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When reading about the Brouwer fixed-point theorem on Wikipedia there are some "real world illustrations" of what the theorem says, one of them being the following:

[T]ake two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up (without ripping or tearing) the other one and place it, in any fashion, on top of the first so that the crumpled paper does not reach outside the flat one. There will then be at least one point of the crumpled sheet that lies directly above its corresponding point (i.e. the point with the same coordinates) of the flat sheet.

Now, what I don't get is the following: Is this example just meant as an illustration of what the mathematical theorem actually says (I haven't gone through the proof mathematically) or is it meant to point to something that is obvious in the real world which then only is formalized mathematically? If it is the later, I don't understand why the given illustration necessarily should be true and am in search of a natural language explanation for it.

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My favorite is the stirred-coffee analogy. There's always a point on the surface that hasn't moved. My guess--and it is only a guess--is that the theorems preceded some of the unconvincing and not-very-useful situations used to illustrate them. – daniel Feb 7 '12 at 21:57
Nothing really related to audio-video production comes to mind. – Will Jagy Feb 7 '12 at 22:00
up vote 8 down vote accepted

The ‘real-world’ illustrations are intended to be more intuitive illustrations of what the theorem says; you’re not expected to find the real-world facts intuitively obvious. In fact, to some extent that’s the point: the mathematics tells you something interesting that isn’t intuitively obvious.

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