A soap film stretched across a closed wire loop pulls itself into the shape of smallest possible area. Surface tension acts equally everywhere, so the film settles where it cannot get any tauter — a minimal surface.

Plateau's problem

Given a closed boundary curve (the wire), find the surface of least area that spans it. This is Plateau's problem, named after the 19th-century physicist Joseph Plateau, who studied soap films experimentally. Real films solve it instantly.

Zero mean curvature

A minimal surface has mean curvature H = 0 at every interior point: wherever the surface bends up in one direction it bends down equally in the perpendicular direction, like a saddle. The two principal curvatures cancel.

Laplace's equation & the small-slope limit

For gentle slopes the height field h(x,y) of a minimal surface satisfies Laplace's equation ∇²h = ∂²h/∂x² + ∂²h/∂y² = 0. The discrete version says each interior height equals the average of its four neighbours:

h[i,j] = (h[i−1,j] + h[i+1,j] + h[i,j−1] + h[i,j+1]) / 4

We fix the boundary heights from the wire and sweep this rule across the interior (Jacobi / Gauss-Seidel relaxation) over and over. The residual — the largest change in any cell — shrinks toward zero as the field converges to the harmonic solution. The simulation stops once the residual drops below a small epsilon or it hits an iteration cap.

Soap films as analog computers

Because the film instantly settles into the harmonic / minimal shape, physicists once used soap films and stretched membranes as analog computers to solve Laplace's and Poisson's equations — for torsion in beams, electrostatic potentials and heat flow — long before digital machines.

Presets to try

• Saddle — two opposite corners high, two low. The film becomes a hyperbolic paraboloid, the textbook minimal-surface look.
• Tilted square — a flat frame raised along one edge.
• Bump / Pinch — a localised hump pulled into the rim.
• Catenoid 1D — a cross-section row whose ends are pinned; it relaxes toward the catenary-like profile of a soap film between two rings.