JOSEPH PLATEAU · GHENT · 1873 — PERFORMED LIVE
Soap films solve it instantly. Dip a wire, and the film finds the least possible surface — no iteration you can see, no error it will admit. Below, the computation runs in the open: a mesh descending its own area, and snapping when no minimum exists.
DRAG THE GAP · KEYS 1 2 3 SWITCH FRAMES
— THE INSTANT SOLVER
Joseph Plateau stared into the sun in 1829 to study afterimages and paid for it with his sight. Blind by 1843, he ran the great film experiments through his family's eyes and published the laws in 1873: films meet in threes, at 120°, always. Their borders meet in fours, at 109.47°, always. Mathematics needed until 1976 — Jean Taylor's proof — to concede he was right. The frame above doesn't illustrate his films; it computes them, every frame, in front of you.
— THREE WIRES, THREE VERDICTS
01 · EULER 1744
TWO RINGS, ONE NECK
The only minimal surface of revolution. Between two rings the film hangs a cosh curve, necking to 0.85 R at a gap of one radius. Pull the rings apart and the neck thins — the film renegotiating its minimum in real time — until the gap passes 1.3255 × R, where the catenoid simply stops existing.
02 · PLATEAU 1873
AN INNER ROOM
A cubic frame refuses the obvious answer. Instead of six faces the film builds twelve panels leaning into a small square suspended at the centre — thirteen surfaces meeting along liquid edges at 120°, exactly as Law I demands. The readout above measures that angle off the live mesh while the cube is dipped.
03 · MEUSNIER 1776
THE ONLY TWIST
A straight wire for an axle, a helix for a rail, and the film spans them with the only ruled minimal surface besides the plane. Every point a saddle, mean curvature exactly zero everywhere. Here it arrives bulged and wrong, and you watch the flow iron it back to the true twist.
— WHEN THERE IS NO ANSWER
1.3255
Beyond that separation no catenoid exists — not a weaker one, not a stretched one; the equation has no solution to offer. The neck accelerates inward, pinches in finite time, and the film retreats to two flat discs — the Goldschmidt solution, on record since 1831. Nothing breaks. The mathematics abandons the surface, and the surface leaves.
Drag the gap slider past the magenta tick and watch the readout: the area never bargains, it only descends. Through the pinch it slides — catenoid, cone, discs — one continuous downhill argument. 2πR², the price of two discs, is where it lands. Then dip again.
— WHAT YOU ARE WATCHING
Every frame, about 3,600 triangles report the gradient of their area to about 1,900 vertices, which step downhill — mass-normalised area descent, the discrete cousin of mean-curvature flow. Boundary vertices stay soldered to the wire; everything else negotiates. The catenoid's snap is not scripted: past the critical gap the descent has no floor, the neck pinches on its own, and the page merely notices.
The colour is a thin-film interference sketch: optical path length grows at grazing incidence, so the fringes crowd the silhouette — magenta trading with cyan the way a real film's micron-thick wall trades wavelengths. The cyan trace in the corner is the solver's confession: total area, sampled live, always falling.