The diversity of living forms led Darwin to state that it is “enough to drive the sanest man mad”. How can we describe this variety? How can we understand the origin and evolution of these “endless forms most beautiful?” And how do these forms link to function and physiology at the organismic level and beyond? Mathematics, and geometry in particular, provides a natural language to express these questions and answer them. Motivated by biological observations on different scales from molecules to organisms to swarms, I will show how a combination of quantitative experiments, physical analogies, mathematical theories and computational models allow us to begin to unravel the mechanistic basis for aspects of morphogenesis and thence towards physiology, pathophysiology and biomimetics.