Eur. Phys. J. E (2021) 44: 95
https://doi.org/10.1140/epje/s10189-021-00099-6

Regular Article - Soft Matter

Nature’s forms are frilly, flexible, and functional

¹ Department of Mathematics, Southern Methodist University, 75275, Dallas, TX, USA
² School of Mathematical Sciences, University of Arizona, 85721, Tucson, AZ, USA
³ Department of Mathematics and Statistics, Wake Forest University, 27109, Winston Salem, NC, USA

^e shankar@math.arizona.edu

Received: 17 March 2021
Accepted: 25 June 2021
Published online: 13 July 2021

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.