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. Plain Language Summary Cold pool (CP) dynamics constitutes a crucial organizing mechanism for midlatitude and tropical clouds—they play a key role in the lead‐up to extreme events and may influence how such events behave in a changing climate. CPs are dense air masses that form under precipitating thunderstorm clouds. Under gravity, CPs spread along the surface and stimulate new precipitation events when they collide with other CPs. We show that CP interaction can be captured by a simple model, where circles grow in space and form new circles when three of them collide. Generalizing to thousands of initial circle centers, the dynamics of these circles gives a steady scale increase over time, similar to the one found in high‐resolution atmospheric simulations. In summary, we introduce a cloud‐organizing mechanism that forms the basis for extreme convective precipitation events, such as those implicated in flash floods. Key Points Convection is often initiated by the collision of three cold pools A model based on expanding and colliding circles captures the convective scale increase This simple model can produce clustering of precipitation cells