# Mathematics Applied to Kirigami Creates Impressive Shapeshifting Sheets

Harvard scientists set out to understand the mathematical principles underlying an old Japanese tradition.

You might not have heard of kirigami, origami's lesser-known cousin. The variation on origami relies on cutting paper rather than folding it - "kiri" stands for "cut" and "kami" for paper - and has long been used in pop-up cards and books.

In a marriage of mathematics and old Japanese decorative traditions, it has just been revealed that Harvard researchers have come up with a mathematical model that allows them to cut a kirigami sheet of paper in just such a way that it can be molded into just about any 3D shape.

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## Mathematical principles of kirigami

The group of Harvard researchers behind the study feel their work can be used to apply the tradition of kirigami to engineering by creating flexible construction shapes.

"We asked if it is 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 and senior author on the paper in a press release.

"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?" said Gary P. T. Choi, a graduate student at SEAS and first author of the paper.

"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 optimization approach to determine the patterns, and then verify this experimentally."

## Engineering applications

Perhaps one way to use these findings is in creating foldable shelter and housing, like Skyshelter.zip, a concept for foldable skyscrapers for disaster zones.

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This research draws from previous work by the Mahadevan lab that studied how origami patterns could be used as building blocks to create a huge variety of 3D shapes.

"Our work draws on inspiration from art, tempered by the rigor of mathematics, and the challenges of engineering shape. Finding kirigami tessellations that can convert a square to a circle, or a flat sheet into a poncho is just the start," said Mahadevan.

The team of scientists sees this as "just the beginning of a class of new ways to engineer shape in the digital age using geometry, topology, and computation."

Next, the researchers will look at ways of linking the useful properties of kirigami and origami to create an even more flexible framework for engineers to draw from.

The research was published in Nature Materials.