Night 012Pendulum Garden

The twelfth night, and the oldest demonstration of chaos I know: a pendulum hanging from a pendulum. Two rods, two masses, no friction, and equations that fit on a postcard. For small swings it is a docile thing. Lift it high and let go, and the second arm flips and tumbles in a way you cannot predict a minute ahead, not because the physics is unknown but because the answer depends on the start more finely than any measurement can resolve.

That sensitivity is hard to see with one pendulum, so tonight there are twenty-four, hung from the same pin. They all start at the same angle except that the lower arm of each one is turned a tenth of a milliradian further than its neighbour, far less than the width of a line on the screen. For the first seconds they move as one, a single bright bob with a single trail. Then the trail thickens, splits into strands, and a few seconds later the bobs are everywhere at once, each drawing its own path in its own colour. The readout counts the seconds until the spread between the furthest two tips passes a tenth of an arm. From the default start it takes a few seconds; from a low start it can take longer than anyone waits, because low swings are not chaotic at all, and the garden stays a single stem.

Drag anywhere to choose a new start; both arms straighten towards your finger and the run begins when you let go. The epsilon slider changes the gap between neighbours from a tenth of a radian down to a ten-millionth, and the split time moves with it in a way that is worth watching: every factor of ten in the start buys only a fixed number of extra seconds of agreement. That linear-in-the-logarithm behaviour is what a Lyapunov exponent is, seen rather than defined.

Built by Claude Opus 5 from my spec, integrated with fourth-order Runge-Kutta at a fixed step so that a given start reproduces exactly from the URL. Decisions I made alone: no damping, so the bloom never settles, and the trails fade on their own instead of accumulating into mud.