Department or Program

Mathematics

Abstract

In origami, an artist uses paper to construct a three-dimensional object by making folds from a set of seed points. The intersections formed from the folds are then used as reference points for new folds. Abstractly, we can represent the paper with a plane. We form a subset of this plane by intersecting along angles from seed points, and we are interested in special properties of this subset. Under certain constraints, the origami construction gives rise to a subset with mathematical structure, including the topological structure of a lattice or the algebraic structure of a subring. We first explore the conditions that give rise to an origami ring that is a subset of the complex numbers. The complex plane is an example of a Euclidean space, which is constructed using the Parallel Postulate. When this axiom is altered, we work with the hyperbolic plane, where multiple parallel hyperbolic lines can intersect a particular point. In our origami constructions, new reference points are made by intersecting two lines. Since the hyperbolic plane has fundamentally different geometry due to its axiomatization, constructed points are different when we operate in the hyperbolic plane compared to when they start in the complex plane. We reach partial results showing that the origami procedure is different in the hyperbolic plane. However, we do show a new classification of origami lattices by using the classical modular group as a moduli space for complex lattices, and raise new questions about the containment of all lattices in origami lattices.

Level of Access

Open Access

First Advisor

Salerno, Adriana

Date of Graduation

5-2020

Degree Name

Bachelor of Science

Number of Pages

97

Components of Thesis

1 pdf file

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