The mysterious 'fairy circles' are vegetation‐free discs that cover vast areas along the pro‐Namib Desert. Despite 30 yr of research their origin remains unknown. Here we adopt a novel approach that focuses on analysis of the spatial patterns of fairy circles obtained from representative 25‐ha aerial images of north‐west Namibia. We use spatial point pattern analysis to quantify different features of their spatial structures and then critically inspect existing hypotheses with respect to their ability to generate the observed circle patterns. Our working hypothesis is that fairy circles are a self‐organized vegetation pattern. Finally, we test if an existing partial‐differential‐equation model, that was designed to describe vegetation pattern formation, is able to reproduce the characteristic features of the observed fairy circle patterns. The model is based on key‐processes in arid areas such as plant competition for water and local resource‐biomass feedbacks.