Symmetry groups of origami structures

TL;DR

This paper explores the mathematical symmetry groups of origami patterns by analyzing their geometric and algebraic properties, classifying them into wallpaper groups or dense patterns through algorithmic methods.

Contribution

It introduces a novel approach to classify origami symmetry groups using complex plane analysis and algorithms, extending understanding of origami pattern symmetries.

Findings

Identifies which wallpaper groups can be realized through origami patterns.

Provides an algorithm for classifying dense, non-periodic origami patterns.

Connects origami symmetries with mathematical wallpaper groups.

Abstract

Origami is the art of folding paper into various patterns without cutting or tearing the paper. By viewing the paper as the complex plane, we iteratively compute and record all intersection points to construct mathematical origami sets. Additionally, we include the various lines to create a repeating pattern that can be viewed as a wallpaper group if the angle set contains fewer than 3 angles. There are 17 wallpaper groups up to isomorphism, so we determine which such groups can be constructed in this way, depending on the rotational and reflectional symmetries present in the given pattern. If the angle set contains more than 4 angles, the resulting pattern will be dense and hence no longer a wallpaper group. In this case, the classification of the symmetry group is done algorithmically.