Why are the patterns of plankton in the ocean so patchy? Why do frequently described geometrical hallucinations tend to fall into one of four different classes of pattern? Why don't we see hallucinations all the time? And why do populations in ecosystems tend to have noisy cycles in abundance? This talk explains how these phenomena all arise from the discreteness of the underlying entities, be they the on-off states of neurons or the numbers of bacteria in a fluid volume of ocean, or the number of signaling molecules in a biofilm. I explain how tools from statistical mechanics can yield insights into these phenomena, and report on a range of studies that include the operation of the primate visual cortex, the behavior of signalling molecules in a forward-engineered synthetic biofilm, and the fluctuating patterns and populations of marine organisms.