The Soap Film

The film. Two rings, one soap film. The side view shows its shape inside an imputed cylinder that just circumscribes it; the shaded silhouette over-reads the volume, so trust the cylinder panel, not the shape.

The cylinder. Imputed from the rings: exactly as wide as the hoops and as tall as their separation. The panel shades its volume-true core, so the shaded share is the real fill: brimful when the rings touch, exactly one half at criticality, empty once it pops.

The panels. The necklace is the same curve hung like a chain; the waist is the rim staying put while the neck narrows; the lawn chair is the cost of every possible waist, with the film as a ping pong ball in the dip; the race pits a linear rabbit against an exponential hare.

The necklace. Two posts, 4 inches apart, and a chain fed in link by link. Pulled tight, the chain matches the film at height zero; feed in slack and it sags into the same curve the film's wall makes, blown up to fit the posts. One inch of extra chain covers the film's whole life: 4.00 inches taut to 5.03 inches at the last film the hoop supports. The chain keeps sagging happily past that point; the film does not, and that difference is a story for the FAQ.

Borrowed time. The film's one rule is spend less. Up to the borrowed time mark, the hourglass is the cheaper shape to keep: it costs less surface than two flat disks would. Past the mark the ledger flips: the two disks are now cheaper, and the film is the expensive option. It survives anyway, because getting to the disks has its own price, the climb over the ridge, called the toll.

The toll. Moving out is never free. Past the borrowed time mark the two disks are the cheaper home, and the savings grow with every inch. A rational tenant would move. The film cannot, because leaving means first building extra surface before saving any, and a film can only ever spend less, never more. It has no savings to hire the movers with. So it stays, overpaying, until something outside pays the movers for it (a draft, a poke), or the pull of the rings lowers the toll to zero and the wall between the two homes ceases to exist. The Numbers card quotes the toll live: about 52 units when the film is young, 10 at the tie, and 0 at the last possible instant.

The chair and the table. Every shape has a bill, measured in surface. We index the two flat disks at 100, so every other bill can be read against them. The chair's fabric is the film's ledger: the ping pong ball rests where the bill is lowest. The table stands at exactly 100, the disks' unchanging price. When the fabric's low point rises level with the table, the ledger has flipped; when the fabric pulls flat, there is nowhere left to rest.

The race. Pull the rings apart and two quantities multiply against each other. The pinch along the bottom counts how many waists, measured across, fit in the height. The straight line is the stretch the film has taken on. The falling curve is the waist itself, measured across, starting at 4 inches when the film is a full cylinder and shrinking as the pinch deepens. The height the film spans is their product, stretch times waist, and a product of a growing line and a shrinking waist can only rise so far. Early, the stretch wins and the height grows fast, because the waist has barely narrowed. Late, the waist collapses faster than the stretch can pay for, and the height falls. The peak sits at a pinch of 1.20, a height of 2.65 inches on the 4 inch hoop. The peak's pinch address is 1.19968, the root of coth x = x; the peak height divided by the hoop's full width is 0.66274, the Laplace limit, a constant astronomers met first. That peak is the critical catenoid. The waist panel tracks that same falling curve as a percent of ring width: it starts at 100% and falls with the blue curve, down to 55.2% at the peak. The curve's far side is not empty country: every height short of the peak is spanned by a second, slimmer film too: the wide film's slim sibling, its ghost twin, and the falling branch is where it lives.

For the full story, see the reader's guide.

Energy labels assume an industrial-strength bubble solution for making the soap film: 100 cost units stand for about 0.8 millijoules, so one unit is roughly 8 microjoules. Every length on this page is read in inches on a 4 inch hoop.  See the engine →