On random locally flat-foldable origami

TL;DR

The paper studies random origami by conditioning a uniform MV assignment on local flat-foldability, and provides polynomial-time sampling guarantees via the face-flip Markov chain for several crease-pattern families. By mapping the MV state space to well-understood combinatorial structures (hypercubes, Cayley graphs, and Glauber dynamics for colorings) and using induced chains and comparison techniques, it proves polynomial mixing times for square twist, square grid, Miura-ori, and single-vertex patterns. A key finding is that global flat-foldability is exponentially rare among locally flat-foldable configurations on the square grid, while the developed samplers yield efficient approximate sampling of MV configurations. The results advance understanding at the intersection of origami combinatorics, Markov chain theory, and statistical physics, enabling practical sampling and providing insights into the complexity landscape of flat-foldability across different crease-pattern families.

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.

Figures (8)

• Figure 1: (a) A single-vertex crease pattern exhibiting the Big-Little-Big Theorem. (b) A degree-6 vertex with all equal angles between consecutive creases.
• Figure 2: (a) The square twist crease pattern. (b) Two ways to tessellate square twists, with isotropic/anisotropic 2-colorings shown. (c) A valid MV assignment for a square twist tessellation, with an illustration of the folded image (made using TessLang). (d) The possible MV assignments for trapezoid and parallelogram faces.
• Figure 3: (a) A $4\times 6$ example of the Miura-ori crease pattern, with MV assignment that folds up in the standard way. (b) Illustrating the bijection between locally-valid MV assignments of an $m\times n$ Miura-ori and proper 3-vertex colorings of the $m\times n$ grid graph (with one vertex pre-colored).
• Figure 4: The non-flat-foldable MV assignment $\sigma_{\hbox{sp}}$ on the crease pattern $S_{2,5}$.
• Figure 5: An example of a neighborhood. Outlined in red is a $1 \times 2$ subset $T$; adding the creases highlighted in blue gives $N(T)$. Moreover, $N(N(T))$ is the entire crease pattern $S_{4,5}$.

Theorems & Definitions (41)

• Theorem 1
• Theorem 2
• Theorem 3: Kawasaki's Theorem
• Theorem 4: Maekawa's Theorem
• Theorem 5: Big-Little-Big Theorem
• Definition 6
• Definition 7
• Definition 8
• Theorem 9
• Lemma 10