Brouwer's fixed-point theorem, permutations and coffee

Question

One of my friends pointed out an interesting application of the Brouwer's fixed-point theorem: You cannot stir a coffee in a mug such that all of the coffee particles have changed their position. Namely there exists one particle that is exactly at the same point from which it started. Does the Brouwer's theorem have something to do with permutations, or are permutations not meaningful to represent such a continuous mappings in this case?

Answer 1

The modern definition of permutation is "a one-on-one correspondence from a set (which could be infinite) to itself". The image of the coffee in the mug suggests a continuous permutation of its atoms or its idealized points.

That said, the Brouwer Fixed-Point Theorem applies even when the continuous map is not a permutation, so I wouldn't say the theorem is about permutations. In the coffee analogy: It still works if you could compress the water into a smaller volume of the mug and if some points magically ended up in the same spot after stirring.