I once had a job for the LEGO company to design a curriculum to teach elementary school mathematics using LEGO. One of the units I came up with used gears of different sizes to introduce the concept of ratios. As I am back in the job of trying to come up with interesting math games for my PhD research, I started talking with my advisor about gears and got interested in them again. What mathematical concepts come out naturally from gears?

Certainly ratios are an intrinsic part of gears. We take advantage of gear ratios when we ride our bikes, by shifting to a lower gear to give ourselves more turns of our crankshaft for every turn of the rear wheel.

But I think what is really interesting about gears from a mathematical perspective is a little different from what is physically useful about them. And in my humble opinion what is really cool about them is that in order to make gears of different sizes mesh together you need to make sure that the teeth are the same size. And this means that smaller gears will have correspondingly fewer teeth and bigger gears will have more teeth. And since bigger gears have more teeth, they make a revolution more slowly than a smaller gear and vice versa.

Now you could ask, and I’m not saying you would, but you could ask, how long does it take two gears to meet back up where they started? It definitely does happen, right, because if it never happened then every tooth on each gear would always be meshing with a different tooth on the other gear, but that would mean that there are infinitely many teeth on the other gear, so that can’t be right! So it definitely does happen that some pair of teeth on each gear meet up again and again, but when is the first time? In the spirit of “show don’t tell” I made a little demo of it using several gears of different sizes. Take a look and count how many teeth mesh before the two little red circles match up! The green gear has 10 teeth and the smaller gear next to it has 5 teeth which is exactly half as many.