The Complexity of Folding Self-Folding Origami

Why is it difficult to refold a previously folded sheet of paper? We show that even crease patterns with only one designed folding motion inevitably contain an exponential number of “distractor” folding branches accessible from a bifurcation at the flat state. Consequently, refolding a sheet requires finding the ground state in a glassy energy landscape with an exponential number of other attractors of higher energy, much like in models of protein folding (Levinthal’s paradox) and other NP-hard satisfiability (SAT) problems. As in these problems, we find that refolding a sheet requires actuation at multiple carefully chosen creases. We show that seeding successful folding in this way can be understood in terms of subpatterns that fold when cut out (“folding islands”). Besides providing guidelines for the placement of active hinges in origami applications, our results point to fundamental limits on the programmability of energy landscapes in sheets.

Popular Summary

Origami, the ancient Japanese art of paper folding, is not just for making decorative cranes. In recent years, researchers have applied principles of origami to create self-deploying devices ranging from nanoscale robots to solar panels on satellites. In this approach, a stiff sheet programmed with creases folds into a desired geometry with a single motion and little supervision. However, other such “bottom-up methods” such as self-assembly of particles and self-folding of polymers sometimes run into problems due to inevitable “traps” that can put the system into an undesired state. Despite anecdotal reports of similar problems and workarounds in specific origami applications, the limits of what can be programmed using self-folding sheets lacks a similar abstracted understanding. We mathematically explore these limits, and we find that the act of programming a desired self-folding behavior inevitably programs an exponential number of undesired behaviors into the same sheet.

Paradoxically, self-folding origami can require delicate care to successfully fold and avoid the many undesired behaviors. We find that these distractors can be described in the same way as the energy landscape of glassy materials—just as in self-assembly and protein folding. We further show how design principles inspired by heuristic algorithms for NP-hard satisfiability problems can alleviate such self-folding problems. The careful placement of multiple actuators, for example, can greatly increase the chances of successful deployment.

Self-folding devices can run into other complications that are specific to the applications and the materials used, but our work shows that there are fundamental limitations to any self-folding device, regardless of implementation.

Figure 1

(a) Structures designed with only one folding motion (“mechanisms”) are thought to be easy to control since any applied force not exactly perpendicular to that motion will actuate it. (b) However, if a mechanism has a branched d.o.f. (bifurcation), the applied force (green) must make a smaller angle with the desired branch than with the undesired branch. (c) We show that programming a stiff sheet with one folding motion inevitably creates an exponential number of other dead-end distractor branches that are of zero energy only to linear order. The applied force needs to be highly aligned with the desired folding motion in order to avoid the distractors. (d) Consequently, we must actuate multiple creases in a carefully selected combination (green) to successfully fold a self-folding crease pattern.

Figure 2

Bifurcations for vertices and chains of vertices. (a) A single vertex has two distinct folding branches, in which three creases form a mountain fold (black line) and one becomes a valley fold (magenta), or vice versa. The distinct branches are identifiable by the odd-one-out crease that folds opposite to the rest, with the two choices marked by blue and red lines. (b) The two branches meet at a bifurcation at the flat state. ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ correspond to the two eigenvectors of folding angles that span the two-dimensional null space of the vertex. Given a folding amplitude (e.g., $\parallel \overrightarrow{\rho }\parallel =0.1$), we blow out the energy of any configuration on a circle of that radius, with two clearly identified zero-energy configurations (and their negatives). (c) When a selected crease is actuated, the vertex chooses the branch in which that actuated crease folds more relative to other creases (rule of mechanical advantage). Since the odd-one-out crease and its transverse crease tend to fold less than the other crease pair, the odd-one-out crease is generally adjacent to the actuated crease. (d) When $N$ vertices are linked together into an open-ended chain, the chain can fold in $2^N$ different folding branches. Given an actuated crease, the resulting MV data can be predicted by applying the branch selection rule of (c) to vertices in sequence, as each successive vertex is actuated through the crease linking it to the prior vertex.

Figure 3

Loops of vertices give rise to a glassy landscape. (a)–(b) When a chain of vertices is closed by adding a final vertex, the resulting branches are no longer of zero energy. For example, if vertices $V_1$, $V_2$, $V_3$ are at one of their two zero-energy states (red dots), $V_4$’s folding state is constrained since the folding states of creases $V_1-V_4$ and $V_3-V_4$ are already set. The resulting energy for $V_4$ is generically not zero (red dot for $V_4$). We computed the energy in the four-dimensional linearized null space at a fixed norm $\parallel \overrightarrow{\rho }\parallel$. (c) The heatmap and surface plot show a two-dimensional projection onto the two top eigenvectors ${\mathrm{\Phi }}_1$, ${\mathrm{\Phi }}_2$ in the linearized null space.

Figure 4

Large patterns have an exponential number of branches (i.e., minima) of decreasing attractor size. We characterized the landscape by sampling random quadrilateral meshes of size up to $A=81$ vertices and folded each one with random torques until no new stable branches were found [${\kappa }_f={10}^{-6}$; see Eq. (1)]. (a) Quadrilateral meshes show an exponential number of distinct folding branches (i.e., local minima) in their energy landscape; the precise scaling depends on the total extent of folding $\parallel \overrightarrow{\rho }\parallel$, reflecting the relative importance of bending and stretching energies (Appendix pp2). (b) The size of attractor basins around different branches for a fixed pattern does not exceed 15% of the total space for a $4\times 4$ mesh.

