Cold pools (CPs) contribute to convective organization. However, it is unclear by which mechanisms organization occurs. By using a particle method to track CP gust fronts in large eddy simulations, we characterize the basic collision modes between CPs. Our results show that CP interactions, where three expanding gust fronts force an updraft, are key at triggering new convection. Using this, we conceptualize CP dynamics into a parameter-free mathematical model: circles expand from initially random points in space. Where two expanding circles collide, a stationary front is formed. However, where three expanding circles enclose a single point, a new expanding circle is seeded. This simple model supports three fundamental features of CP dynamics: precipitation cells constitute a spatially interacting system, CPs come in generations, and scales steadily increase throughout the diurnal cycle. Finally, this model provides a framework for how CPs act to cause convective self-organization, clustering, and extremes.