richard vodden’s blog

It is a well known fact (although it has yet to be elegantly proven) that if you have a map, no matter how complicated, then you only need 4 colours to colour it in in such a way that no two touching countries are the same colour. The proof is nasty, but some things can be found about it here.

That is all well and good but I (being awkward as ever) wondered what would happen if we were looking at a three dimensional map. Another way of stating the four colour map theorem is: It is impossible to construct a map in such a way that four or more countries touch any single others.

Rather dissapointingly it turns out that there is no limit to the number of colours which are required for a three dimensional map. The proof (or counterexample) is quite simple. Take a rope and put it on the ground in a straight line, then take another rope and lie it down next to it, so that its touching all the way along one edge. Take a third rope and lie that next to the second in the same way. Now take all three ropes and fold them over each other into a V-shape, so that you get a shape which when viewed from the top like this:

               +   +   +
              / \ / \ / \
             /   \   \   \
            /   / \ / \   \
           /   /   \   \   \
          /   /   / \   \   \
         /   /   /   \   \   \
        /   /   /     \   \   \

With a little thought you can see that all the ropes are touching all of the others. This means that you need three colours to colour in this 3D map. You can probably also see that if we add a fourth rope, it will also touch all the other ropes, so we need at least four colours. We can keep adding ropes as long as we like and we will keep needing more colours. Hence there is no limit to the number of colours required to colour in a map in three dimensions.

Ho hum… It is nice to occasionally demonstrate to myself that I can still do maths