# The Atomic Theory of Origami

In 1970, an astrophysicist named Koryo Miura conceived what would become one of the most well-known and well-studied folds in origami: the Miura-ori. The pattern of creases forms a tessellation of parallelograms, and the whole structure collapses and unfolds in a single motion—providing an elegant way to fold a map. It also proved an efficient way to pack a solar panel for a spacecraft, an idea Miura proposed in 1985 and then launched into reality on Japan’s Space Flyer Unit satellite in 1995.

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Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.

Back on Earth, the Miura-ori has continued to find more uses. The fold imbues a floppy sheet with form and stiffness, making it a promising metamaterial—a material whose properties depend not on its composition but on its structure. The Miura-ori is also unique in having what’s called a negative Poisson’s ratio. When you push on its sides, the top and bottom will contract. But that’s not the case for most objects. Try squeezing a banana, for example, and a mess will squirt out from its ends.

Researchers have explored how to use Miura-ori to build tubes, curves and other structures, which they say could have applications in robotics, aerospace and architecture. Even fashion designers have been inspired to incorporate Miura-ori into dresses and scarves.

Now Michael Assis, a physicist at the University of Newcastle in Australia, is taking a seemingly unusual approach to understanding Miura-ori and related folds: by viewing them through the lens of statistical mechanics.

Assis’ new analysis, which is under review at *Physical Review E*, is the first to use statistical mechanics to describe a true origami pattern. The work is also the first to model origami using a pencil-and-paper approach that produces exact solutions—calculations that don’t rely on approximations or numerical computation. “A lot of people, myself included, abandoned all hope for exact solutions,” said Arthur Evans, a mathematical physicist who uses origami in his work.

Traditionally, statistical mechanics tries to make sense of emergent properties and behaviors arising from a collection of particles, like a gas or the water molecules in an ice cube. But crease patterns are also networks—not of particles, but of folds. Using these conceptual tools normally reserved for gases and crystals, Assis is gaining some intriguing insights.

### Hot Folds

In 2014, Evans was part of a team that studied what happens to Miura-ori when you throw in a few defects. The researchers showed that by inverting a few creases, by pushing on a convex segment to make it concave and vice versa, they could make the structure stiffer. Instead of being a flaw, they found, defects could be a feature. Just by adding or subtracting defects, you can configure—and reconfigure—a Miura-ori to be as stiff as you want.

This drew the attention of Assis. “No one had really thought about defects until this paper,” he said.

His expertise is in statistical mechanics, which applies naturally to a lattice pattern like Miura-ori. In a crystal, atoms are linked by chemical bonds. In origami, vertices are linked by creases. Even with a lattice as small as 10 units wide, Assis said, such a statistical approach can still capture its behavior fairly well.

Defects appear in crystals when you crank up the temperature. In an ice cube, for example, the heat breaks the bonds between water molecules, forming defects in the lattice structure. Eventually, of course, the lattice breaks down completely and the ice melts.

Similarly, in Assis’ analysis of origami, a higher temperature causes defects to appear. But in this case, temperature doesn’t refer to how hot or cold the lattice is; instead, it represents the energy of the system. For example, by repeatedly opening and closing a Miura-ori, you’re injecting energy into the lattice and, in the language of statistical mechanics, increasing its temperature. This causes defects because the constant folding and unfolding might cause one of the creases to bend the wrong way.

But to understand how defects grow, Assis realized that it’s better not to view each vertex as a particle, but rather each defect. In this picture, the defects behave like free-floating particles of gas. Assis can even calculate quantities like density and pressure to describe the defects.

At relatively low temperatures, the defects behave in an orderly fashion. And at high enough temperatures, when defects cover the entire lattice, the origami structure becomes relatively uniform.

But in the middle, both the Miura-ori and another trapezoidal origami pattern appear to go through an abrupt shift from one state to another—what physicists would call a phase transition. “Finding that origami can have a phase transition to me was very, very exciting,” Assis said. “In a sense, it shows origami is complex; it has all the complexities of real-world materials. And at the end of the day, that’s what you want: real-world metamaterials.”

Without doing experiments, Assis said, it’s hard to say exactly how the origami changes at this transition point. But he hypothesizes that as defects multiply, the lattice steadily becomes more disordered. Beyond the transition point, there are so many defects that the whole origami structure becomes awash in clutter. “It’s almost as if you’ve lost all order, and globally, it’s behaving kind of randomly,” he said.

Yet phase transitions don’t necessarily show up in all types of origami. Assis also studied a tessellation of squares and parallelograms called Barreto’s *Mars*. This pattern doesn’t undergo a phase transition, which means you can add more defects without generating widespread disorder. If you want a metamaterial that can withstand more defects, this pattern might be the way to go, Assis said.

Defects also grow much faster on the Miura-ori and trapezoid patterns than on Barreto’s *Mars*. So if you’d rather have a metamaterial on which you can fine tune the number of defects, the Miura-ori or a trapezoid would be a better design.

### Flat Faces

Whether these conclusions actually apply to real-world origami is up for debate. Robert Lang, a physicist and origami artist, thinks that Assis’ models are too idealized to be of much use. For example, Lang said, the model assumes the origami can be made to fold flat even with defects, but in reality, defects can prevent the sheet from flattening. The analysis also doesn’t incorporate the angles of the folds themselves, nor does it forbid the sheet from intersecting with itself as it folds, which can’t happen in real life. “This paper doesn’t really come close to describing the behavior of actual origami with these crease patterns,” Lang said.

But the assumptions in the model are reasonable and necessary, especially if we want exact solutions, Assis said. In many engineering applications, such as the folding of a solar panel, you want the sheet to fold flat. The act of folding can also force defects to flatten. The angles of the folds may be important around defects, especially when you also consider that the faces of the lattice can warp. Assis plans to address such “face bending” in subsequent work.

Unfortunately, the question of global flat-foldability is one of the hardest mathematics problems around, which is why most researchers in the field assume local flat foldability, said Thomas Hull, a mathematician at Western New England University and a co-author of the 2014 study. These kinds of assumptions, he said, make sense. But he admits that the gap between theory and designing real metamaterials and structures remains wide. “It’s still not clear whether work like Michael’s is going to help give us things that we can do in practice,” he said.

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To find out, researchers will need to carry out experiments to test Assis’ ideas and gauge whether the models can actually inform the design of origami structures, or if they’re toy models of interest only to theorists in statistical mechanics. Still, this kind of study is a step in the right direction, Hull said. “These are the basic building blocks we need in order to use this stuff for real.”

Christian Santangelo, a physicist at the University of Massachusetts, Amherst, who also collaborated on the 2014 paper, agrees. Not enough researchers are tackling the problem of defects in origami, in his opinion, and if anything, he hopes this work will get more people to think about the problem. “Of the people who are actually building things, it doesn’t seem to be on their radar,” he said. Whether it is or not, origami technology will require a careful consideration of defects. “These structures,” he said, “aren’t just going to fold themselves.”

*Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.*

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