Face flips in origami tessellations
Hugo A. Akitaya, Vida Dujmovi, David Eppstein, Thomas C. Hull, and Kshitij Jain, Anna Lubiw

TL;DR
This paper investigates how flipping faces in origami tessellations affects foldability, providing algorithms for certain patterns and proving complexity results for others.
Contribution
It characterizes the foldability changes caused by face flips in tessellated origami and develops polynomial algorithms for specific patterns, while proving NP-hardness in others.
Findings
Face flips can alter foldability in tessellated origami.
Polynomial algorithms exist for square grid and Miura-ori patterns.
NP-hardness is shown for triangle lattice crease patterns.
Abstract
Given a flat-foldable origami crease pattern (a straight-line drawing of a planar graph on a region of the plane) with a mountain-valley (MV) assignment indicating which creases in bend convexly (mountain) or concavely (valley), we may \emph{flip} a face of to create a new MV assignment which equals except for all creases bordering , where we have . In this paper we explore the configuration space of face flips for a variety of crease patterns that are tilings of the plane, proving examples where results in a MV assignment that is either never, sometimes, or always flat-foldable for various choices of . We also consider the problem of finding, given two foldable MV assignments and of a given crease pattern , a minimal sequence of face flips to turn into . We…
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