# Rigidity percolation and geometric information in floppy origami

^{a}John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138;^{b}Department of Physics, Harvard University, Cambridge, MA 02138;^{c}Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138;^{d}Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, MA 02138

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Edited by Martin van Hecke, Leiden University, Leiden, The Netherlands, and accepted by Editorial Board Member John A. Rogers March 12, 2019 (received for review December 1, 2018)

## Significance

Origami structures are a particularly interesting class of thin-sheet–based mechanical metamaterials that rely on folds for their morphology and mechanical properties. Here, we study how excess folds in a simple origami pattern control the rigidity of the structure. Furthermore, we show that the onset of geometrical cooperativity in the system allows for information storage in a scale-free manner. Understanding how mechanical rigidity and geometric information can be simultaneously controlled in folded sheets has implications for structures on a range of scales, from graphene to architecture.

## Abstract

Origami structures with a large number of excess folds are capable of storing distinguishable geometric states that are energetically equivalent. As the number of excess folds is reduced, the system has fewer equivalent states and can eventually become rigid. We quantify this transition from a floppy to a rigid state as a function of the presence of folding constraints in a classic origami tessellation, Miura-ori. We show that in a fully triangulated Miura-ori that is maximally floppy, adding constraints via the elimination of diagonal folds in the quads decreases the number of degrees of freedom in the system, first linearly and then nonlinearly. In the nonlinear regime, mechanical cooperativity sets in via a redundancy in the assignment of constraints, and the degrees of freedom depend on constraint density in a scale-invariant manner. A percolation transition in the redundancy in the constraints as a function of constraint density suggests how excess folds in an origami structure can be used to store geometric information in a scale-invariant way.

Origami’s artistic origins harken back to the ancient art of paper folding, but it is also found in many natural settings, such as insect wings, leaves, vertebrate guts, flower petals, etc. (1⇓–3). The beauty and complexity of these origami folding patterns arise from permutations and combinations of a few modules, of which the simplest is a unit cell with four quads and four folds intersecting at a vertex. The classic Miura-ori consists of periodic repetition of this unit cell (Fig. 1*A*). It is highly symmetric and has three important geometric properties (1). It is rigid foldable; i.e., the folding process from a flat sheet is continuous without bending any quads (2). It has only one degree of freedom (DoF) (see *SI Appendix*, section 2 for details) (3). It is also flat foldable; i.e., a flat sheet can be folded to a state where all of the planes become coplanar. These geometric properties, together with the fact that these patterns arise spontaneously from simple physical processes (2⇓–4), have sparked much interest in the design of origami-inspired objects such as satellite sails and self-folding robots (5, 6), while also inspiring work on the mathematics and mechanics of these objects (7⇓⇓⇓–11). Simultaneously, from a technological perspective, origami has become a paradigm for programming geometry (12⇓⇓–15). Most studies focus on origami with rigid quads (rigid-foldable origami) or elastic quads with an associated bending energy, neither of which have more than a few floppy (zero energy) degrees of freedom (16). The exceptions are the studies of configurations near the unfolded state of triangulated origami (14, 15), but the general question of the interplay between fold geometry, topology, and rigidity in origami remains open. We address this question here by studying how fold topology and geometry allow for the control of rigidity in origami structures, with the potential for storage of geometric information or making reconfigurable structural materials for use in nanotechnology, soft robotics, and architecture.

We start with a flat sheet of paper that is inextensible and unshearable, made up of quadrilateral unit cells that can all fold along their edges, and with a total of *A*) (7, 8, 17) and 4-coordinated vertices. This is the classical Miura-ori pattern that has a single zero-energy (floppy) global DoF associated with an overall rigid folding motion (5). Topologically, 4-coordinated vertices are generic in systems like crumpled paper and rigid origami (5- or higher-coordinated vertices are degenerate and will spontaneously split into two or more 4-coordinated vertices, and 3-coordinated vertices are impossible). However, generating nonperiodic folding patterns with 4-coordinated vertices is hard because of the presence of topological obstructions (13, 16). Therefore, we focus on the simple periodic Miura-ori structure, but our results should generalize to any 4-coordinated origami folded pattern which usually has a very small number of internal degrees of freedom.

To make the folded structure floppy we introduce additional folds in some of the quads by allowing them to also fold along one of their two diagonals. If the additional fold is introduced to every quad, the resulting sheet will have a large number of zero-energy DoF. A number of natural questions then pose themselves: How does the number of distinguishable geometric states in such an origami structure change as a function of the number, location, and type of excess folds or constraints? How redundant are these constraints? When and how does geometric (mechanical) cooperativity arise in the system? And how does rigidity arise in the system? Here we answer these questions and show that there is a percolation transition that heralds the onset of cooperativity in the system as a function of the density of constraints. Equivalently, we show that the number of states increases exponentially as the density of new folds increases past a critical threshold. Together these results show how we might manipulate the information storage capacity of the system by exploiting the distance to the critical threshold.

