The Complexity of Folding Self-Folding Origami

Menachem Stern,Arvind Murugan,Matthew B Pinson

doi:10.1103/physrevx.7.041070

Abstract

Why is it difficult to refold a previously folded sheet of paper? We show that even crease patterns with only one designed folding motion inevitably contain an exponential number of `distractor' folding branches accessible from a bifurcation at the flat state. Consequently, refolding a sheet requires finding the ground state in a glassy energy landscape with an exponential number of other attractors of higher energy, much like in models of protein folding (Levinthal's paradox) and other NP-hard satisfiability (SAT) problems. As in these problems, we find that refolding a sheet requires actuation at multiple carefully chosen creases. We show that seeding successful folding in this way can be understood in terms of sub-patterns that fold when cut out (`folding islands'). Besides providing guidelines for the placement of active hinges in origami applications, our results point to fundamental limits on the programmability of energy landscapes in sheets.

Highlights

Single-degree-of-freedom (d.o.f.) mechanical structures are attractive in a range of fields, as almost any force will actuate that specific designed mode
Much like an umbrella or a folding chair, such “self-folding” structures can be reliably deployed even in uncertain environments with unreliable actuation forces. This principle has found wide use in kinetic or deployable architecture, heart stents, microelectromechanical systems (MEMS), sensors, and robots on a range of length scales [1,2,3,4]; recently, self-folding origami has become a popular framework for such applications [5,6,7,8,9,10]
Sheets with crease patterns designed to exhibit exactly one folding behavior are difficult to fold. We traced this difficulty to the fact that stabilizing one folding behavior using frustrated interactions between binary d.o.f. inevitably stabilizes an exponential number of other distractor behaviors, i.e., a complex or glassy landscape [43]

Summary

Introduction

Single-degree-of-freedom (d.o.f.) mechanical structures are attractive in a range of fields, as almost any force will actuate that specific designed mode. Much like an umbrella or a folding chair, such “self-folding” structures can be reliably deployed even in uncertain environments with unreliable actuation forces This principle has found wide use in kinetic or deployable architecture, heart stents, microelectromechanical systems (MEMS), sensors, and robots on a range of length scales [1,2,3,4]; recently, self-folding origami has become a popular framework for such applications [5,6,7,8,9,10].

Methods

Results

Conclusion