# Programming shape using kirigami tessellations

## TL;DR

This paper develops a novel inverse design framework for kirigami tessellations, enabling the creation of shape-changing sheets that conform to arbitrary 2D and 3D target shapes through optimized cut patterns.

## Contribution

It introduces a constrained optimization approach to design kirigami patterns that deploy into specified shapes, extending capabilities to complex geometries and curved surfaces.

## Key findings

- Validated designs with physical models confirming shape accuracy
- Demonstrated deployment to various 2D target shapes
- Extended framework to approximate 3D curved surfaces

## Abstract

Kirigami tessellations, regular planar patterns formed by cutting flat, thin sheets, have attracted recent scientific interest for their rich geometries, surprising material properties and promise for technologies. Here we pose and solve the inverse problem of designing the number, size, and orientation of cuts that allows us to convert a closed, compact regular kirigami tessellation of the plane into a deployment that conforms approximately to any prescribed target shape in two and three dimensions. We do this by first identifying the constraints on the lengths and angles of generalized kirigami tessellations which guarantee that their reconfigured face geometries can be contracted from a non-trivial deployed shape to a novel planar cut pattern. We encode these conditions in a flexible constrained optimization framework which allows us to deform the geometry of periodic kirigami tesselations with three, four, and sixfold symmetry, among others, into generalized kirigami patterns that deploy to a wide variety of prescribed boundary target shapes. Physically fabricated models verify our inverse design approach and allow us to determine the tunable material response of the resulting structures. We then extend our framework to create generalized kirigami patterns that deploy to approximate curved surfaces in $\mathbb{R}^3$. Altogether, this work illustrates a novel framework for designing complex, shape-changing sheets from simple cuts showing the power of kirigami tessellations as flexible mechanical metamaterials.