An Odd Kind of Sympathy — an interactive explanation of synchronization

An Odd Kind of Sympathy

How metronomes, bridges, fireflies and heartbeats fall into step — an interactive explanation

A note before you read: this entire page — the text, the simulations, the demos — was written and built by Claude (Anthropic's AI). It is also, quite blatantly, an attempt to write in the style of Bartosz Ciechanowski, whose interactive essays are the gold standard of this genre and are far better than this one. If you enjoy anything here, you owe it to yourself to read the real thing — start with Sound, or Mechanical Watch, and prepare to lose an afternoon. They're amazing.

In February 1665, Christiaan Huygens — inventor of the pendulum clock — lay sick in bed and noticed something odd about two of his clocks hanging from the same wooden beam. No matter how they started, within half an hour their pendulums were swinging in perfect opposition — each swinging left just as the other swung right, the ticks interleaved. He wrote to his father about an "odd kind of sympathy" between the clocks.

You can recreate his discovery with a few metronomes and a board resting on two cans. Wind them up, start them out of step, and set them on the board. Here is the real thing, filmed by the UCLA physics department:

“Spontaneous synchronization” — UCLA Physics

The same experiment is easy to simulate, and a simulation lets us slow the process down, speed it up, and take it apart. Below, five simulated metronomes start at random points in their swing. Press play and watch what they do:

They always end up ticking together, although nothing connects them but the board they stand on. Curiously, they agree in step, where Huygens's clocks agreed in opposition — we'll come back to that. In the rest of this article we'll build the experiment up from its parts and see how it happens. One suggestion before we start: the speaker button in the corner of the page adds ticking sounds to every demo, and with it on you can hear the moment agreement arrives.

A metronome

At its heart a metronome is a pendulum: a rod swinging on a pivot, with a small weight that slides along the rod to set the tempo. Moving the weight away from the pivot makes the swing slower, and moving it closer makes it faster. Each tick — each beat — is one half of a full swing.

The dial on a real metronome runs from about 40 beats per minute to 208. Everything in this article works the same way at any tempo, so we'll mostly leave it near 144.

Keeping time

A pendulum on its own winds down. Friction at the pivot and air resistance take a little energy from every swing, and after ten or twenty seconds it hangs still. You can see this happen by switching the escapement off:

The escapement is the mechanism that keeps this from happening. A wound spring stores energy, and a toothed wheel passes a small push to the pendulum at the same point in every swing — the tick you hear is that hand-off. The push replaces what friction took, so the pendulum settles into a steady swing of fixed width, marked above by the dashed lines.

One property of this arrangement matters more than any other for what follows. Press the nudge button above and watch what recovers. The shove throws the swing off its usual width, and within a couple of beats the escapement has trimmed it back — that is the escapement doing its job. But the faint gray pendulum behind the colored one keeps the timing the swing would have kept, and the nudged pendulum never drifts back to it. The escapement polices how wide the swing is; nothing polices when it happens. After a disturbance the metronome swings exactly as before, permanently a little earlier or later than it would have been. That freedom in timing is what the rest of the story turns on.

Throwing its weight around

Now place the metronome on a board, and rest the board on two cans so that it can roll. The pendulum's weight accelerates left and right on every swing, and by Newton's third law the case — and the board beneath it — is pushed the opposite way. The arrow below the ground line shows the sideways force the metronome exerts on the board at each instant; the pale band behind it shows how large that force has recently been.

Try making the base heavier with the slider: the recoil shrinks, because the same force now has more mass to move. The board's motion is drawn at twice its true size here and throughout — in reality the sway is a centimeter or two at most.

The board pushes back

So each metronome shakes the board. The influence also runs the other way: a pendulum whose pivot accelerates feels an extra push, the same push you feel standing in a bus as it brakes. Through the board, every metronome is gently pushing on every other.

What can that push actually change? Not the width of the swing — the escapement defends that. What nothing defends is timing. To see the effect in isolation, let's take the other metronomes away and shake the board...