Branching Paths Problem I was drawing some shapes during class, and I came across the following problem. If one takes steps of constant length, and one must deviate a constant angle $\alpha$ from one's previous step either left or right on the next step, for an angle $\alpha$ what is the set of all possible points I can reach on my path? A point is defined as the end of a step.
Here is a poorly drawn diagram of the problem:


NOTE: This is perhaps assuming you don't have to turn every time, i.e. you can just travel one unit in the current direction. It depends on the angle. If the angle is $2 \pi, \pi, \pi / 2, 3 \pi / 2$,$2 \pi / 3$,$4 \pi / 3$, $\pi / 3$ or $5 \pi / 3$ then the set of points you can reach will form what's called a "lattice," and "regular tiling" which you can look up but it basically means your points are regularly spaced from one another and form a regular tiling pattern that has some form of rotational symmetry (and for $2\pi$ or $\pi$, it will be a regular pattern along a horizontal line). If the angle is $\alpha \pi$ where $\alpha$ is irrational, then I believe the set of points is hard to describe with some form of closed form parametric formula or nice description, but the set of points you can reach should be dense in the plane (but not all the plane, because the set of points you can reach is countable whereas the set of points in the plane is uncountable). If $\alpha$ is rational not one of the values I described earlier, then I think it's more complicated, and the set of points you can reach may not dense. Maybe someone can leave a comment and then I can update with a more definitive descriptive answer for that case.
