Soap films span wire frames by minimizing area. The mean curvature H=0 everywhere characterizes minimal surfaces. Solved variationally via the Euler-Lagrange equations.
Plateau's problem (1847): Given a closed wire curve, find the surface of minimum area spanning it.
Catenoid: Only minimal surface of revolution. Parametric: x=a·cosh(v/a)·cos(u), y=a·cosh(v/a)·sin(u), z=v
Helicoid: Simply connected minimal surface. Conjugate to catenoid via Bonnet rotation.
Enneper: Self-intersecting algebraic minimal surface.
Scherk: Doubly periodic, models two intersecting planes at infinity.
Weierstrass–Enneper: every minimal surface given by holomorphic f,g.