Four Colour Theorem

Ugliness
80x60cm 2005
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THE FOUR COLOUR THEOREM

In Greek mythology, the Sirens were sisters whose song lured sailors to their deaths on a treacherous reef. The Four Colour Theorem is to mathematicians what the song of the Sirens was to the sailors of ancient times. Many a mathematician has foundered in attempting its proof .

The problem is deceptively simple. Take a map of the world, a few coloured pens and start colouring in the countries. There's one rule--adjacent countries cannot be the same colour. The Four Colour Theorem is the hypothesis that the maximum number of colours needed to fill the map is four.

It's the kind of problem that, for a mathematician, has the smell of simplicity and elegance about it. You just know that somebody somewhere will have come up with an elegant method that makes the answer apparent. 

Except they haven't. In place of an elegant proof is a monstrous machine created by Kenneth Appel and Wolfgang Haken, both mathematicians at the University of Illinois, who cracked the problem in 1976. But instead of insight and panache, they used force.

Appel and Haken first worked out that they could reduce the number of possible maps from infinity to just 1,936. They then checked each one against the theory. The task turned out to be so intensive that they needed a computer to do it. Even then, it took hundreds of hours. In a sense, they squeezed the life out of the problem until it gave up its answer, a method known as brute force.

Many mathematicians were deeply unsettled by this result because it offered no new insight into the problem, it could not be checked by a human and above all it offended their aesthetic sense. It was ugly.

Twenty years later, Neil Robertson, Daniel Sanders, Paul Seymour and Robin Thomas produced another proof that whittled down the number of maps to 633, which they then checked on a computer.  The piece above shows them all. They are reproduced with the kind permission of Robin Thomas.

The search for an elegant proof of the Four Colour Theorem is ongoing. And while computer-aided proofs have begun to gain acceptance, largely thanks to the Four Colour Theorem, there remains the feeling that beauty, elegance and insight should triumph over the horror of a computer-generated proof.

 FURTHER READING
A discussion of the four colour theorem is at:
http://en.wikipedia.org/wiki/Four_color_theorem

A brief summary of proof by Robertson et al is at:
http://www.math.gatech.edu/~thomas/FC/fourcolor.html

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