Higher dimensional origami constructions

TL;DR

This paper extends the mathematical framework of origami-inspired constructions from the complex plane to higher dimensions, analyzing the algebraic and geometric properties of the resulting point sets and their lattice structures.

Contribution

It generalizes previous two-dimensional origami constructions to higher dimensions and investigates how angles influence the lattice structures formed.

Findings

Extended origami constructions to higher dimensions.

Analyzed the relationship between angles and lattice generators.

Determined how new angles modify lattice structures.

Abstract

Origami is an ancient art that continues to yield both artistic and scientific insights to this day. In 2012, Buhler, Butler, de Launey, and Graham extended these ideas even further by developing a mathematical construction inspired by origami -- one in which we iteratively construct points on the complex plane (the "paper") from a set of starting points (or "seed points") and lines through those points with prescribed angles (or the allowable "folds" on our paper). Any two lines with these prescribed angles through the seed points that intersect generate a new point, and by iterating this process for each pair of points formed, we generate a subset of the complex plane. We extend previously known results about the algebraic and geometric structure of these sets to higher dimensions. In the case when the set obtained is a lattice, we explore the relationship between the set of angles and the generators of the lattice and determine how introducing a new angle alters the lattice.