In the bacterium Escherichia coli, the Min proteins oscillate between the cell poles to select the cell center as division site. This dynamic pattern has been proposed to arise by self-organization of these proteins, and several models have suggested a reaction-diffusion type mechanism. Here, we found that the Min proteins spontaneously formed planar surface waves on a flat membrane in vitro. The formation and maintenance of these patterns, which extended for hundreds of micrometers, required adenosine 5'-triphosphate (ATP), and they persisted for hours. We present a reaction-diffusion model of the MinD and MinE dynamics that accounts for our experimental observations and also captures the in vivo oscillations.