  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.
