Exactly solvable flat-foldable quadrilateral origami tilings

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 eight-vertex model as well as Baxter's exactly solved three-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 density, 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.

Figure 1

Mapping of mountain folds to solid lines (left) and valley folds to dashed lines (right).

Figure 2

Odd eight-vertex model weights, with bond states shown in terms of line type, dashed or solid, representing valley or mountain creases, respectively.

Figure 3

Four of the five origami CPs we consider: (a) the Miura-ori CP, (b) the trapezoid CP, (c) Barreto's Mars CP, and (d) the kite CP; the simple square tiling is not shown.

Figure 4

(a) Single vertex with angles labeled and (b) dependence of the allowed flat-foldable vertex weights on the angle pattern examples.

Figure 5

Schematic representation of the valid crease reversals around vertices, with crease reversals indicated by solid red lines. All CPs admit the seventh type. In addition, the kite CP admits the first two types; Miura-ori, trapezoid, and Barreto's Mars CPs admit the first four types; and Barreto's Mars CP admits the first two only on one sublattice and the next two on the other sublattice only. The simple square CP admits all seven types.

Figure 6

Schematic representation of face-flip defect configurations, where solid red lines represent crease reversals for Miura-ori, trapezoid, Barreto's Mars, and the simple square CPs. The first two rows are at low density and the bottom row is at high density. The top middle configuration cannot occur except for the simple square CP, and Barreto's Mars face-flip defects only occur for its diamond faces.

Figure 7

Graphical interpretation of the factor of 2 in the free-fermion condition of the Miura-ori and trapezoid models. Shown on top is the interpretation around each vertex with four crease reversals and on the bottom is an example with two face-flip defects.

Figure 8

Crease assignment weight notation convention on a unit of four vertices for crease weights $a,...,h$, each of which represents two weights for the mountain and valley assignment separately, e.g., $a_{\mathrm{m}}$ and $a_{\mathrm{v}}$ for the $a$ crease notation, respectively. The notation for the four vertex weights $t_i, u_i, v_i$, and $w_i$ are also shown.

Figure 9

From top left to bottom right, the vertex weight notation conventions for each vertex unit of Miura-ori, trapezoid, Barreto's Mars, and kite CPs.

Figure 10

Shown on the left are conventions for changing the face color across creases for the Miura-ori (top) and trapezoid (bottom) CPs. Following the arrow direction, a valley crease increases the face color number while mountain creases decrease the face color number ($\text{mod}3$). Shown on the right is one of the three possible color ground states the Miura-ori (top) and trapezoid (bottom) CPs map to. The other two possible ground states are found by globally increasing or decreasing all face color values by one ($\text{mod}3$).

Figure 11

Shown on the left is the convention for changing the face color across creases for the kite CP. Following the arrow direction, a valley crease increases the face color number while mountain creases decrease the face color number ($\text{mod}3$). Shown on the right is one of the three possible color ground states the kite model maps to. The other two possible ground states are found by globally increasing or decreasing all face color values by one ($\text{mod}3$).

Figure 12

Comparison of the crease-reversal defect densities as a function of (a) $y$ and (b) $z$ for Miura-ori and trapezoid CPs (solid black line) and Barreto's Mars CPs (blue dashed line). Also shown in (b) is the Miura-ori and trapezoid layer defect density as a function of layer defects $z$ (dotted red line). Phase transition points are indicated by circles.

Figure 13

Comparison of the derivative of the crease-reversal defect density as a function of (a) $y$ and (b) $z$ for Miura-ori and trapezoid CPs (solid black line) and Barreto's Mars CPs (blue dashed line). Also shown in (b) is the Miura-ori and trapezoid layer defect density derivative as a function of layer defects $z$ (dotted red line).

Figure 14

Equations of state as a function of crease-reversal densities (a) $\rho \left(y\right)$ and (b) $\rho \left(z\right)$ for Miura-ori and trapezoid CPs (solid black line) and Barreto's Mars CPs (blue dashed line). Also shown in (b) is the Miura-ori and trapezoid equation of state as a function of layer defect density $\rho \left(z\right)$ (dotted red line). Phase transition points are indicated by circles.

Figure 15

Even eight-vertex model weights, with bond states shown in terms of line type, dashed or solid.

Figure 16

Correspondence between the odd eight-vertex model weights and dimer coverings.

Figure 17

(a) Vertex and bond weight definitions around a cluster for the odd eight-vertex model. (b) Orientation graph convention on a column staggered lattice with four dimer cluster units for the odd eight-vertex model. The numbering convention for each of the four cluster units is shown.