Origami rings

TL;DR

This paper investigates the algebraic structure of points generated by origami folding rules with specific angles, showing they form rings related to roots of unity, which answers a key origami construction question.

Contribution

It proves that sets of points generated by origami folds with angle groups form rings, specifically identifying these rings as cyclotomic rings for equally spaced angles.

Findings

Generated sets form subrings of the complex plane.

For equally spaced angles, the sets are rings like lgebraic integers in cyclotomic fields.

The structure depends on whether the number of angles is prime or composite.

Abstract

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 $\RU$ 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 $\RU$ 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 $k\pi/n$, $0 \le k < n$, then $\RU$ is the ring $\Z[\zeta_n]$ if $n$ is prime, and the ring $\Z[1/n,\zeta_{n}]$ if $n$ is not prime, where $\zeta_n := \exp(2\pi i/n)$ is a primitive $n$-th root of unity.