Chemotaxic receptors in bacteria form clusters at cell poles and also laterally, and this clustering plays an important role in signal transduction. These clusters were found to be periodically arranged on the surface of the bacterium Escherichia coli, independent of any known positioning mechanism. In this work we extend a model based on diffusion and aggregation to more realistic geometries and present a means based on "bursty" protein production to distinguish spontaneous positioning from an independently existing positioning mechanism. We also consider the case of isotropic cellular growth and characterize the degree of order arising spontaneously. Our model could also be relevant for other examples of periodically positioned protein clusters in bacteria.