Face flips in origami tessellations
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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→{-1,1} indicating which creases in E bend convexly (mountain) or concavely (valley), we may 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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