You are delegated the task of designing a new motorway system to connect three towns that form the vertices of an equilateral triangle. Don’t worry, it’s not difficult. All you must do is connect the towns in such a way that you use the minimal length of motorway.
What instantly comes to mind is to connect the towns in a triangle:
If the distance between each town was 1 (for simplicity), then the total length of motorway that you have used is 3. However, you may realise that the third side is not necessary since you can travel via the middle town, thereby reducing the length to 2:
The shortest possible length requires a tiny bit of thinking outside the box, and the solution is to connect the motorways so that they meet in the centre:
This solution has a length of approximately 1.3.
Your employer is so proud of your cost-cutting abilities that you’re given another assignment, this time four towns in a square. You’ve learnt from your first assignment that connecting the towns in a V or a U-shape is not particularly efficient, and instead it’s better to try and connect them at the centre. Perhaps a cross then?
This solution has a length of approximately 2.82. You begin to feel smug, until you have a hunch that there may be an even better solution. You mess about with lines and curves until you have THE solution:
This is the best possible design, with a length of approximately 2.73.
This problem is one of a series named the Steiner tree problem. Steiner trees concern the optimal interconnects between a given set of objects, and has clear real-world applications, one of which we have explored above. (Of course, the motorway system that you may be asked to design will most likely be more complex and many other factors must be considered.)
An curious method to solve these problems is through the use of bubbles. Bubbles tend to take the shape that has the minimal surface area, as you can find more about here.
In any case, your employer is very proud.