Le 07/05/2012 23:09, Orchid Win7 v1 nous fit lire :
> Question: Why can't you extend the complex numbers to 3D space?
> 
> Answer: Because the hairy ball theorem says so.
> 
> Yes, that's right. There really is a theorem called "the hairy ball
> theorem". Isn't that wonderful? :-D
> 
> Perhaps even more satisfyingly, what this theorem /says/ isn't some
> obscure exotic thing that only a mathematician could understand.
> Actually, it just says you can't comb the hair on a ball flat. You
> always end up with at least one tufty bit. (You /can/ comb the hair flat
> on a flat plane, a torus, or a number of other 3D shapes. Just not a
> sphere.)

It's even more deeper. The theorem says you cannot comb a sphere of odd
dimension, but that even dimension is ok.
(2D sphere == circle, 4D sphere is left as an exercise)

That's also why, even if complex cannot be extended in 3D, they can in
4D. (look at quaternion...

Notice that there is a meteorological application of the theorem: there
is always on earth at least one place where the horizontal speed of wind
is null.

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