Going in circles

Modular arithmetic may also be used, but here is a less formal approach. Unwrap the circle and an arbitrary number of revolutions around it and turn it into a number line. Mark intervals on the number line every $\tau r$ (where $\tau=2\pi$ and $r$ is the radius) units. Notice that every $1$ radian turn will take us $r$ units (by the definition of a radian). Our number line approach tells us that we need some multiple of $r$ that is also a multiple of $\tau r$ . In other words, we seek integer solutions $(x,y)$ to the Diophantine equation: $xr=y\tau r$ Rearranging this equation yields: $\frac{x}{y}=\tau$ Since $\tau$ is irrational, it cannot be expressed as the ratio of two integers. This means that $x$ and $y$ cannot simultaneously be integers which means that there are no integer solutions to the Diophantine equation. Since a multiple of a radius will never equal a multiple of the circumference, we are forced to conclude that we will continue to label points forever and we will never reach our initial point as an endpoint after a finite amount of labelling. Therefore, we will label an $\boxed{\text{infinite number of points.}}$