Geometry and design of popup structures

TL;DR

This paper investigates the geometry of popup structures inspired by origami and kirigami, developing a design method to create deployable shapes with controllable curvature for applications like architecture and packaging.

Contribution

It introduces a geometric framework and design pipeline for popup structures combining origami and kirigami principles, enabling shape customization and curvature control.

Findings

A geometric description of popup units based on four-bar mechanisms.

A design method for creating structures with prescribed shapes.

Demonstration of shape transitions from negative to positive curvature.

Abstract

Origami and Kirigami, the famous Japanese art forms of paper folding and cutting, have inspired the design of novel materials & structures utilizing their geometry. In this article, we explore the geometry of the lesser known popup art, which uses the facilities of both origami and kirigami via appropriately positioned folds and cuts. The simplest popup-unit resembles a four-bar mechanism, whose cut-fold pattern can be arranged on a sheet of paper to produce different shapes upon deployment. Each unit has three parameters associated with the length and height of the cut, as well as the width of the fold. We define the mean and Gaussian curvature of the popup structure via the discrete surface connecting the fold vertices and develop a geometric description of the structure. Using these definitions, we arrive at a design pipeline that identifies the cut-fold pattern required to create popup structure of prescribed shape which we test in experiments. By introducing splay to the rectangular unit-cell, a single cut-fold pattern is shown to take multiple shapes along the trajectory of deployment, making possible transitions from negative to positive curvature surfaces in a single structure. We demonstrate application directions for these structures in drag-reduction, packaging, and architectural facades.