Background:
Paint the origin $(0,0)$ black in $\mathbb{R}^2$.
Let $S$ be a set $\{ (x,y) \in \mathbb{R}^2 ~|~ x^2 + y^2 =1 \}$.
Paint $S$ black.
Paint $(u,v) +S$ black for all $(u,v) \in S$.
(Then we can easily check that all points with $\|(x,y)\| \leq 1$ is painted.)
Paint $(u',v') + S$ black for all the previous black painted points $(u',v')$.
If we repeat this process, then we can paint all points in the plane.
Especially, we can find out that every points $(x,y)$ is painted in the finite process.
Question:
I shall do almost same thing.
Denote $O=(0,0)$, $A=(1,0)$.
Let $S'$ be a set $\{ B \in S ~|~ \frac{\angle BOA}{2\pi} \in \mathbb{Q} \}$.
Paint the origin $(0,0)$ black.
Paint $(u,v) + S'$ black for all the previous black painted points $(u,v)$.
Repeat this process.
Then, can we paint all points in the plane? i.e.,
For every point $(x,y) \in \mathbb{R}^2$, is $(x,y)$ painted in the finite process?