In deciding where the stepping stones should be laid in our garden, I thought it would be fun to minimise the total length of the three legs between our two sheds and the steps onto the ‘lawn’.
There is a lovely analytical solution for this simple case which involves determining the Fermat-Torricelli point, i.e. the point that minimizes the sum of the distances to the 3 vertices of a triangle (with none of its angles greater than 120° – see also Fermat point ).
This can be found using a model and soap films, but the method shown allows each of the endpoints to be weighted in terms of importance. I decided that the sheds and steps were of equal priority and thus weighted the milk bottles with the same amounts of water as they were hung over the wheelie bin wheels.
The final path we chose looks nothing like the minimal length version, which was just ugly.