. The rule. "right − left" is the original game; "right + left" counts arrivals regardless of direction, which is what a sensor that can't tell direction would see; the others isolate one direction or take the magnitude.