Origami rings

Motivated by a question in origami, we consider sets of points in the complex plane constructed in the following way. Let $L_{\alpha }(p)$ be the line in the complex plane through $p$ with angle $\alpha$ (with respect to the real axis). Given a fixed collection $U$ of angles, let $\text{RU}$ be the points that can be obtained by starting with $0$ and $1$, and then recursively adding intersection points of the form $L_{\alpha }(p)\cap L_{\beta }(q)$, where $p,q$ have been constructed already, and $\alpha ,\beta$ are distinct angles in $U$. Our main result is that if $U$ is a group with at least three elements, then $\text{RU}$ is a subring of the complex plane, i.e., it is closed under complex addition and multiplication. This enables us to answer a specific question about origami folds: if $n\ge 3$ and the allowable angles are the $n$ equally spaced angles $k\pi \mathrm{/}n$, $0\le k<n$, then $\text{RU}$ is the ring $\mathbb{Z}[{\zeta }_n]$ if $n$ is prime, and the ring $\mathbb{Z}[1\mathrm{/}n,{\zeta }_n]$ if $n$ is not prime, where ${\zeta }_n:=\exp (2\pi i\mathrm{/}n)$ is a primitive $n$-th root of unity.