While classical liquid droplets are rounded, transitions have recently been discovered which render polyhedral water-suspended droplets of several oils. Yet, the mechanism of these transitions and the role of the droplets’ interfacial curvature in inducing these transitions remain controversial. In particular, one of the two mechanisms suggested mandates a convex interface, in a view from the oil side. Here we show that oil-suspended water droplets can spontaneously assume polyhedral shapes, in spite of their concave interface. These results strongly support the alternative mechanism, where the faceting in both oil and water droplets is driven by the elasticity of a crystalline monolayer, known to self-assemble at the oil–water interface, independent of its curvature. The faceting transitions in the water droplets allow the fundamental elastic properties of two-dimensional matter to be probed, enable new strategies in faceted nanoparticle and nanoshell synthesis, and provide insight into the molecular mechanisms of morphogenesis.

While the curvature of the classical liquid surfaces exhibits only a weak temperature dependence, we demonstrate here a reversible temperature-tunable concave–convex shape switching in capillary-contained, surfactant-decorated, oil–water interfaces. The observed switching gives rise to a concave–convex shape transition, which takes place as a function of the width of the containing capillary. This apparent violation of Young’s equation results from a hitherto-unreported sharp reversible hydrophobic–hydrophilic transition of the glass capillary walls. The transition is driven by the interfacial freezing effect, which controls the balance between the competing surfactants’ adsorption on, and consequent hydrophobization of, the capillary walls and their incorporation into the interfacially frozen monolayer. Since capillary wetting by surfactant solutions is fundamental for a wide range of technologies and natural phenomena, the present observations have important implications in many fields, from fluid engineering to biology, and beyond.