Suppose you have a random point inside a square and you set the point moving in a random direction. When the point hits a side of the square, it bounces off the side like a billiard ball, i.e. angle in $=$ angle out.
Will the point always end up in a fixed pattern, i.e. will it always come to a position it has been at before, moving in the same direction it did at that earlier time?
I suspect the answer is No, based on computer simulations, but then computers have limited accuracy. It is clear that there are many cases where a fixed pattern does occur. One example is if the initial angle of movement $\theta$ is a multiple of $45^\circ$. In fact, it seems to me that if $\theta = \text{arctan}(\frac{1}{n})$, where $n$ is a positive integer, a fixed pattern will occur.
But will a fixed pattern occur for any starting point and angle? If not, why not?
A fixed pattern occurs if and only if the slope of the first direction is rational. Otherwise, it doesn't occur, and moreover, you can prove that the trajectory is dense in the square (which means that it approaches every point of the square with every precision you can choose).