On random locally flat-foldable origami
Thomas C. Hull, Marcus Michelen, Corrine Yap

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.
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Taxonomy
TopicsAdvanced Materials and Mechanics · Structural Analysis and Optimization · Computational Geometry and Mesh Generation
