Brouwer and his fixed point theorem

    

The Brouwer fixed-point theorem is one of my favourite mathematical theorems. It is named after the Dutch mathematician Luitzen Egbertus Jan Brouwer (above right). Brouwer is also known for his work in Intuitionism. I have mentioned the Brouwer fixed-point theorem before.

The theorem states that any continuous function f on a compact convex set (and specifically, on a disc in the plane) will have at least one fixed point – that is, there will be at least one point p such that f(p) = p. The picture below is intended to illustrate the theorem; it is explained further down.

In the case of a disc, the theorem can be proved by contradiction. Assume that f(p) ≠ p for every point p. Then the pair of f(p) and p always defines a continuous mapping g from p to the boundary of the disc, as illustrated above (left). However, such a continuous mapping is impossible (for complex reasons, but in simple terms, because it creates a hole, which continuous mappings cannot do).

So what about that picture? It shows a continuous function f from the disc to itself, combining an irregular rotation about the centre (rotating least towards the east of the disc) with a “folding” operation that leaves the centre and boundary untouched. The picture below shows a cross-section of the folding in action. The shades of blue in the picture above show how far each point p is from f(p), with lighter colours representing smaller values. Arrows show the action of the function on 6 randomly chosen points. There are two fixed points, marked with black dots: the centre and one other point where the folding and the irregular rotation cancel each other out.

The three-dimensional version of the theorem tells us that, when I stir my morning cup of coffee, at least one speck of liquid will wind up exactly where it started.


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