Figure 5

Spatial distribution of actuators determines folding success. (a) A $4\times 4$ quadrilateral mesh pattern, with their designed soft branch indicated by line colors (black for mountain, red for valley). (b) If standard actuators are placed on randomly chosen creases of the pattern, at least 18 actuators ($\sim 30\%$ of creases) are needed to have a 50% chance of successful folding. (c) If just one crease is pressed, the resulting branches typically have a decreasing folding magnitude for creases away from the actuated crease. (d) The folding island of a crease is the largest subpattern that folds correctly when cut out from the full pattern and actuated at that crease. The area of the union of folding islands (UFI) relative to the entire pattern provides a simple design heuristic. (e) Actuated crease sets with $\mathrm{UFI}<1$ generally do not successfully fold the pattern while (f) actuators with $\mathrm{UFI}=1$ are successful. (g) However, successful actuation can sometimes be ruined by actuating an additional crease (orange) with a small folding island. (h) Actuated crease sets of a given size are dramatically more likely to fold successfully if their $\mathrm{UFI}=1$ (green) rather than $\mathrm{UFI}<1$ (red). (i) Using the data in (b), we find that UFI is a sharper predictor of success than the number of actuators. (j) Folding islands also explain a counterintuitive effect where successful actuation can sometimes be ruined by actuating an additional crease (orange) with a small folding island. The obtained branch is different than the designed one primarily in creases close to the “bad” actuator (thick lines).

Figure 6

Energy landscape of a quad loop pattern at a fixed norm of folding angles $\parallel \rho \parallel$. (a) A generic quad loop pattern has three to seven folding branches seen as minima in the energy landscape. Left: Heat map. Right: Corresponding three-dimensional surface. Generic minima have energies that scale as $E\sim {\rho }^4$. (b) By solving loop equations [38, 39], we can design one specific folding branch to be qualitatively softer than all others (identified as the much deeper minimum). The rest of the branches retain their original energy scaling and thus become distractors in the energy landscape.

Figure 7

Stretching energy scaling in a 4-vertex $E(\parallel \overrightarrow{\rho }\parallel )$ depends on the direction of $\overrightarrow{\rho }$. For random directions, we have $E\sim {\rho }^2$, while for directions within the linearized null space of the system, $E\sim {\rho }^4$. Only two special directions (not shown) exist for which $E=0$ to all orders in folding.

Figure 8

Number of folding branches for patterns of different sizes, folded to magnitude $\parallel {\overrightarrow{\rho }}_c\parallel$. At large folding angles, the same patterns have many more folding branches due to the lesser importance of face bending compared to stretching. In the main text, Fig. 4 shows vertical slices of the same data, illustrating the exponential size scaling of the branch number (${\kappa }_f={10}^{-6}$).

Figure 9

Finite-element models of realistic sheets, simulated with comsol Multiphysics, show that the mechanical advantage rule identifies the right folding branch. When a torque is applied to just one crease, the resulting branch is predicted by the mechanical advantage rule [Fig. 2]. We used the comsol shell model to simulate folding a realistic origami vertex (experiencing elastic strains). The folded configurations match those expected from our simplified vertex model defined only by vertex constraints, even though the comsol model accounts for many real-world complications not accounted for by our simple model (e.g., finite crease stiffness and thickness, delocalized bending and stretching). The pattern parameters are length approximately 0.2 m, crease width 0.015 m, thickness ${10}^{-4}\mathrm{m}$. Material parameters of the faces are density $\rho =1760\mathrm{kg}/{\mathrm{m}}^3$, Young’s modulus $Y=8\times {10}^8\mathrm{Pa}$. Material parameters of the creases are density $\rho =930\mathrm{kg}/{\mathrm{m}}^3$, Young’s modulus $Y=5\times {10}^6\mathrm{Pa}$.

Figure 10

Varying material models changes landscape details but maintains underlying structure of exponential minima. (a) Simulating a 4-vertex with stiff creases results in the same two modes of an idealized vertex, as long as the creases are not overwhelmingly stiff. The shift in minima position causes a moderate deviation from the mechanical advantage rule for sufficiently stiff creases. (b) and (c) Comparing a pattern with free-folding faces and creases to the same pattern with stiff faces and creases, the number of branches remains similar. Furthermore, the distribution of branch energies and attractor sizes is statistically indistinguishable, implying that the glassy structure retains the same statistics. (d) When branches of the patterns with free-folding and stiff faces or creases are paired up, we find that most random attempts to fold the system enter the same branch for both patterns, showing how corresponding branches remain “close” in configuration space. (e) Realistic origami structures are expected to have manufacturing errors in vertex placement. The blue and orange pattern’s vertices differ in location by 2% of the mean lattice spacing. (f)–(h) The same metrics as in (b)–(d) show how the landscape of patterns with manufacturing errors retain the same statistics as the originals.

Figure 11

Large meshes require the applied vector of torques $\overrightarrow{\tau }$ to be closely aligned with the folding angles ${\overrightarrow{\rho }}_{\text{desired}}$ of the desired branch for successful folding. By folding random quadrilateral meshes of different sizes with random $\overrightarrow{\tau }$ and determining the success of folding, we find that the dot product $D=\overrightarrow{\tau }·{\overrightarrow{\rho }}_{\text{desired}}/\parallel \overrightarrow{\tau }\parallel \parallel {\overrightarrow{\rho }}_{\text{desired}}\parallel$ in a large quadrilateral mesh must be close to 1. Since most vectors in high-dimensional space are orthogonal, only a vanishingly small fraction of applied vectors of torques can successfully fold the desired branch.