Atomic theory of origami

Imagining that the folds and curves of origami are atoms in a lattice, the researchers discover strange behavior hidden in simple structures

Michael Assis discovered that origami can experience a phase transition



In 1970, astrophysicist Koryo Miura [Koryo Miura] conceived a scheme that was destined to become one of the most well-known and well-studied schemes for folding origami: Miura-ori . The fold pattern creates a mosaic of parallelograms, and the whole structure is folded and unfolded in one motion, creating a great way to fold the card. This is also a great way to fold the spacecraft's solar panel - this idea was proposed by Miura in 1985, and then it was carried out in reality on the Japanese satellite Space Flyer Unit in 1995.



On earth, Miura-ori finds more and more applications. The folding system gives the flexible sheet its form and strength, creating a promising metamaterial — a material whose properties depend not on its chemical composition, but on its structure. Also Miura-ori is characterized by a negative Poisson ratio . If you push it from the sides, the top and bottom of the origami will move. But for most objects this does not happen - if you try to squeeze, say, a banana, then the contents will start crawling from its ends.



Researchers studied how to create pipes, curves, and other structures that could be used in robotics, aerospace, and architecture using Miura-ori. Even fashion designers were inspired by this system, including it in dresses and scarves.



Now Michael Assis [Michael Assis], a physicist from Newkaslov University in Australia, is working on an unusual approach to understanding Miura-ori and similar origami: he views them through the prism of statistical mechanics.



The new Assis analysis, which is now being tested by experts for the journal Physical Review E, will be the first work using statistical mechanics to describe origami. This work also simulates origami for the first time using an approach that uses “pencil and paper”, which produces exact solutions — solutions that are independent of approximate computer calculations. “Many people, and I among them, have lost hope for exact solutions,” says Arthur Evans, an expert in mathematical physics, using origami in his work.



Usually, experts in statistical mechanics try to describe the appearing properties and behavior of a set of particles, for example, gas or water molecules, existing in an ice cube. But sets of folds are also networks, only consisting not of particles, but of folds. Using the conceptual tools commonly used for gases and crystals, Assis gets very interesting ideas.

Hot folds

In 2014, Evans worked in a team that studied what happens to Miura-ori when defects are added to it. Researchers have shown that by inverting several folds, pressing in the convexity and squeezing out the concavities, the structure can be made more durable. Defects, instead of serving as imperfections, became virtues. Adding or removing defects, you can reconfigure Miura-ori, achieving the desired strength.



This caught the attention of Assis. “Before this work, no one thought about defects,” he said.



He is versed in statistical mechanics, naturally applied to lattice circuits like Miura-ori. In a crystal, atoms are bound by chemical bonds. In origami, vertices are wrinkled. Even in a grid that contains only 10 repeating units, the statistical approach, according to Assis, can quite accurately describe its behavior.



Defects appear in crystals if the temperature is raised. For example, in an ice cube, heat destroys the bonds between water molecules, which forms defects in the lattice. As a result, the grid is completely destroyed and the ice melts.



Similarly, in the analysis of origami made by Assis, the high temperature leads to the appearance of defects. But in this case the temperature does not mean how cold or warm the lattice is; it denotes the energy of the system. For example, constantly closing and opening Miura-ori, you add energy to the grid, and in the language of statistical mechanics, increase its temperature. This leads to the appearance of defects, since permanent disclosures and coagulation can lead to the fact that one of the folds will develop in the other direction.



To understand how defects grow, Assis decided that it would be better to treat not every vertex as individual particles, but each defect. In this case, the defects behave like free-flowing gas particles. Assis can even calculate parameters such as density and pressure.





Defect in the grid Miura-ori



At relatively low temperatures, defects behave as usual. At high temperatures, when defects cover the entire lattice, the origami structure becomes relatively homogeneous.



And in the interval between these states, Miura-ori, like the other trapezoidal origami addition scheme, goes through a sharp transformation from one state to another - what physicists call a phase transition occurs. “I was surprised and delighted when I was able to detect a phase transition in origami,” says Assis. - In a sense, this demonstrates its complex structure. He has the complexity of a real material. And in the end, this is what we need - metamaterials of the real world.



Without experiments, it’s hard to say how origami changes at a transition point. He assumes that as the number of defects increases, the grid gradually becomes less and less organized. After the transition point, there are already so many defects in it that the whole origami structure is wallowing in interference. “It seems that the whole order disappears and origami behaves randomly,” he says.



However, phase transitions are not necessarily inherent in all types of origami. Assis also studied a mosaic of squares and parallelograms called Mars Barreto . This grid does not experience a phase transition, so you can add more defects to it and not cause confusion. If you need material that can withstand more defects, says Assis, then this may be useful for you.





Assis shows how to use defects to tweak Miura-ori

Flat edges

Whether these conclusions apply to real origami is a moot point. Robert Lang, an origami physicist and sculptor, thinks Assis’s models are too perfect to be used. For example, this model assumes that origami can be made to fold into a flat figure, even if there are defects, but in fact, defects can prevent the sheet from folding flat. The analysis does not include the corners of the folds, it does not prohibit the sheet to self-intersect when added - but this cannot be in real life. “The work doesn't even come close to describing real origami with such folds,” says Lang.



But Assis says that the model is supposed to be reasonable and necessary, especially when you need to get accurate solutions. In many practical cases, for example, when folding solar panels, you need the sheet to fold flat. Folding can smooth defects. The corners of the folds can play an important role if they are located close to the defects, especially if one considers that the faces of the lattice can also bend. Assis plans to consider bending faces in the next work.



Unfortunately, the question of the possibility of global addition to a flat figure is one of the most difficult mathematical problems, therefore most of the researchers assume only the presence of a local addition to a flat figure. So says Thomas Hull, a mathematician at Western New England University and co-author of the 2014 study. He says such assumptions make sense. But he acknowledges that the difference between theory and the development of real metamaterials and structures remains significant. “It is still not clear whether such work, which Michael presented, will help us to do something in practice,” he said.



To find out, researchers will need to independently conduct experiments to test the ideas of Assis and evaluate whether models can actually make sense of origami, or they can only be played by theoreticians in statistical mechanics. Yet such a study is a step in the right direction, says Hull. "We need basic building blocks that can be used for practical use."



Christian Santangelo, a physicist at the University of Massachusetts at Amherst who has been involved in writing the 2014 work, agrees with him. In his opinion, not enough researchers work on origami defects, and he hopes that the presented work will attract more scientists to this field. "Apparently, these problems are not a priority for people who actually create something." Like it or not, origami technology requires careful study of the effect of defects. "These structures," he said, "will not add up themselves."



You can fold Miura-ori yourself by downloading and printing a PDF file .