Thomas C. Hull

Using origami design principles to fold reprogrammable mechanical metamaterials

Folding robots and metamaterials The same principles used to make origami art can make self-assembling robots and tunable metamaterials—artificial materials engineered to have properties that may not be found in nature (see the Perspective by You). Felton et al. made complex self-folding robots from flat templates. Such robots could potentially be sent through a collapsed building or tunnels and then assemble themselves autonomously into their final functional form. Silverberg et al. created a mechanical metamaterial that was folded into a tessellated pattern of unit cells. These cells reversibly switched between soft and stiff states, causing large, controllable changes to the way the material responded to being squashed. Science, this issue p. 644, p. 647; see also p. 623 Origami folded sheets can be structurally altered by adding defects to control the mechanical properties. [Also see Perspective by You] Although broadly admired for its aesthetic qualities, the art of origami is now being recognized also as a framework for mechanical metamaterial design. Working with the Miura-ori tessellation, we find that each unit cell of this crease pattern is mechanically bistable, and by switching between states, the compressive modulus of the overall structure can be rationally and reversibly tuned. By virtue of their interactions, these mechanically stable lattice defects also lead to emergent crystallographic structures such as vacancies, dislocations, and grain boundaries. Each of these structures comes from an arrangement of reversible folds, highlighting a connection between mechanical metamaterials and programmable matter. Given origami’s scale-free geometric character, this framework for metamaterial design can be directly transferred to milli-, micro-, and nanometer-size systems.

Programming Reversibly Self‐Folding Origami with Micropatterned Photo‐Crosslinkable Polymer Trilayers

Self-folding microscale origami patterns are demonstrated in polymer films with control over mountain/valley assignments and fold angles using trilayers of photo-crosslinkable copolymers with a temperature-sensitive hydrogel as the middle layer. The characteristic size scale of the folds W = 30 μm and figure of merit A/ W (2) ≈ 5000, demonstrated here represent substantial advances in the fabrication of self-folding origami.

Origami structures with a critical transition to bistability arising from hidden degrees of freedom

Origami is used beyond purely aesthetic pursuits to design responsive and customizable mechanical metamaterials. However, a generalized physical understanding of origami remains elusive, owing to the challenge of determining whether local kinematic constraints are globally compatible and to an incomplete understanding of how the folded sheets material properties contribute to the overall mechanical response. Here, we show that the traditional square twist, whose crease pattern has zero degrees of freedom (DOF) and therefore should not be foldable, can nevertheless be folded by accessing bending deformations that are not explicit in the crease pattern. These hidden bending DOF are separated from the crease DOF by an energy gap that gives rise to a geometrically driven critical bifurcation between mono- and bistability. Noting its potential utility for fabricating mechanical switches, we use a temperature-responsive polymer-gel version of the square twist to demonstrate hysteretic folding dynamics at the sub-millimetre scale.

Modelling the folding of paper into three dimensions using affine transformations

Abstract We model the folding of ordinary paper via piecewise isometries R 2 → R 3 . The collection of crease lines and vertices in the unfolded paper is called the crease pattern. Our results generalize the previously known necessity conditions from the more restrictive case of folding paper flat (into R 2 ); if the crease pattern is foldable, then the product (in a non-intuitive order) of the associated rotational matrices is the identity matrix. This condition holds locally in a multiple vertex crease pattern and can be adapted to a global condition. Sufficiency conditions are significantly harder, and are not known except in the two-dimensional single-vertex case.

Solving Cubics With Creases: The Work of Beloch and Lill

Abstract Margharita P. Beloch was the first person, in 1936, to realize that origami (paperfolding) constructions can solve general cubic equations and thus are more powerful than straightedge and compass constructions. We present her proof. In doing this we use a delightful (and mostly forgotten?) geometric method due to Eduard Lill for finding the real roots of polynomial equations.

Self-foldability of Rigid Origami

When actuating a rigid origami mechanism by applying moments at the crease lines, we often confront the bifurcation problem where it is not possible to predict the way the model will fold when it is in a flat state. In this paper, we develop a mathematical model of self-folding and propose the concept of self-foldability of rigid origami when a set of moments, which we call a driving force, are applied. In particular, we desire to design a driving force such that a given crease pattern can uniquely self-fold to a desired mode without getting caught in a bifurcation. We provide necessary conditions for selffoldability that serve as tools to analyze and design self-foldable crease patterns. Using these tools, we analyze the unique self-foldability of several fundamental patterns and demonstrate the usefulness of the proposed model for mechanical design. [DOI: 10.1115/1.4035558]

Rigid origami vertices: conditions and forcing sets

We develop an intrinsic necessary and sufficient condition for single-vertex origami crease patterns to be able to fold rigidly. We classify such patterns in the case where the creases are pre-assigned to be mountains and valleys as well as in the unassigned case. We also illustrate the utility of this result by applying it to the new concept of minimal forcing sets for rigid origami models, which are the smallest collection of creases that, when folded, will force all the other creases to fold in a prescribed way.

Corrigendum: Origami structures with a critical transition to bistability arising from hidden degrees of freedom

Corrigendum: Origami structures with a critical transition to bistability arising from hidden degrees of freedom

Box Pleating is Hard

Flat foldability of general crease patterns was first claimed to be hard for over twenty years. In this paper we prove that deciding flat foldability remains NP-complete even for box pleating, where creases form a subset of a square grid with diagonals. In addition, we provide new terminology to implicitly represent the global layer order of a flat folding, and present a new planar reduction framework for grid-aligned gadgets.