The devil lies in the details here. Brouwer's fixed point theorem says that for a continuous function f that maps a _compact, convex_ set to itself, there must

The devil lies in the details here. Brouwer's fixed point theorem says that for a continuous function f that maps a _compact, convex_ set to itself, there must exist a point x₀ such that f(x₀) = x₀. This theorem is only valid for sets that are compact (i.e., closed and bounded) and homeomorphic to convex. I think the 3-dimensional analogy (similar to the stirring of coffee and sugar Brouwer observed) is rather unintuitive. I think some of the proofs are quite beautiful, but I like this analogy from Wikipedia: > Take an ordinary map of a country, and suppose that that map is laid out on a table inside that country. There will always be a "You are Here" point on the map which represents that same point in the country.