Morphing flat materials into any shape you want

18 November 2019

How the maths behind the art of kirigami may engineer future change

Kirigami, a variation of origami, is a Japanese decorative tradition that relies on cutting paper rather than folding it. The cutting process changes the flexibility of a flat paper sheet and allows it to morph into 3D shapes, as commonly seen in pop-up cards and paper models.

Inspired by this craft, a research team working in the Mahadevan Lab at Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS), including doctoral candidate Gary PT Choi (Croucher Scholarship 2016), has developed a mathematical framework that can turn any sheet of material into a 3D shape.

Choi, first author of the paper published in Nature Materials, and his fellow researchers set out to uncover the underlying mathematical principles in kirigami. They then sought to create algorithms that would enable the design of the number, size, and orientation of the cuts in a flat sheet so it could morph into any given shape.

“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?” Choi said. “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 draws on previous work by the Mahadevan Lab that characterised how origami-based patterns could be used as building blocks to create almost any three-dimensional curved shape.

“Finding kirigami tessellations that can convert a square to a circle, or a flat sheet into a poncho, is just the start,” said Prof L Mahadevan, de Valpine Professor of Applied Mathematics, Physics, and Organismic and Evolutionary Biology and senior author for the Nature Materials paper. “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.”

The next step for the research team is to explore how to combine cuts and folds to achieve any shape with a given set of properties, thus linking origami and kirigami.

Gary Choi is an applied mathematics PhD candidate in the John A. Paulson School of Engineering and Applied Sciences at Harvard University, where he works with Prof L Mahadevan and Prof Chris Rycroft. He obtained his BSc and MPhil in Mathematics from The Chinese University of Hong Kong in 2014 and 2016 respectively, and his Master of Science in Applied Mathematics from Harvard University in 2019. He received a Croucher Scholarship in 2016.

To view Choi’s Croucher profile, please click here