Abstract
Adaptive avian radiations associated with the diversification of bird beaks into a multitude of forms enabling different functions are exemplified by Darwin's finches and Hawaiian honeycreepers. To elucidate the nature of these radiations, we quantified beak shape and skull shape using a variety of geometric measures that allowed us to collapse the variability of beak shape into a minimal set of geometric parameters. Furthermore, we find that just two measures of beak shape-the ratio of the width to length and the normalized sharpening rate (increase in the transverse beak curvature near the tip relative to that at the base of the beak)-are strongly correlated with diet. Finally, by considering how transverse sections to the beak centreline evolve with distance from the tip, we show that a simple geometry-driven growth law termed 'modified mean curvature flow' captures the beak shapes of Darwin's finches and Hawaiian honeycreepers. A surprising consequence of the simple growth law is that beak shapes that are not allowed based on the developmental programme of the beak are also not observed in nature, suggesting a link between evolutionary morphology and development in terms of growth-driven developmental constraints.