The mathematical framework transforms any sheet of material into any shape using kirigami slices

Researchers at the Harvard School of Engineering and Applied Sciences John A. Paulson have developed a mathematical framework to transform any sheet of material into any form prescribed, inspired to cut and kami, which means paper).

Unlike its more well-known cousin origami, which uses folds to shape paper, the kirigami relies on a pattern of cuts in a sheet of flat paper to change its flexibility and allow it to transform into three-dimensional shapes. Artists have long used this form of art to create everything from pop-up cards to castles and dragons.

"We asked if it was possible to discover the basic mathematical principles underlying the kirigami and use them to create algorithms allowing us to design the number, size and orientation of the slices in a flat sheet so that it can turn into a given shape, "said L. Mahadevan, Valpine Professor of Applied Mathematics, Physics and Biology of Organisms and Evolution, lead author of the document.

"Specifically, if we get a general shape in two or three dimensions, how should we design the cut patterns in a reference shape in order to be able to deploy it to the final shape in one motion?" said Gary P. T. Choi, graduate student at SEAS and first author of the newspaper. "In this work, we solve this problem by identifying the constraints to satisfy to obtain this cutting pattern, use a numerical optimization approach to determine the patterns, and then experimentally verify this."

The research is published in Nature's materials.

This research follows earlier work by the Mahadevan lab that described how origami-based designs could be used as building blocks to create almost all three-dimensional curved shapes.

"We were able to do a little more with kirigami than with origami," said Levi Dudte, a graduate student at Mahadevan's lab and co-author of the journal. "The presence of cuts and holes inside the material gives the kirigami the opportunity to change shape significantly."

"Our work is inspired by the art, tempered by the rigor of mathematics and the challenges posed by the engineering of form. Find tilings of kirigami able to convert a square into a circle or a sheet. Flat in poncho is only a beginning.We think it's only the beginning of a class of new engineering methods from the form to the era using geometry, topology and calculation, "said Mahadevan.

The researchers then aim to explore ways to combine cuts and folds to obtain any shape with a given set of properties, thus connecting origami and kirigami.