  A ubiquitous motif in nature is the self-similar hierarchical buckling of a
thin lamina near its margins. This is seen in leaves, flowers, fungi, corals,
and marine invertebrates. We investigate this morphology from the perspective
of non-Euclidean plate theory. We identify a novel type of defect, a
branch-point of the normal map, that allows for the generation of such complex
wrinkling patterns in thin elastic hyperbolic surfaces, even in the absence of
stretching. We argue that branch points are the natural defects in hyperbolic
sheets, they carry a topological charge which gives them a degree of
robustness, and they can influence the overall morphology of a hyperbolic
surface without concentrating elastic energy. We develop a theory for branch
points and investigate their role in determining the mechanical response of
hyperbolic sheets to weak external forces.