## Mathematical Formulation

To calculate the (folding) degrees of freedom (m) of a sheet with a prescribed number of floppy edges that allow for out-of-plane bending, we replace the sheet by an equivalent network, where the *i*) E, the four peripheral edges have constant length since the material is inextensible; (*ii*) D, one of the diagonals has constant length (to prevent any internal shear/rotation since the material is also unshearable); and (*iii*) P (quad planarity), i.e., prevention of out-of-plane folding about this internal diagonal fold, this last constraint being optional.

Constraint counting allows us to calculate the maximum DoF of the structure (18), which occurs when all of the quads are allowed to fold about the internal diagonal [we choose all of the diagonal folds along the closest pair of diagonal vertices of the quad, but the results are invariant to random choices of the diagonals (*SI Appendix*, Fig. S1)]. For a sheet with

To calculate the DoF of the system with a given number of planarity constraints, we generalize the constraint-counting argument to an infinitesimal consideration of how constraints affect the rigidity of the network defined in terms of the coordinates of all its nodes,

If each node suffers an infinitesimal displacement defined by a vector *A*, which is the number of independent constraints, so that*SI Appendix*, section 3 for more details).

## Analysis

For concreteness of our calculations, we start with a periodic partially folded Miura-ori structure defined by a rhombus of side *SI Appendix*, Fig. S2)]. Defining c as the number of planar constraints in the floppy quads and *A*), so that the remaining *A*). Eq. **4** shows that calculating **1** and **2**. Therefore, m reaches the maximum *C* shows that

### Planarity Constraints on Boundary Quads Can Rigidify the Origami.

Since *B*). When L is odd, there are two DoF, i.e., one additional DoF besides the planar expanding/folding mode (*SI Appendix*, Fig. S3). When L is even, there is only one DoF corresponding to the expanding–folding mode. The difference comes from the structure of the center quads. When L is odd, there is a single center quad. In the infinitesimal mode corresponding to the extra DoF, as shown in *SI Appendix*, Fig. S3, one side of the sheet folds while the other side expands. It involves the bending of the single center quad. However, when L is even, there are four center quads, and their infinitesimal bending will in general not be mutually compatible. No matter whether L is odd or even, placing constraints on the boundary is the most efficient way to rigidify the system, since it requires only the minimum number of planarity constraints.

There are likely two reasons for how rigidification arises from planarity constraints on the boundary quads in origami. It is known that in 2D square lattices (22), if the constraint pattern satisfies the one-per-row and one-per-column condition, the whole system becomes rigid. Constraining the boundary quads in origami to be planar satisfies exactly the same condition and might thus give an intuitive explanation for our observations of how boundary-driven rigidification arises in floppy origami. Additionally, as in purely 2D systems, floppy modes in origami are more likely to involve boundary quads (22, 23); indeed, corner quads are an extreme example as they can bend without involving any quads in the bulk. Therefore, rigidifying the boundaries first might be the most efficient way to rigidify the whole system.

### DoF Decreases First Linearly and Then Sublinearly As Constraint Density Increases.

If we place the planarity constraints randomly on any quad in the system, we expect a certain inefficiency in their action—some will reduce the number of degrees of freedom, while others will be redundant. To understand this, we note that since there are *A*, we see that the DoF indeed decreases linearly when the constraint number density is small and increasing.

However, when the density of constraints increases sufficiently, some added constraints become redundant and thus do not reduce m. If a constraint pattern includes more redundant constraints, the corresponding DoF is larger, and vice versa. Different constraint patterns in this regime can thus lead to different m. In the neighborhood of a critical constraint density (that is to be determined), some patterns have redundant constraints while most patterns do not; in the latter case the DoF m still follows the simple rule *B*, *i*). As ρ further increases past a threshold, the mean DoF decreases sublinearly with ρ as the constraints are more likely to be redundant. Different constraint pattern realizations in this regime lead to a distribution of m that is close to being normal (Fig. 2 *B*, *ii* and *iii*). The SD (*A*) and the interquartile range Q of the distribution of m (*SI Appendix*, Fig. S4) are nonzero. We note that *B*, *iv*). Finally, when

The transition from linear to sublinear behavior in the dependence of the DoF as a function of the constraint density is similar for all system sizes L. Indeed if we rescale the DoF by its maximum (corresponding to no constraints), we see that the transition from the linear to the sublinear regime shifts to smaller ρ when L increases (Fig. 2*C*). We use the interquantile range Q of the distribution of m to define the critical transition density *SI Appendix*, Fig. S4), a definition that is more robust than using the SD (*SI Appendix*, section 5 for more information), and find that the best fit to this dependence is *SI Appendix*, Fig. S5). To understand this, we note that *SI Appendix*, Fig. S5).

