# The difficulty of folding self-folding origami

## TL;DR

Refolding self-folding origami is inherently complex due to an exponential number of folding pathways, requiring precise actuation and revealing fundamental limits on origami programmability.

## Contribution

This paper demonstrates that even simple crease patterns have exponentially many folding pathways, linking origami refolding difficulty to NP-hard problems and energy landscape complexity.

## Key findings

- Exponential number of distractor folding branches exist in simple crease patterns.
- Refolding involves navigating a glassy energy landscape with many local minima.
- Successful refolding can be guided by identifying 'folding islands' or sub-patterns.

## 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.