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 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.