When two gears agree to meet again

2026-06-03

Two gears mesh. Paint one tooth on each - here, a red dot and a green dot - and start them off kissing at the point where the gears touch. Now turn them. The two painted teeth swing apart. When, if ever, do they come back to meet at that same point?

They do come back - but not after one turn. The left gear has twelve teeth; pick the right gear and watch the counter. With eight teeth on the right, the marks meet again only after twenty-four teeth have gone through the mesh: two full turns of the left gear, three of the right. With seven teeth, you wait eighty-four.

Counting teeth, not turns

Gears mesh tooth-for-tooth. Whatever happens, the same number of teeth pass through the contact point on both gears - that is what meshing means. So if k teeth have gone by, the gear with A teeth has turned k / A of a full revolution, and the gear with B teeth has turned k / B.

A painted tooth is back at the contact point exactly when its gear has done a whole number of turns. So both marks are back together only when k / A and k / B are both whole numbers - that is, when k is a common multiple of A and B. The first time it happens is the smallest such number:

k = lcm(A, B)

the lowest common multiple. That single number is the whole story of the dance.

Why some pairs are friendly

Twelve and eight share a factor of four, so their lowest common multiple is small - just twenty-four - and the marks reunite quickly. Twelve and sixteen share four as well: forty-eight. But twelve and seven share nothing at all; they are coprime. The only way to make both counts come out whole is to multiply them together, so you wait the full 12 × 7 = 84 teeth before the marks meet again.

This is exactly why machinists sometimes add a single extra "hunting" tooth to a gear. A gear that shares no factor with its partner forces every tooth on one to eventually touch every tooth on the other, spreading wear evenly instead of letting the same two teeth grind together every turn. Coprimeness, doing useful work.

The same clock everywhere

Two things going round at steady but different rates, drifting apart and snapping back when their cycles share a common multiple - that pattern is everywhere once you have seen it in the teeth. It is why a row of pendulums returns to unison after a fixed time, why planets fall into orbital resonances, and why two notes sound consonant when their frequencies are a tidy ratio like 3:2 and restless when they are not. The gears just let you hear the click.

Set the right gear to 7 and let it run. The marks wander for a long, long time. Set it to 12 and they meet every single turn - the friendliest ratio of all, 1:1.