  Pattern formation is a ubiquitous phenomenon observed in nonlinear and
out-of-equilibrium systems. In equilibrium, quantum ferrofluids formed from
ultracold atoms were recently shown to spontaneously develop coherent density
patterns, manifesting a supersolid. We theoretically investigate the phase
diagram of such quantum ferrofluids in oblate trap geometries and find an even
wider range of exotic states of matter. Two-dimensional supersolid crystals
formed from individual ferrofluid quantum droplets dominate the phase diagram
at low densities. For higher densities we find honeycomb and labyrinthine
states, as well as a pumpkin phase. We discuss scaling relations which allow us
to find these phases for a wide variety of trap geometries, interaction
strengths, and atom numbers. Our study illuminates the origin of the various
possible patterns of quantum ferrofluids and shows that their occurrence is
generic of strongly dipolar interacting systems stabilized by beyond mean-field
effects.
