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Hi Jorgen,

I'm confused about the Brouwer fixed-point theorem, or at least its explanation. You offer three illustrations of the theorem: two pieces of grid paper, a map and the placed mapped, and the before and after of a stirred liquid. In all three cases, there will be (at least) one point in one (the crumpled grid paper, the map, and the stirred liquid) that corresponds exactly to a point in the other (the smooth grid paper, the place mapped, and the pre-stirred liquid).

But this seems mistaken, at least in cases 2 & 3, unless there's a condition forbidding rotation. Suppose that the table inside the country straddles the border between two states - let's say North Dakota and South Dakota. One flips the map so that map-South Dakota is on the north side of the border and map-North Dakota is on the south side, and the map-border is as thick as the real border. This seems to be a counterexample to the theorem (or at least to the illustration of the theorem).

Likewise in the case of the liquid: why can't all the liquid molecules on the north side of the glass wind up on the south side of the glass and vice-versa (let's assume that there is an exact halfway point between its north and south sides, and that this halfway point is itself an unoccupied boundary). If this condition is possible, then there won't be any point in the liquid that is in exactly the same place as it was before the glass was stirred.

The examples were helpful in letting me understand the Brouwer fixed-point theorem, so I appreciate them, even though - or perhaps in part because! - they may have oversimplified the theorem so non-topologists like myself could understand something.

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