  Cooled oil emulsion droplets in aqueous surfactant solution have been
observed to flatten into a remarkable host of polygonal shapes with straight
edges and sharp corners, but different driving mechanisms - (i) a partial phase
transition of the liquid bulk oil into a plastic rotator phase near the droplet
interface and (ii) buckling of the interfacially frozen surfactant monolayer
enabled by drastic lowering of surface tension - have been proposed. Here,
combining experiment and theory, we analyse the hitherto unexplored initial
stages of the evolution of these 'shape-shifting' droplets, during which a
polyhedral droplet flattens into a polygonal platelet under cooling and
gravity. Using reflected-light microscopy, we reveal how icosahedral droplets
evolve through an intermediate octahedral stage to flatten into hexagonal
platelets. This behaviour is reproduced by a theoretical model of the phase
transition mechanism, but the buckling mechanism can only reproduce the
flattening if surface tension decreases by several orders of magnitude during
cooling so that the flattening is driven by buoyancy. The analysis thus
provides further evidence that the first mechanism underlies the
'shape-shifting' phenomena.
