# Polygonal Folding

## Findings related to paper folding and computational geometry.

### January 2021 to Present | Mathematics Research

I was first introduced to research in the field of mathematics under the Northwestern Undergraduate Research Assistant Program (URAP) during my first year. I saw a prompt to study geometry and paper-folding under guidance of Dr. Shuyi Weng. After submitting my documents and participating in an interview, I heard back that I was accepted and I was going to work on this project with another student, Jade Zhang. We started by learning the theory and background material with lectures, reading papers, and, of course, folding paper.

The research topic studies the relationship between the starting two-dimensional shape and the final three-dimensional shape that it becomes, when glued to a companion shape. How is this companion shape generated? My mentor developed what is called the "cap construction algorithm" to take any polygon, distribute uniform curvature to each vertex, and generate a "polygonal cap curve" with the same number of sides and side lengths. Thus, corresponding sides get glued together to create a three-dimensional polyhedron. We are particularly interested in which kinds of starting shapes allow this construction to create a valid, closed polyhedron. My mentor's previous work revelead that equilateral triangles are the only three-sided shapes which work, and parallelograms are the only four-sided shapes which work. Thus, it was natural for us to start examining five-sided shapes: pentagons.

While exploring test cases in software such as Mathematica, we eventually stumbled on quite an interesting phenomenon: it appeared as if, as one moved any of the vertices of the pentagon along a line, the endpoint of the cap curve would also appear to move along a line. This raised suspicion as to whether there was a linear (or affine) relationship regarding dependencies of the points of these curves. For the next school year, Dr. Weng and I continued to explore this phenomenon and its implications. We found that the endpoint of the polygonal cap curve indeed was affine dependent on any of the vertices of the starting polygon. Furthermore, we were able to show that there is a clean linear relation which precisely defines when the closed cap condition is satisfied for a given number of vertices. We spent much of our time formalizing these conclusions in proofs and a paper which was recently submitted to be reviewed for publication.

I had a wonderful time presenting these findings at the Joint Mathematics Meeting in Boston, in January 2023. I am so grateful for all of the help I received from Jade, my mentor, and my various math professors. It feels so rewarding to see my work for the past few years come to fruition.

To wrap up this project, I am currently revising the paper and preparing it for submission to *Involve*.

#### Awards & Presentations

- AMS-PME Joint Mathematics Meeting, 10-minute oral presentation, 1/5/2023
- Summer Undergraduate Research Grant of $4000, 4/11/2022
- Northwestern Undergraduate Research Exposition, 10-minute oral presentation, 5/26/2021
- Northwestern Undergraduate Research Assistant Program funding up to $1500, Winter 2021 to Spring 2021

#### Media

- My slide deck for my presentation at JMM, 1/5/2023
- Our research paper on arXiv, 10/1/2022
- Our presentation at the Northwestern Undergraduate Research Exposition, 5/27/2021