Minimal surfaces bounded by elastic lines

TL;DR

This paper explores the shapes of minimal surfaces bounded by elastic, flexible lines, combining experiments, analysis, and simulations to extend classical Plateau problem concepts to elastic boundaries.

Contribution

It introduces the Euler-Plateau problem, coupling minimal surfaces with elastic boundary curves, and provides a comprehensive analysis through experiments, scaling, and numerical methods.

Findings

Identification of new equilibrium shapes for elastic boundary-spanned soap films

Development of a combined experimental and numerical framework for the problem

Insights into the geometric and physical properties of elastic-boundary minimal surfaces

Abstract

In mathematics, the classical Plateau problem consists of finding the surface of least area that spans a given rigid boundary curve. A physical realization of the problem is obtained by dipping a stiff wire frame of some given shape in soapy water and then removing it; the shape of the spanning soap film is a solution to the Plateau problem. But what happens if a soap film spans a loop of inextensible but flexible wire? We consider this simple query that couples Plateau's problem to Euler's Elastica: a special class of twist-free curves of given length that minimize their total squared curvature energy. The natural marriage of two of the oldest geometrical problems linking physics and mathematics leads to a quest for the shape of a minimal surface bounded by an elastic line: the Euler-Plateau problem. We use a combination of simple physical experiments with soap films that span soft filaments, scaling concepts, exact and asymptotic analysis combined with numerical simulations to explore some of the richness of the shapes that result. Our study raises questions of intrinsic interest in geometry and its natural links to a range of disciplines including materials science, polymer physics, architecture and even art.