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

TL;DR

This paper studies the structure of the flip graph of valid mountain-valley assignments in the $2 imes n$ Miura-ori origami pattern, revealing its size, degree distribution, and diameter.

Contribution

It introduces the origami flip graph for the Miura-ori, enumerates its vertices and edges, and determines its diameter using combinatorial and graph-theoretic methods.

Findings

Number of vertices and edges in the flip graph are explicitly characterized.

Vertex degrees follow polynomial patterns and recurrence relations.

The diameter of the flip graph is $oxed{rac{n^2}{2}}$.

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 $\mu:E\to\{-1,1\}$. An MV assignment $\mu$ is locally valid if the faces around each vertex in $C$ can be folded flat under $\mu$. In this paper, we investigate locally valid MV assignments of the Miura-ori, $M_{m,n}$, an $m\times n$ 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 $0\leq a$) are both described by polynomials of particular degrees. We then prove that the diameter of $OFG(M_{2,n})$ is $\lceil \frac{n^2}{2}\rceil$ using techniques from 3-coloring reconfiguration graphs.