On random locally flat-foldable origami

TL;DR

This paper develops a probabilistic framework for analyzing random flat-foldable origami crease patterns, introducing a Markov chain approach and demonstrating rapid mixing for certain tessellations, while also exploring local versus global flat-foldability.

Contribution

It introduces a Markov chain method to sample random flat-foldable origami configurations and proves rapid mixing for key origami tessellations, advancing understanding of origami folding probabilities.

Findings

The face-flip Markov chain mixes rapidly for several origami tessellations.

On the square grid, locally flat-foldable configurations are exponentially unlikely to be globally flat-foldable.

Theoretical analysis of flat-foldability conditions for various crease patterns.

Abstract

We develop a theory of random flat-foldable origami. Given a crease pattern, we consider a uniformly random assignment of mountain and valley creases, conditioned on the assignment being flat-foldable at each vertex. A natural method to approximately sample from this distribution is via the face-flip Markov chain where one selects a face of the crease pattern uniformly at random and, if possible, flips all edges of that face from mountain to valley and vice-versa. We prove that this chain mixes rapidly for several natural families of origami tessellations -- the square twist, the square grid, and the Miura-ori -- as well as for the single-vertex crease pattern. We also compare local to global flat-foldability and show that on the square grid, a random locally flat-foldable configuration is exponentially unlikely to be globally flat-foldable.