Face flips in origami tessellations

Given a flat-foldable origami crease pattern $G=(V,E)$ (a straight-line drawing of a planar graph on a region of the plane) with a mountain-valley (MV) assignment $μ:E\to\{-1,1\}$ indicating which creases in $E$ bend convexly (mountain) or concavely (valley), we may \emph{flip} a face $F$ of $G$ to create a new MV assignment $μ_F$ which equals $μ$ except for all creases $e$ bordering $F$, where we have $μ_F(e)=-μ(e)$. In this paper we explore the configuration space of face flips for a variety of crease patterns $G$ that are tilings of the plane, proving examples where $μ_F$ results in a MV assignment that is either never, sometimes, or always flat-foldable for various choices of $F$. We also consider the problem of finding, given two foldable MV assignments $μ_1$ and $μ_2$ of a given crease pattern $G$, a minimal sequence of face flips to turn $μ_1$ into $μ_2$. We find polynomial-time algorithms for this in the cases where $G$ is either a square grid or the Miura-ori, and show that this problem is NP-hard in the case where $G$ is the triangle lattice.

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