On random locally flat-foldable origami (2502.04279v1)

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.

• The paper analyzes random locally flat-foldable origami and proposes the face-flip Markov chain for sampling crease patterns.
• It introduces the face-flip Markov chain and demonstrates its polynomial mixing times for various patterns like the square grid and Miura-ori, making it an efficient sampling tool.
• The study examines local versus global flat-foldability in specific patterns, noting the computational challenge in identifying globally flat-foldable configurations.

An Overview of "On Random Locally Flat-foldable Origami"

The paper, titled "On Random Locally Flat-Foldable Origami," authored by Thomas C. Hull, Marcus Michelen, and Corrine Yap, offers a comprehensive theoretical analysis of random flat-foldable origami and the potential of the face-flip Markov chain as a suitable method for efficient sampling in origami tessellations. By focusing on distinct crease patterns such as the square twist, square grid, and Miura-ori, the authors delve into the complexities and intricacies of both local and global flat-foldability, providing a nuanced understanding of origami from a mathematical and probabilistic viewpoint.

Key Themes and Findings

1. Local vs. Global Flat-Foldability: In origami, a crease pattern comprises edges and vertices with specific mountain-valley assignments. A pattern is flat-foldable locally if it can be folded around each vertex without clashes, while global flat-foldability ensures that the entire pattern can be flattened in three-dimensional space without self-intersection. The paper postulates exponential improbability in the transition from local to global flat-foldability within the square grid, underscored by Theorem 4.2, highlighting the computational challenges in identifying globally flat-foldable configurations.
2. The Face-Flip Markov Chain: The authors propose the face-flip Markov chain as a means to sample from locally flat-foldable configurations. This chain involves randomly selecting a face of the origami pattern and flipping its mountain and valley assignments uniformly, conditioned on the face being flippable. The authors demonstrate that this Markov chain results in polynomial mixing times for various origami patterns, establishing it as an effective tool for sampling flat-foldable crease patterns.
3. Origami Patterns Examined: Various established origami crease patterns such as the square twist, square grid, Miura-ori, and single-vertex crease patterns are scrutinized through the lens of local flat-foldability and its uniform random configurations. Their mixing times under the face-flip Markov chain are explored, confirming feasible polynomial bounds, particularly with $O(n \log n)$ for the square twist and square grid and $O(m^4n^9)$ for the Miura-ori using associations with vertex coloring models.
4. Sampling Approaches for Single-Vertex Crease Patterns: The authors develop an efficient sampling method for single-vertex crease patterns by leveraging the structural properties of such configurations where sector angles influence feasible mountain-valley assignments. This exact sampler has a complexity of $O(n)$.
5. Future Explorations and Complexity: The paper emphasizes the need for further investigations into origami crease patterns where global flat-foldability remains computationally intensive. Specific state spaces like the kite crease pattern demonstrate non-flippable complexities unattainable through the face-flip methodology, suggesting alternate approaches are needed for sampling flat-foldable configurations.

Implications and Further Research Directions

The theoretical model developed in this paper provides a foundation for understanding the likelihood of various crease pattern foldabilities in practical and computational contexts. This work paves the way for using Markov chain methods in designing algorithms capable of efficiently probing the space of flat-foldable configurations, which holds potential applications in fields such as materials science, aerospace engineering, and robotics, where compact deployable structures are vital.

Future work should consider exploring complexity classes of new origami models and examining whether alternative Markov chain Monte Carlo techniques might yield more efficient convergence or whether entirely new sampling strategies are warranted. Additionally, integrating deeper geometric analysis into this probabilistic framework could enhance our understanding of origami's potential as a form of engineering material.

In conclusion, by synthesizing complex mathematical principles and probabilistic models, this paper enriches the dialogue surrounding computational origami, offering novel insights and proposing compelling methodologies for the exploration of random flat-foldable configurations.