We consider several quadrilateral origami tilings, including the Miura-oricrease pattern, allowing for crease-reversal defects above the ground statewhich maintain local flat-foldability. Using exactly solvable models, we showthat these origami tilings can have phase transitions as a function of creasestate variables, as a function of the arrangement of creases around vertices,and as a function of local layer orderings of neighboring faces. We use theexactly solved cases of the staggered odd 8-vertex model as well as Baxter'sexactly solved 3-coloring problem on the square lattice to study these origamitilings. By treating the crease-reversal defects as a lattice gas, we findexact analytic expressions for their density, which is directly related to theorigami material's elastic modulus. The density and phase transition analysishas implications for the use of these origami tilings as tunable metamaterials;our analysis shows that Miura-ori's density is more tunable than Barreto'sMars, for example. We also find that there is a broader range of tunability asa function of the density of layering defects compared to as a function of thedensity of crease order defects before the phase transition point is reached;material and mechanical properties that depend on local layer orderingproperties will have a greater amount of tunability. The defect density ofBarreto's Mars, on the other hand, can be increased until saturation withoutpassing through a phase transition point. We further consider relaxing therequirement of local flat-foldability by mapping to a solvable case of the16-vertex model, demonstrating a different phase transition point for thiscase.
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