Universal Hinge Patterns to Fold Orthogonal Shapes

Abstract

An early result in computational origami is that every polyhedral surface can be folded from a large enough square of paper [2]. But each such folding uses a different crease pattern. Can one design a hinge pattern that can be folded into various different shapes? Our motivation is developing programmable matter out of a foldable sheet. The idea is to statically manufacture a sheet with specific hinges that can be creased in either direction, and then dynamically program how much to fold each crease in the sheet. Thus a single manufactured sheet can be programmed to fold into anything that the single hinge pattern can fold. We prove a universality result: a single n×n hinge pattern can fold into all face-to-face gluings of O(n) unit cubes. Thus, by setting the resolution n sufficiently large, we can fold any 3D solid up to a desired accuracy. The proof is by construction: we describe an algorithm for generating the crease pattern for a given polycube. We also show how an implementation of the algorithm can be used to automate experimentation and design of geometric paper origami using a cutting plotter or laser cutter to score the paper. We describe three hinge-pattern variants, which are equivalent in order of growth properties, but have different applications. One robotics-motivated variant uses a box-pleated lattice—only 90◦ and 45◦ angles. Another variant is more size efficient and symmetrical but requires additional hinges with arctan( 2) angles and is thus less efficient in number of hinges. A final robotics-motivated variant is as efficient as the arctan(12) angle variant, and uses only