Robert J. Lang

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.

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

Folding a Better Checkerboard

Folding an n ×n checkerboard pattern from a square of paper that is white on one side and black on the other has been thought for several years to require a paper square of semiperimeter n 2. Indeed, within a restricted class of foldings that match all previous origami models of this flavor, one can prove a lower bound of n 2 (though a matching upper bound was not known). We show how to break through this barrier and fold an n ×n checkerboard from a paper square of semiperimeter

Circle Packing for Origami Design Is Hard

{1 \over 2} n^2 + O(n)

Facet Ordering and Crease Assignment in Uniaxial Bases

. In particular, our construction strictly beats semiperimeter n 2 for (even) n > 16, and for n = 8, we improve on the best seamless folding.