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.

Assis at the University of Newcastle in Australia.


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.

A defect in a Miura-ori pattern.

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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…

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