[Paper Review] Double-line rigid origami

Thomas C. Hull, Tomohiro Tachi|arXiv (Cornell University)|Sep 11, 2017

Advanced Materials and Mechanics9 references3 citations

TL;DR

This paper introduces the double-line rigid origami method, which transforms higher-degree or non-flat-foldable rigid origami vertices into equivalent systems of degree-4 flat-foldable vertices by replacing each crease with two parallel creases connected via a polygonal frame. The method preserves the original kinematics and enables full rigid folding without panel collision, even with thick materials, by distributing fold angles across double lines and ensuring rigid-foldability through symbolic analysis of symmetric 2n-degree vertices.

ABSTRACT

In this paper, we will show methods to interpret some rigid origami with higher degree vertices as the limit case of structures with degree-4 supplementary angle vertices. The interpretation is based on separating each crease into two parallel creases, or \emph{double lines}, connected by additional structures at the vertex. We show that double-lined versions of degree-4 flat-foldable vertices possess a rigid folding motion, as do symmetric degree-$2n$ vertices. The latter gives us a symbolic analysis of the original vertex, showing that the tangent of the quarter fold angles are proportional to each other. The double line method is also a potentially useful in giving thickness to rigid origami mechanisms. By making single crease into two creases, the fold angles can be distributed to avoid $180^\circ$ folds, when panels can easily collide with each other. This can be understood as an extension of the crease offset method of thick rigid origami with an additional guarantee of rigid-foldability.

Motivation & Objective

• To develop a method for interpreting higher-degree or non-flat-foldable rigid origami vertices as limits of degree-4 flat-foldable vertex systems.
• To enable full rigid folding of thick-panel origami mechanisms by distributing fold angles across double lines to prevent 180° folds and panel collisions.
• To extend the symbolic analysis of fold angle proportionality from degree-4 flat-foldable vertices to symmetric 2n-degree vertices via the double-line transformation.
• To provide a systematic way to construct rigid-foldable, thickened origami mechanisms with guaranteed kinematic behavior.

Proposed method

• Replace each crease in a vertex with two parallel creases by constructing a small polygon (the 'double-line frame') around the vertex, connected perpendicularly to each original crease.
• Define the double-line version of a vertex, DL(V), by extending two parallel lines from each corner of the frame, forming a new crease pattern composed only of degree-4 vertices.
• Preserve the original sector angles of the vertex while separating them across the double-line structure, ensuring geometric consistency.
• Use symbolic analysis to show that symmetric 2n-degree vertices with equal sector angles have fold angles whose tangents of the quarter angles are proportional, mirroring the behavior of degree-4 flat-foldable vertices.
• Ensure rigid-foldability of multi-vertex systems by adjusting folding modes and fold angles to maintain consistent double-line ratio at shared edges.
• Demonstrate that serially connected double-line vertices form a rigidly foldable mechanism, and that symmetric tessellations like Miura-ori and elongated Yoshimura patterns can also be adapted into double-line forms with rigid-foldability.

Experimental results

Research questions

• RQ1 Can higher-degree or non-flat-foldable rigid origami vertices be interpreted as limits of degree-4 flat-foldable vertex systems through a geometric transformation?
• RQ2 How can the double-line method preserve the kinematic behavior of the original crease pattern while enabling full rigid folding with thick panels?
• RQ3 What conditions ensure that the double-line version of a symmetric 2n-degree vertex remains rigidly foldable and maintains proportional fold angle dynamics?
• RQ4 Can the double-line method be extended to multi-vertex systems with cycles, such as tessellated patterns, while preserving rigid-foldability?
• RQ5 What is the role of fold angle distribution across double lines in preventing panel collisions during full folding of thick rigid origami?

Key findings

• The double-line version of a degree-4 flat-foldable vertex is rigidly foldable, preserving the original fold angle proportionality and motion characteristics.
• Symmetric 2n-degree vertices with equal sector angles exhibit fold angle dynamics where the tangents of the quarter fold angles are proportional, enabling symbolic analysis similar to degree-4 flat-foldable vertices.
• The double-line method allows for full rigid folding of thick-panel origami by distributing fold angles across two parallel creases, avoiding 180° folds that cause panel collisions.
• Multi-vertex systems composed of serially connected double-line vertices remain rigidly foldable, with the ability to connect new vertices by tuning the folding mode and angle of the new vertex.
• Symmetric tessellations such as Miura-ori and elongated Yoshimura patterns can be adapted into double-line forms that maintain rigid-foldability, demonstrating the method’s applicability to practical deployable structures.
• The double-line method provides a systematic, geometrically consistent way to thicken rigid origami mechanisms while preserving their kinematic behavior and deployability.