When is an origami set a ring?

TL;DR

This paper characterizes the set of points constructible through origami folds with slope restrictions and identifies conditions under which this set forms a ring, addressing questions in geometric origami theory.

Contribution

It provides an explicit description of constructible points under slope-restricted folds and criteria for when this set forms a ring, advancing geometric origami understanding.

Findings

Explicit characterization of constructible points

Criteria for the constructible set to be a ring

Answers to open questions in origami geometry

Abstract

Starting with a flat sheet of paper, points can be constructed as the intersection of two folds. The set of constructible points clearly depends on which folds are admissible. In this paper, we study the situation where a fold is admissible if its slope is admissible and it contains an already constructed or a generator point. We give an explicit characterization of the set of constructible points. We also state several criteria for this set to be a ring. This answers questions originally raised by Erik Demaine and discussed by Butler et al. and Buhler et al.