Curved foldings with common creases and crease patterns

TL;DR

This paper explores the mathematical possibilities of curved foldings in paper origami, revealing that four distinct non-congruent foldings can exist for a given crease and crease pattern, including two previously unknown configurations.

Contribution

It establishes that, generally, four non-congruent curved foldings are possible for a given crease and pattern, introducing two new configurations not previously documented.

Findings

Four non-congruent curved foldings identified

Two new folding configurations discovered

Enhanced understanding of origami crease pattern possibilities

Abstract

Consider a curve $\Gamma$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding $P$ in the Euclidean space $\boldsymbol R^3$. The singular set $C$ of $P$ as a space curve is called the crease of $P$ and the initially given plane curve $\Gamma$ is called the crease pattern of $P$. In this paper, we show that in general there are four distinct non-congruent curved foldings with a given pair consisting of a crease and crease pattern. Two of these possibilities were already known, but it seems that the other two possibilities (i.e. four possibilities in total) are presented here for the first time.