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Origami rings

Motivated by a question in origami, we consider sets of points in the complex plane constructed in the following way. Let $L_α(p)$ be the line in the complex plane through $p$ with angle $α$ (with respect to the real axis). Given a fixed collection $U$ of angles, let $\RU$ be the points that can be obtained by starting with $0$ and $1$, and then recursively adding intersection points of the form $L_α(p) \cap L_β(q)$, where $p, q$ have been constructed already, and $α, β$ are distinct angles in $U$. Our main result is that if $U$ is a group with at least three elements, then $\RU$ 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 $kπ/n$, $0 \le k < n$, then $\RU$ is the ring $\Z[ζ_n]$ if $n$ is prime, and the ring $\Z[1/n,ζ_{n}]$ if $n$ is not prime, where $ζ_n := \exp(2πi/n)$ is a primitive $n$-th root of unity.