It is shown that the air-liquid interface can be made to display the same rich curvature phenomena as common lyotropic liquid crystal systems. Through mixing an insoluble, naturally occurring, branched fatty acid, with an unbranched fatty acid of the same length, systematic variation in the packing constraints at the air-water interface could be obtained. The combination of atomic force microscopy and neutron reflectometry is used to demonstrate that the water surface exhibits significant tuneable topography. By systematic variation of the two fatty acid proportions, ordered arrays of monodisperse spherical caps, cylindrical sections, and a mesh phase are all observed, as well as the expected lamellar structure. The tuneable deformability of the air-water interface permits this hitherto unexplored topological diversity, which is analogous to the phase elaboration displayed by amphiphiles in solution. It offers a wealth of novel possibilities for the tailoring of nanostructure.