Geometry and elasticity of strips and flowers

TL;DR

This paper investigates how surfaces with prescribed metrics deform, revealing that long strips tend to form twisted helices and that surfaces are determined by their metrics and boundary conditions, with singularities indicating shape deviations.

Contribution

It introduces a new formulation for surface metric problems using Lagrangian methods and extends evolution equations to analyze surface shape determination and singularity formation.

Findings

Long strips form twisted helical shapes at low energy.

Evolution equations determine surface shape from metric and boundary data.

Singularities indicate non-differentiability or metric deviations.

Abstract

We solve several problems that involve imposing metrics on surfaces. The problem of a strip with a linear metric gradient is formulated in terms of a Lagrangean similar to those used for spin systems. We are able to show that the low energy state of long strips is a twisted helical state like a telephone cord. We then extend the techniques used in this solution to two--dimensional sheets with more general metrics. We find evolution equations and show that when they are not singular, a surface is determined by knowledge of its metric, and the shape of the surface along one line. Finally, we provide numerical evidence that once these evolution equations become singular, either the surface is not differentiable, or else the metric deviates from the target metric as a result of minimization of a suitable energy functional .