Four Colours Solution?

Four Colours Solution?
27 Oct 2003

I was just now listening to Simon Singh's Radio 4 broadcast about the Four Colours problem. No doubt a little over-stimulated by having just finished Cryptomicron (cracking book, if you haven't read it), I'm now sitting in a Little Chef having a scribbled a potential, mathematical solution to the problem all over the backs of the menu sheets. It goes something like this: rather than trying to work out all the possible maps, it instead tries to enumerate all the potential relationships between two two-dimensional shapes.

First, an assumption - that the four-colour issue doesn't apply for shapes that meet at a point where several (more than four) lines meet, like the roads leading off from the Arc de Triomphe. If this assumption were wrong, and there were ten lines meeting at the point, this would require 9 colours. Sorry - what I'm referring to are the shapes that are diametrically opposite (across the point) as opposed to the shapes which are next to each other and happen to meet at the point.

Given this assumption, the steps to a proof are as follows.

1. Prove that there are only 6 ways in total that two shapes can join, based on whether the connection is made with an edge, a vertex or one object completely inside the other. This results in the proof also that no shape can touch any more than three already touching shapes.
2. Prove that, given an infinitely large shape as a starting point (represented as a line that stretches to infinity in either direction), and given that the shape has one colour and the space not occupied by the shape has another (visualise the sea and the sky, with the line being the horizon), you can add another two shapes into the resulting visualisation with only two more colours. This should be true however the additional shapes are positioned with relation to each other (there are 6 ways, remember).
3. Prove that you can add another shape without requiring any additional colour, based on 6 available relationships between shapes:
   a. If new shape touches one or two existing shapes (including the infinite line), use the colour of the shape it does not touch
   b. If the shape doesn't touch any other shapes, use either of two available colours
   c. If new shape touches 3 or more shapes, you may have to switch colours with existing shapes to accommodate new relationship between shapes. As a shape cannot contact any more than 3 others (proved in (1) ), you should always have one colour to spare
4. Finally, prove that the act of joining the two points of the infinite line has no effect on the number of relationships that can exist between the existing shapes. The same is true if the plane is linked, for example by forming a sphere or a cylinder.

There may be another way to prove this, going purely on the 6 relationships basis. Any shape can be made into a triangle just by straightening out the lines. It appears to my limited brain that there are only so many ways that the planar sides of 2, 3, 4, - n triangular shapes can come into contact with each other. The central triangle can of course be connected to stacks of triangles, it's just that they can't all touch each other (for a start, they're all pointing away from the centre!) As no 2-dimensional object can have less than three sides, then topologically, any part of a map can be reduced to a 3-sided object with a number of 3-sided objects connected to it. As any shape can be positioned as the central shape, topologically, prove it for one and you've proved it for them all.

Maybe I should have stuck with languages ;-)