An involute curve (specifically, an involute of a circle) is very commonly used to define the shape of the teeth on a gear. Apparently this idea goes back to Euler.
Why is this? What special mathematical properties of an involute curve make it suitable for use in gears?
As far as I know, an involute is a curve formed by "unwinding string" from a circular hub. Is this imaginary piece of string somehow related to the functioning of the gear.
This Wikipedia page has a nice animation of involute gears, but it gives no insight into the mathematics (for me, anyway).
For easy reference, the involute of a circle of radius $a$ has parametric equations \begin{align} x(t) = &a(\cos t+t \sin t) \\ y(t) = &a(\sin t-t \cos t) \end{align} The equations give a spiral shape, but a gear tooth would use only a small portion of the spiral, with values of $t$ ranging from $0$ up to around $\pi/6$, maybe.