Number of ways a laser will bounce on a circular reflective material in 4 times

Question

Is there a formula for getting the number of ways a laser will bounce on a circular reflective material in 4 times, returning to its original position?

So the correct answer is 4, but is there a formula?

Here's a picture showing the 4 ways:

Answer 1

There will be $\phi(n\mathord{+}1) $ options for $n$ bounces as illustrated, where $\phi()$ is Euler's totient function, counting all numbers less than $n$ and coprime to $n$.

Taking the leftmost illustration as the base case, the beam needs to hit those same points to find its way back to the start. However if the initial point chosen is a divisor of the total number of points on the base case polygon, you will not hit every point before reach the start point again.

Where $n\mathord{+}1$ is a prime, as for your example, the answer will be $n$.

Answer 2

The points where the beam first hits the circle are at angles $\frac{2k\pi}{n+1}$ from the initial point. If $\gcd(k,n+1)\gt1$, then the beam will return to the start point after $\frac{n+1}{\gcd(k,n+1)}-1$ bounces, so we only want to count the number of $1\le k\le n$ so that $\gcd(k,n+1)=1$. This is $\phi(n+1)$ where $\phi$ is the Euler Totient Function.

Since $\phi(4+1)=4$, we get $4$ ways.