Steiner Networks and Soap Films

What’s going on

Four towns stand at the corners of a square, 10 km apart. They want roads that let you drive from any town to any other, using as little road as possible. Roads may meet at new junctions out in the fields.

The first idea is the two diagonals, meeting in the middle. That is 2 × 14.14 ≈ 28.3 km. Roads along three sides of the square are worse: 30 km. But look at the scene above: the shortest network has two junctions joined by a short middle road, and it needs only 27.3 km. That is 10 + 10 × √3, almost a whole kilometre less than the X.

Drag the towns around and watch the network change. One rule keeps showing up:

At every junction, exactly three roads meet, and they meet at 120°.

Why 120°?

Start with three towns. Where should one junction go so that the three roads are as short as possible? Imagine three strings tied together in a knot. Each string runs through a hole in a table, at a town, and pulls with the same weight underneath. The knot settles where the weights hang lowest, which is where the string on the table is shortest. Three equal pulls balance only when they are 120° apart.

This special point is called the Fermat–Torricelli point of the triangle. Pierre de Fermat posed the puzzle, and Evangelista Torricelli solved it around 1640. Switch the scene to Three towns and look at the marks around the junction.

There is a catch. Choose Flat triangle, or drag one town until the angle of the triangle at that town grows past 120°. Now no point inside can see all three towns at 120°, and the junction slides into the town itself. If a triangle has an angle of 120° or more, the shortest network is just its two shorter sides.

Why the X loses

At the centre of the X, four roads meet at 90°. Any junction where roads meet at less than 120° can be improved: split it in two, pull the pieces apart a little, and the total length drops. Keep improving and you arrive at the network with two junctions. For a square there are two such networks, one turned by 90° from the other, and they are equally short. Tick Other junction pattern to see the second one.

A shortest network like this is called a Steiner network, after the Swiss geometer Jakob Steiner. With more towns there are many ways to place the junctions, and finding the best one quickly becomes very hard, even for computers.

Let soap do the work

Put two clear plastic sheets a few centimetres apart, joined by a few pins standing between them, and dip the whole thing into soapy water. When you lift it out, soap films stretch between the pins. A soap film pulls itself as small as it can, so it draws a network of short walls between the pins, with three films meeting at 120°. Tick Soap film in the scene to see what it looks like.

Soap is not perfect: it finds a network that small changes cannot shorten, but a better one may exist further away. Dip it again and you may get a different answer.

Try it yourself

1. A rectangle of towns

Four towns sit at the corners of a 3 km × 4 km rectangle. The two diagonals are 5 km each. Can you beat 10 km? Which way should the middle road point?

Answer

Yes. Put the middle road along the long side: the best network is 4 + 3 × √3 ≈ 9.20 km. Pointing it the other way gives 3 + 4 × √3 ≈ 9.93 km, still better than the diagonals. Three sides of the rectangle also give 3 + 4 + 3 = 10 km. Try it: choose Square and drag the towns into a 3 by 4 shape.

2. Three towns in a perfect triangle

Three towns form an equilateral triangle with sides of 10 km. How long is the shortest network, and how much shorter is it than two of the sides?

Answer

The junction is the centre of the triangle, 10 ÷ √3 ≈ 5.77 km from each town. The three roads add up to 10 × √3 ≈ 17.3 km, compared with 20 km for two sides. That saves about 2.7 km.

3. The wide-angle village

Three villages: A and B are 10 km apart, and C is close to the middle of the road between them, so the angle at C is 150°. Where should the junction go?

Answer

At C itself. The angle at C is more than 120°, so no junction can do better than the two roads A–C and C–B. Any junction away from C would add length. Choose Flat triangle in the scene to see this case.

Go deeper

• The 120° rule for soap films is one of Plateau’s laws, found by the Belgian physicist Joseph Plateau in the 19th century: soap films meet three at a time along an edge, at angles of 120°.
• Steiner networks matter wherever wire or pipe is expensive: in laying out cables and pipelines, and in wiring the circuits of computer chips.
• For many towns the problem is famously hard: no one knows a fast method that is guaranteed to find the best network every time.