Fundamental domains of Dihedral groups

Question

Let $D_n$ be dihedral group of order $2n$, it acts on plane $\mathbb{R}^2$ in a standard way, by rotations and reflections. How one can find fundamental domains for such action?

Answer 1

There's probably a better way that this, but my answer would be "by guesswork". Let's look at $D_3$ as an example. I'd start by guessing "in polar coordinates, a wedge $E$ from $0$ to $2\pi/3$." Clearly the order-3 subgroup $\{e, a, a^2\}$of $D_3$ sends this to the entire plane (i.e., $E \cup aE \cup a^2E = R^2$). If we say that the order-two subgroup $\{e, b\}$ is generated by reflection through the $x$-axis (i.e., $(x, y) \mapsto (x, -y)$, then $bE \cap a^2E$ is a wedge of size $2\pi/6$. That's bad. So let's cut it in half! Declare $U$ to be the wedge from $\theta = 0$ to $2\pi/6$ in polar coordinates.

There's a small problem: all elements of $D_3$ send the origin to itself, so $U \cap gU$ will always contain the origin. So my answer is really an answer for the case of $D_n$ acting on the punctured plane. (Perhaps I also have a problem with whether we use $0 \le \theta \le 2\pi/6$, or whether one or both needs to be a "<" instead of $\le$.)