Nature's forms are frilly, flexible, and functional

TL;DR

This paper explores the complex wrinkling patterns in natural hyperbolic surfaces, introducing a new defect type called branch points that influence morphology without elastic energy concentration.

Contribution

It identifies and develops a theory for branch points as natural defects in hyperbolic sheets, explaining their topological robustness and role in morphology.

Findings

Branch points are a new defect type in hyperbolic sheets.

Branch points carry a topological charge, providing robustness.

They influence morphology without elastic energy concentration.

Abstract

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.