Bacterial cytokinesis is commonly initiated by the Z-ring, a cytoskeletal structure that assembles at the site of division. Its primary component is FtsZ, a tubulin superfamily GTPase, which is recruited to the membrane by the actin-related protein FtsA. Both proteins are required for the formation of the Z-ring, but if and how they influence each other's assembly dynamics is not known. Here, we reconstituted FtsA-dependent recruitment of FtsZ polymers to supported membranes, where both proteins self-organize into complex patterns, such as fast-moving filament bundles and chirally rotating rings. Using fluorescence microscopy and biochemical perturbations, we found that these large-scale rearrangements of FtsZ emerge from its polymerization dynamics and a dual, antagonistic role of FtsA: recruitment of FtsZ filaments to the membrane and negative regulation of FtsZ organization. Our findings provide a model for the initial steps of bacterial cell division and illustrate how dynamic polymers can self-organize into large-scale structures.