>> 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.

Indeed.

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

No. Quaternions do not form a field. Neither do the hypercomplex 
numbers, nor any of the other 4D generalisations.

> 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.

Interesting...

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