On random locally flat-foldable origami

2025-02-06

AI Summary

This paper investigates the theoretical properties of randomly assigned mountain-valley (M/V) crease patterns in locally flat-foldable origami: given a crease pattern satisfying local flat-foldability constraints at each vertex, what is the probability that a uniformly random M/V assignment yields global flat-foldability? The authors first establish polynomial mixing time for the face-flip Markov chain on square-twist, Miura-ori, square-grid, and single-vertex origami models—providing an efficient algorithmic foundation for random sampling. They then rigorously prove an exponential separation: on the square grid, asymptotically almost all locally flat-foldable M/V assignments are *not* globally flat-foldable. Unifying Markov chain mixing theory, combinatorial geometry, and constraint propagation analysis, the work not only demonstrates rapid mixing for multiple canonical origami structures but also quantifies a fundamental gap between local and global flat-foldability—yielding critical theoretical support for probabilistic modeling and feasibility analysis in computational origami.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Develops theory of random flat-foldable origami.
• Analyzes face-flip Markov chain for origami tessellations.
• Compares local and global flat-foldability probabilities.

Innovation

• random flat-foldable origami theory
• face-flip Markov chain sampling
• local vs global flat-foldability comparison