Fifteen pendulums that agree to disagree

2026-05-20

A row of fifteen pendulums hangs from a beam. Each one is a little shorter than the one before. Pulled aside together, released together, they start the same way every pendulum starts: in phase, swinging as one.

They stay that way for about a second. Then it falls apart.

Watch a full thirty seconds. The first few swings look synchronised. A travelling wave appears almost immediately - a slow ripple that walks down the row and back. The wave splits. Pairs of pendulums knot together, neighbours separate. By the middle of the run the shape has no shape at all - fifteen independent bobs, looking random.

Wait. The mess unbuilds itself. Two waves form again, running in opposite directions. They meet. They flatten. And at exactly thirty seconds every pendulum is back at its starting point, swinging in perfect unison, as if nothing happened.

Why it isn't really chaos

The trick is in the lengths. A pendulum's period - the time for one full swing - depends only on how long it is. So if you pick the longest pendulum and say "this one will swing exactly ten times in the next thirty seconds", you've fixed its length. The next pendulum is set to swing eleven times in the same thirty seconds. The next, twelve. And so on, up to twenty-four for the shortest.

Ten, eleven, twelve, ..., twenty-four. Every one is a whole number. That is the entire secret. After thirty seconds, each pendulum has completed a whole number of swings - so each is back exactly where it started, moving exactly how it started. They are forced to agree.

Where the wave comes from

In between the agreements they disagree, but in an exquisitely organised way. After one second the shortest pendulum has had time to get a little further into its swing than its neighbour to the left, which has had time to get a little further than its neighbour, and so on along the row. The result is a smooth phase gradient along the line of bobs - a travelling wave, with the wavelength set by how much faster each pendulum is than the one before.

As time goes on the phase gradient steepens, then folds back on itself, then steepens again from the other side. The "chaos" in the middle is just many wavelengths laid on top of each other - the same kind of interference you get when several sounds at different pitches play at once. It looks random because your eye cannot count fifteen frequencies. The numbers can.

The deeper trick

The pendulum wave is what happens when you let several simple periodic things run side by side. Two pendulums of different lengths drift apart and return when their periods' lowest common multiple comes around. With fifteen, the LCM is thirty seconds. With a hundred, picked the same way, you could run for an hour. With any rational ratio at all, the dance always closes.

The same idea sits underneath gear teeth that line up again after a fixed number of turns, planets that almost-align every few centuries, and chords that sound consonant because their frequencies share small whole-number ratios. The pendulum wave is the most direct version of that idea you can look at - one beam, fifteen strings, and a clock.

Hold the slider at twelve seconds. Look at the row. It is genuinely impossible to see the order. Drag to thirty. It arrives anyway.