Origami rings
Joe Buhler, Steve Butler, Warwick de Launey, Ron Graham

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 be the line in the complex plane through with angle (with respect to the real axis). Given a fixed collection of angles, let be the points that can be obtained by starting with and , and then recursively adding intersection points of the form , where have been constructed already, and are distinct angles in . Our main result is that if is a group with at least three elements, then 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 and the allowable angles are the equally spaced angles , , then is the…
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Taxonomy
TopicsAdvanced Materials and Mechanics
