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 can find those junctions by itself. Drag the pins below to explore a numerical model.

This numerical model can miss the shortest network. The angle readout omits merged vertices.

The target 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 can settle into this shape. Surface tension drives it toward a minimum of area, which can be local rather than absolute, and the area of a vertical film is its length on the plate times the gap. At a fixed gap, least area means least length. The film tackles a geometry problem with no maths at all.

Why balanced three-way junctions are 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 The readout above shows angles at separate three-way junctions. Numerical junctions need not balance exactly, and vertices can merge with each other away from the pins, leaving four-way meetings. Vertices on near-zero edges are omitted from the angle readout.

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. The model can also return a crossing; that output is not evidence of a stable film.

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 convex function, but its bottom need not be a single point. Convexity alone does not guarantee that this numerical iteration reaches the bottom. And 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 can become trapped in a suboptimal configuration. Press Dip again on a lopsided scatter and the model will sometimes return a longer network, occasionally one longer than the obvious network it was supposed to beat. That length comparison does not test physical stability. Scott Aaronson dipped parallel glass plates joined by copper rods and found that real films do not always reach the optimum.⁴