The Soap Film

Dip two rings of wire in soapy water and a film spans the gap. Surface tension pulls it to the smallest possible area — and the only such shape is the catenoid, a hanging-chain curve spun around. Pull the rings apart and the film thins… until, past a sharp threshold, it can’t hold — it snaps into two flat discs. Drag the stage to rotate.

The wire frame

Ring radius R 1.00

Separation 2h / R 0.80

Show

Frames

neck radius a—

film area—

two-disc area—

max mean curvature—

The profile is a catenary’s sibling: a chain hangs as a cosh; spin that curve and you get the only minimal surface of revolution. Self-test proves zero mean curvature, the closed-form area, and the Goldschmidt snap. · ↗ The First Integral — the neck radius a is a conserved quantity, the same law a chain and a slide obey.