[BorisTheBrave]

A better example I've seen for the theorem is that if you take a paper map of a country, messily schrunch it up into a ball and drop it somewhere in the country, there will be at least one point on the map that is exactly above the corresponding real point in the country.

As others have said, it's meant for mappings of the space to itself. So stirring the water, but not moving the glass.

But anyway, the theorem works only for continiuous mappings. The moment they started mentioning "water particles", instead of some hypothetical fluid that is a continuous block, the theorem no longer applied. You could break it by mirroring the position of every particle. There's still a fixed point (the line of mirroring), but there's no obligation that there's a particle on that line.

[drc500free]

Another practical example is that you're guaranteed to have an annoying spot on your head where the hair sticks straight out and you can't brush it down.

That's actually the one that helped me visualize the theorem. If you look at your scalp from above, you can divide all the hairs into "points left" or "points right" and draw a boundary between them of hairs that point neither left or right. Then you can do the same thing with "points up" and "points down." Where the two kinds of boundaries cross, you have a hair that doesn't point up, down, left, or right - it points straight out of your scalp.

[chacham15]

This still doesnt make sense to me. Imagine a continuous line whose positions map to the real numbers between 0 and 1. If I "move" the line over 0.1 wrapping the end back to the beginning (i.e. x2 = (x1 + 0.1) % 1), there will be no points that are in the same position as they were in before.

EDIT: If you need a continuous function, wouldnt expanding the space to a line from -Inf to +Inf and then using x2 = x1 + 0.1 do the trick?

[contravariant]

Brouwer's fixed point theorem only applies to compact convex sets.

Infinite lines don't work, as they are not compact.

Similarly a circle would not work as it is not convex (you're close with your example, you just need to glue together the endpoints to turn it into a circle and make the map continuous).

[frutiger]

The theorem only applies to continuous functions.