  We consider several quadrilateral origami tilings, including the Miura-ori
crease pattern, allowing for crease-reversal defects above the ground state
which maintain local flat-foldability. Using exactly solvable models, we show
that these origami tilings can have phase transitions as a function of crease
state variables, as a function of the arrangement of creases around vertices,
and as a function of local layer orderings of neighboring faces. We use the
exactly solved cases of the staggered odd 8-vertex model as well as Baxter's
exactly solved 3-coloring problem on the square lattice to study these origami
tilings. By treating the crease-reversal defects as a lattice gas, we find
exact analytic expressions for their density, which is directly related to the
origami material's elastic modulus. The density and phase transition analysis
has implications for the use of these origami tilings as tunable metamaterials;
our analysis shows that Miura-ori's density is more tunable than Barreto's
Mars, for example. We also find that there is a broader range of tunability as
a function of the density of layering defects compared to as a function of the
density of crease order defects before the phase transition point is reached;
material and mechanical properties that depend on local layer ordering
properties will have a greater amount of tunability. The defect density of
Barreto's Mars, on the other hand, can be increased until saturation without
passing through a phase transition point. We further consider relaxing the
requirement of local flat-foldability by mapping to a solvable case of the
16-vertex model, demonstrating a different phase transition point for this
case.
