# Mathematical framework helps to shape any sheet of material

Image credit: Tamara Kulikova | Dreamstime.com

Researchers at Harvard University have developed a mathematical framework that can turn any sheet of material into any prescribed shape, inspired by the Japanese paper craft kirigami.

Unlike its better-known cousin origami, which uses folds to shape paper, kirigami (from the Japanese, kiri, meaning to cut and kami, meaning paper) relies on a pattern of cuts in a flat paper sheet to change its flexibility and allow it to morph into 3D shapes.

The team, from Harvard John A Paulson School of Engineering and Applied Sciences (SEAS), investigated the ways in which this technique can be applied to any given material other than paper, through such models.

“We asked if it’s possible to uncover the basic mathematical principles underlying kirigami and use them to create algorithms that would allow us to design the number, size and orientation of the cuts in a flat sheet so that it can morph into any given shape,” said L Mahadevan, de Valpine professor of applied mathematics, physics, and organismic and evolutionary biology.

Gary P T Choi, a graduate student at SEAS added: “Specifically, if we are given a general shape in two or three dimensions, how should we design the cut patterns in a reference shape so that we can get it to deploy to the final shape in one motion?

“In this work, we solve that problem by identifying the constraints that have to be satisfied in order to achieve this cut pattern, use a numerical optimisation approach to determine the patterns, and then verify this experimentally.”

The research follows previous work by the Mahadevan lab that characterised a fundamental origami fold, or tessellation, that could be used as a foundation in creating almost any 3D curved shape – ranging from nanostructures to buildings, the team said.

The folding pattern used in this experiment, known as the Miura-ori, is a periodic way to tile the plane using the simplest mountain-valley fold in origami. A folded Miura can be packed into a flat, compact shape and unfolded in one continuous motion, making it ideal for packing rigid structures such as solar panels.

According to the team, this simple origami fold could pave the way to designing pop-up furniture, medical devices and scientific tools.

“We were actually able to do a little more with kirigami than we were able to do with origami,” said Levi Dudte of the latest research, a graduate student in the Mahadevan lab. “The presence of cuts and holes in the interior of the material gives kirigami the ability to change its shape significantly.”

“Our work draws on inspiration from art, tempered by the rigour of mathematics, and the challenges of engineering shape,” added Mahadevan.

“Finding kirigami tessellations that can convert a square to a circle, or a flat sheet into a poncho is just the start.

“We think that this is just the beginning of a class of new ways to engineer shape in the digital age using geometry, topology, and computation.”

Moving forward, the researchers aim to use both kirigami and origami techniques simultaneously, and to explore how to combine cuts and folds to achieve any shape with a given set of properties.

The research is published in Nature Materials.

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