# Discrete symmetries control geometric mechanics in parallelogram-based origami

### Significance

Most materials from rubber bands to steel beams thin out as they are stretched, but engineers can use origami’s interlocking ridges and precise folds to reverse this tendency and build devices that grow wider as they are pulled apart. Geometric compatibility constraints dictate the mechanical response of soft systems that can be utilized for the design of mechanical metamaterials such as the negative Poisson’s ratio Miura-ori origami crease pattern. Researchers increasingly use this kind of technique, drawn from the ancient art of origami, to design spacecraft components, medical robots and antenna arrays. However, much of the work has progressed via instinct and trial and error. There is a need to develop a general formula that analyzes how structures can be configured to thin, remain unaffected, or thicken as they are stretched, pushed or bent.

In a new study published in the Proceedings of the National Academy of Sciences, Professor Glaucio Paulino and his colleagues from Princeton University lay out their general rule for the way a broad class of origami responds to stress. The rule applies to origami formed from parallelograms (such as a square, rhombus or rectangle) made of thin material. In their article, the researchers use origami to explore how structures respond to certain kinds of mechanical stress, for example, how a rectangular sponge swells in a bowtie shape when squeezed in the middle of its long sides. Of particular interest was how materials behave when stretched, like a stick of chewing gum that thins as it is pulled at both ends. The ratio of compression along one axis with stretching along the other is called the Poisson ratio. Most materials have a positive Poisson ratio. If, for example, you pick up a rubber band and stretch it, it will become thinner and thinner before it breaks, Cork has a zero Poisson ratio, and that is the only reason you can put the cork back in a wine bottle. Otherwise, it will break the bottle. The researchers were able to write a set of equations to predict how origami-inspired structures will behave under this kind of stress. They then used the equations to create origami structures with a negative Poisson ratio, origami structures that grew wide instead of narrower when their ends were pulled, or structures that snapped into dome shapes when bent instead of sagging into a saddle shape.

The process was more complex than defining the symmetry rules because some of the folds resulted in deformations that did not obey the rules. Generally, the deformations made in the same plane as the paper (or thin material being folded) obeyed the rules, and those out of the plane broke the rules. Usually, if you take a thin sheet or slab and you pull on it, it will retract in the middle. Some materials instead thicken when you pull on them, and those always form domes rather than saddles. The amount of thinning always predicts the amount of bending. The bending of these origami is exactly the opposite of all conventional materials.

Researchers have spent years seeking to define rules governing different classes of origami, with different folding patterns and shapes. The research team discovered the class of origami was not important. It was the way the folds interacted that was key. To understand why origami seemed to defy movement usually defined by Poisson’s ratio growing wider when pulled, for example the authors needed to understand how the interaction affected the movement of the entire structure. When artists fold the sheet so that it moves along its plane, for example, corrugating it so it can expand and contract they also introduce a bend that moves the sheet into the saddle shape.

The findings will create new tools and paths for the technical community to harness and pursue that will further elevate the functionalities of advanced origami and metamaterials. Future studies will focus on build on their work by examining more complex systems. The authors will validate this for different patterns, different configurations; to make sense of the theory and validate it.

Glaucio H Paulino

Margareta Engman Augustine Professor of Engineering
The Department of Civil and Environmental Engineering
Princeton University

Civil engineers lead design innovations for buildings, bridges and roads, which are critical elements of human settlements. Research discoveries in mechanics, materials and structures lay the foundations for innovation in infrastructure and the built environment.

Faculty in this department focus on structural optimization techniques; smart, kinetic, deployable structures; energy-efficient architecture; structural health monitoring; sustainable engineering materials; materials for CO2 capture; architected materials and additive manufacturing; and 3D printing of buildings. Research also includes the connection of engineering to society, art and aesthetics, which enables faculty to be at the forefront in the study of both modern design methods and heritage structures.

### Reference

James McInerney, Glaucio H. Paulino, and D. Zeb Rocklin. Discrete symmetries control geometric mechanics in parallelogram-based origami, Proceedings of the National Academy of Sciences (2022). DOI: 10.1073/pnas.2202777119