In the sublinear regime, we also note that the mean DoF decreases exponentially with the density of constraints, independent of L, with *D*, before eventually becoming asymptotic to unity corresponding to

Additionally, we see that the range in the number of DoF m, shown in a lighter shading in Fig. 2*A*, is much larger than the variance in *E* we show a system with

### Information Storage in Origami.

The results of the previous section suggest mechanisms for maximizing geometric information storage in origami structures as a function of the variability in *D*, *Inset*). Once *SI Appendix*, section 5).] For larger origami, the peak of *A*), which is similar to the shift of *C*. The peak *A*, *Inset*), suggesting that there is an increasing range of m associated with different constraint patterns, which results from the increasing number of constraint patterns around

In this regime, the number of floppy configurations is large, so that origami can be used to store information geometrically. Assume there are *B*, we see that for origami with large L, the Shannon information peaks for *SI Appendix*, section 5).] Due to the sampling limit (200 patterns per ρ), there is an upper limit in the information capacity in the system:

### Scale-Free Control of DoF in Origami.

Strikingly, once *D*) and information capacity (Fig. 3*B*) are completely determined by the density of coplanar constraints regardless of how big the origami structure is (Fig. 2*D*), i.e., in a scale-invariant form. Furthermore, when ρ becomes sufficiently large (*C*, we show that this probability is a function of ρ alone. All of the results above suggest that when

### Percolation Behavior of Redundancy Near ρ c .

To understand the nature of the transition from the linear to sublinear decrease in DoF in the neighborhood of *A*), by comparing the current DoF and the DoF for a pattern with an additional planarity constraint on one of the free quads (not dark blue, which are already constrained). We iterate this process until all of the quads are constrained (

When *A*. When *SI Appendix*, Fig. S6), but the mean redundancy [the blue curve in Fig. 4*A*, averaged from the 100 simulations (gray lines)] keeps increasing and approaches 1. (In Movie S1, we show the process of adding constraints sequentially and the resultant change in redundancy.) To quantify the potential percolation transition in the redundancy r as a function of constraint density ρ, we explore system sizes in the range *B*, we see a signature of percolation at different constraint density as a function of L, with *C*) (see *SI Appendix*, Figs. S7 and S8 for more details); indeed, redundancy percolation happens at exactly the same time as Q deviates from zero. Redundancy not only increases sharply near *A*), which means that randomly generated constraint patterns may contain a significantly different number of redundant quads. The difference in the number of redundant quads leads to different DoF and thus nonzero Q. All of the quantities m, Q, and *Mathematical Formulation*, the planarity constraint (type P) in the flat quads of Miura-ori is fundamentally different from the diagonal edge-length constraint (type D). The former prevents volume change while the latter prevents length/area change. If we generalize our geometry of Miura-ori to more general folding patterns, with some quads bent along the diagonal, these two types of constraints will be equivalent, and we might recover the results of rigidity percolation in 3D random networks as discussed in ref. 25. See *SI Appendix*, section 2 for more details.

## Discussion

The geometric complexity associated with origami has long been an artist’s playground. But this ancient art form is in equal measure a rich source of mathematics and an inspiration for technology. Here, we have focused on the role of excess folds, or complexity (Latin cognate: *com + plicare =* fold together), in determining how excess folds can influence the rigidity of these structures. Beginning with a minimally complex geometric origami pattern associated with Miura-ori that has just a single folding degree of freedom, we have investigated how adding extra diagonal folds allows us to ask and answer questions about its potential for reconfigurability and ability to store geometric information.

When we start with a maximally floppy Miura-ori structure that is fully triangulated and introduce coplanarity constraints, the mean DoF initially decreases linearly until the constraint density

The presence of a percolation transition points toward some intriguing applications that include (*i*) a framework for multibit mechanical information storage, going beyond a recent 1-bit origami-based storage device (28); (*ii*) strategies for the optimal control of the number of DoF in a floppy origami structure in the neighborhood of the transition as well as deep in the scale-free regime; and (*iii*) an exploration of mechanical cooperativity in origami, similar to that in 2D networks (29, 30). Given that these chimeric denizens can move between two and three dimensions, there is clearly a lot that still remains to be explored.

## Acknowledgments

We thank Levi Dudte for discussions of the project during its early stages and Chris Rycroft for advice on algorithms. We also acknowledge NSF Grant DMR 14-20570 MRSEC, NSF Grant DMR 15-33985 Biomatter, and NSF Grant EFRI 18-30901 for partial financial support.

## Footnotes

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^{1}To whom correspondence should be addressed. Email: lmahadev{at}g.harvard.edu.

Author contributions: S.C. and L.M. designed research, conceived mathematical models, analyzed data, interpreted results, and wrote the paper. S.C. performed numerical simulations.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. M.v.H. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1820505116/-/DCSupplemental.

Published under the PNAS license.

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