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Comment by GeneralMayhem | Hacker News Reader

It only applies when you're mapping a space onto itself (plus a couple other qualifications), so you have to put the glass back in the same place.

I think of it as a generalization of the intermediate value theorem - some things are going left, some things are going right, one thing must be sitting still in between.

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And: it has to be ‘continuous’. E.g. in ‘real’ water, the molecule of H2O ‘colloquially’ known as ‘number 12345678998776165441’ which sat next to molecule known as ‘number 1’, still has to sit next to it. All be it in a different location. I think something like mayonaise would have been a better example, as water molecules that are ‘next’ to each other can in reality split up quite easily by itself. While something like mayonaise would not.

*off course, quantum mechanics and all that suggesting that it would be impossible to label the individual molecules.

QM doesn't say that it's impossible to label the individual moleccules. However, hydrogens are labile, it makes more sense to identify the unique oxygens.

Are you trying to say that for every point P there exists a neighborhood U around it for which the each transformed point T(P) is contained in the neighborhood T(U)?

What's the conceptual difference between this and the hairy ball theorem?

Loosely, Hairy ball theorem is about the derivative of a mapping at a single point in time. (Imagine slowly deforming/mixing the input configuration t to obtain the output configuration. Brouwer's theorem can be thought of as being about the integral of a continuous family of hairy balls over a time interval.

It's not exactly the same because the assumptions about differentiability are different.

Interesting, thanks!

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