Digital Origami From Geometrically Frustrated Tiles

Abstract

This paper describes “digital origami” from geometrically frustrated tiles: arrays of structures that cannot attain an energetically favorable flat state because of internal constraints. Each tile can typically snap between two symmetric energy-minimizing states, and neighboring tiles are coupled so that a collection of binary tile states determines the local curvature of the entire sheet. Modular structures like these tiles give great advantages in manufacturing and in predictive simulation, and their discrete nature is a good match for digital readout and control of self-folding systems. The digital origami concept applies to materials from the nanoscale to the macroscale. An example from previous researchers is a metal sheet with an array of dimples that can flex up or down. In this paper we investigate more general techniques that can develop planar sheets into bistable structures. Such methods include installing compressed pieces into a flat sheet of material, or tying together parts of a sheet (smocking). These methods work with a large variety of technologically important materials including circuit boards and semiconductor substrates. While there are clear benefits to such structures, significant obstacles to design exist in manufacturing, in evaluating their mechanical properties, and in choosing the best arrangement of tile states to match a desired shape. Determining the optimal flipping order of tile states to change the sheet from one shape to another is a sequencing problem analogous to protein folding, and origami from non-planar surfaces is a little-explored area in the fine arts. The paper discusses algorithms for curve-matching with one-dimensional arrays by error diffusion, and shape prediction for two-dimensional sheets with pre-programmed tile states. Low computational cost is required for creating structures that can predict, detect, and even change their own three-dimensional shapes using low-power onboard microprocessors. Motivators for this challenge include shape measurement over a large size range — for example, detecting the changing shapes of biological microstructures or endowing robots with a spatial sense similar to human proprioception — and self-modeling of structural properties for lightweight morphing structures that can strengthen for impact in a given direction using a limited amount of material.