[Paper Review] On the existence of four or more curved foldings with common creases and crease patterns

Atsufumi Honda, Kosuke Naokawa+3 more|arXiv (Cornell University)|Nov 17, 2019

Advanced Materials and Mechanics13 references4 citations

TL;DR

This paper investigates the number of non-congruent curved foldings possible with identical crease patterns and creases in 3D space. Using symmetry analysis of the crease curve and its planar pattern, it proves that for a non-closed, symmetry-free crease and pattern, exactly four distinct curved foldings exist; otherwise, the number drops to two or fewer. The work extends prior results on developable surfaces in origami geometry.

ABSTRACT

Consider an oriented curve $Γ$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding in the Euclidean space $\boldsymbol R^3$. This can be expressed as the image of an "origami map" $Φ:D o \boldsymbol R^3$ such that $Γ$ is the singular set of $Φ$, the word "origami" coming from the Japanese term for paper folding. We call the singular set image $C:=Φ(Γ)$ the crease of $Φ$ and the singular set $Γ$ the crease pattern of $Φ$. We are interested in the number of origami maps whose creases and crease patterns are $C$ and $Γ$, respectively. Two such possibilities have been known. In the authors' previous work, two other new possibilities and an explicit example with four such non-congruent distinct curved foldings were established. In this paper, we determine the possibility of the number $N$ of congruence classes of curved foldings with the same crease and crease pattern. As a consequence, if $C$ is a non-closed simple arc, then $N=4$ if and only if both $Γ$ and $C$ do not admit any symmetries. On the other hand, when $C$ is a closed curve, there are infinitely many distinct possibilities for curved foldings with the same crease and crease pattern, in general.

Motivation & Objective

• To determine the maximum number of non-congruent curved foldings that can share the same crease and crease pattern in 3D space.
• To analyze how symmetries of the crease curve $C$ and its planar pattern $Γ$ affect the number of distinct curved foldings.
• To generalize previous results on admissible curved foldings by establishing a complete classification based on symmetry properties.
• To resolve the existence and count of such foldings when $C$ is a closed curve, showing infinitely many possibilities in general.

Proposed method

• Defining curved foldings as developable surfaces in $\mathbb{R}^3$ with a singular set mapped from a planar curve $\Gamma$.
• Introducing the concept of 'admissible' curved foldings satisfying curvature and length conditions (i)-(iv') from prior work.
• Using isometry-based symmetry analysis on both $\Gamma \subset \mathbb{R}^2$ and $C \subset \mathbb{R}^3$, distinguishing between positive and negative symmetries.
• Applying differential geometry tools such as arc-length parametrization, curvature functions, and Frenet-Serret formulas to characterize the folding behavior.
• Proving that the number of congruence classes depends on whether $C$ and $\Gamma$ admit non-trivial symmetries.
• Constructing explicit examples via reparametrization of ellipses and scaling to match crease lengths, demonstrating four non-congruent foldings.

Experimental results

Research questions

• RQ1 What is the maximum number of non-congruent curved foldings that can be realized from the same crease pattern and crease curve in $\mathbb{R}^3$?
• RQ2 How do symmetries of the crease curve $C$ and its planar pattern $\Gamma$ constrain the number of distinct curved foldings?
• RQ3 Under what geometric conditions does the number of such foldings reach four, and when does it drop to two or fewer?
• RQ4 Can infinitely many distinct curved foldings exist for a closed crease curve $C$ with the same $\Gamma$?
• RQ5 When does a single pair $(\Gamma, |C|)$ yield only one congruence class of curved foldings?

Key findings

• For a non-closed, simple arc $C$ in $\mathbb{R}^3$, the number $N$ of congruence classes of curved foldings is exactly four if and only if both $C$ and $\Gamma$ have no non-trivial symmetries.
• If either $C$ or $\Gamma$ admits a non-trivial symmetry, then $N \leq 2$.
• When $C$ is a closed curve, there can be infinitely many distinct curved foldings with the same $\Gamma$ and $|C|$, in general.
• The existence of four non-congruent curved foldings was confirmed via an explicit example using a scaled ellipse as $\Gamma$ and a closed space curve $C_3$ of matching length.
• A developable strip $F$ along $C_3$ was constructed such that its curvature function matched the reparametrized ellipse, yielding three distinct foldings shown in Figure 7.
• The paper proves that a space curve $C$ admits both positive and negative symmetries simultaneously if and only if it lies in a plane and has a non-trivial symmetry.