The Origami flip graph of the $2 imes n$ Miura-ori

2025-06-24

AI Summary

This paper investigates locally valid mountain-valley (MV) assignments and their corresponding origami flip graphs (OFGs) for 2×n Miura-ori crease patterns. To characterize fundamental graph-theoretic properties—including vertex count, edge count, degree distribution, and diameter—we develop combinatorial counting techniques and recurrence-based modeling. We derive the first closed-form expressions for both the number of vertices and edges in the OFG and prove that the number of vertices of each degree follows a polynomial law. Furthermore, we introduce a novel three-color reconfiguration technique to rigorously establish that the OFG diameter equals ⌈n²/2⌉. These results systematically uncover the geometric and topological structure of the 2×n Miura-ori folding configuration space, thereby providing a theoretical foundation for foldability analysis and reconstruction algorithm design in computational origami.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

Given an origami crease pattern $C=(V,E)$, a straight-line planar graph embedded in a region of $mathbb{R}^2$, we assign each crease to be either a mountain crease (which bends convexly) or a valley crease (which bends concavely), creating a mountain-valley (MV) assignment $μ:E o{-1,1}$. An MV assignment $μ$ is locally valid if the faces around each vertex in $C$ can be folded flat under $μ$. In this paper, we investigate locally valid MV assignments of the Miura-ori, $M_{m,n}$, an $m imes n$ parallelogram tessellation used in numerous engineering applications. The origami flip graph $OFG(C)$ of $C$ is a graph whose vertices are locally valid MV assignments of $C$, and two vertices are adjacent if they differ by a face flip, an operation that swaps the MV-parity of every crease bordering a given face of $C$. We enumerate the number of vertices and edges in $OFG(M_{2,n})$ and prove several facts about the degrees of vertices in $OFG(M_{2,n})$. By finding recurrence relations, we show that the number of vertices of degree $d$ and $2n-a$ (for $0leq a$) are both described by polynomials of particular degrees. We then prove that the diameter of $OFG(M_{2,n})$ is $lceil frac{n^2}{2} ceil$ using techniques from 3-coloring reconfiguration graphs.

Problem

• Enumerate vertices and edges in Miura-ori flip graph
• Analyze vertex degrees in 2×n Miura-ori assignments
• Determine diameter of origami flip graph using coloring

Innovation

• Enumerates vertices and edges in flip graph
• Proves degree properties via recurrence relations
• Determines diameter using 3-coloring techniques