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Elastic sheets offer a path to encapsulating a droplet of one fluid in another that is different from that of traditional molecular or particulate surfactants. In wrappings of fluids by sheets of moderate thickness with petals designed to curl into closed shapes, capillarity balances bending forces. Here, we show that, by using much thinner sheets, the constraints of this balance can be lifted to access a regime of high sheet bendability that brings three major advantages: ultrathin sheets automatically achieve optimally efficient shapes that maximize the enclosed volume of liquid for a fixed area of sheet; interfacial energies and mechanical properties of the sheet are irrelevant within this regime, thus allowing for further functionality; and complete coverage of the fluid can be achieved without special sheet designs. We propose and validate a general geometric model that captures the entire range of this new class of wrapped and partially wrapped shapes.

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 sheet's 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.

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