Artificial Wasteland · a playable instrument

The Shortest Network

Connect a handful of points with the least possible length of road. The obvious answer — run the road along the shortest links between them — is not the shortest. Adding new junctions, where exactly three roads meet at 120°, makes the whole network shorter. A soap film finds those junctions by itself. Drag the pins below and watch it.

That is the Euclidean Steiner tree — the shortest network that joins a set of points, allowed to invent new junction points wherever that helps. A film of soap stretched between two glass plates with pins between them settles into exactly this shape, because surface tension pulls it to the least possible area, and the area of a vertical film is just its length on the plate times the gap. Least area means least length. The film is solving a geometry problem with no maths at all.

Why every junction is 120°

Look at any point where three sheets of film meet. Each sheet is pulling along itself with the same tension — it is the same soap. Three equal pulls can only balance if they are spread evenly, 120° apart: any tighter and two of them would gang up and drag the junction sideways until the angles opened back out. Writing the three equal unit pulls as vectors, balance means they sum to zero, and three unit vectors sum to zero only when each pair sits at cos⁻¹(−½) = 120°. This is Plateau's law, named for the blind physicist Joseph Plateau who catalogued it from soap in 1873.¹ You can read the angle straight off the film above — it holds at 120° no matter how you drag the pins, right up until a junction is no longer worth having and collapses onto a pin.

The square that hides the surprise

Press Square. Four pins at the corners of a unit square. The obvious network — three sides — has length 3. The next guess, an X of both diagonals crossing in the middle, is 2√2 ≈ 2.828. But the soap film grows two junctions, joined by a short bridge, every angle 120°, for a total of 1 + √3 ≈ 2.732 — shorter than either.² Dip the film a few times and watch the bridge flip between horizontal and vertical: the square is so symmetric it has two equally-best answers, and the film simply picks one as it forms. That picking-one is called symmetry breaking, and you are watching a soap film do it.

The catch — and why it matters

It is tempting to think a soap film is a free supercomputer: dip it, read off the perfect answer. It isn't, and the reason is deep. For a fixed wiring of the junctions, the total length is a tidy bowl-shaped function with a single bottom, and the film slides straight down to it. But which wiring to use is a separate, brutal question: the number of possible wirings explodes with the number of pins, and choosing the best is NP-hard — the same wall that guards thousands of the hardest problems in computer science.³ A real film just freezes into whatever wiring it happened to nucleate into. Press Dip again on a lopsided scatter and you will sometimes catch it settling for a longer network — occasionally one longer than the obvious network it was supposed to beat. The film is honest about finding a good answer fast; it never promises the best one. Scott Aaronson dipped real wire frames to make exactly this point.⁴