2025/02/06 by Thomas C. Hull, Hull, Thomas C., Marcus Michelen +2
Computer Science · Engineering · #68U05 #Advanced Materials and Mechanics #Combinatorics (math.CO) #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #FOS: Computer and information sciences #FOS: Mathematics #Primary: 60J10 #Probability (math.PR) #Secondary: 52C45 #Structural Analysis and Optimization
paper · pdf · doi:10.48550/arxiv.2502.04279
openalex publication_date 2025/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.