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Offlineisaacein
exp(ix) = cosx + isinx


Registered: 05/21/08
Posts: 1,141
Last seen: 6 years, 7 months
The Hairy Ball Theorem
    #11194647 - 10/06/09 05:39 PM (8 years, 13 days ago)

I thought some people here would enjoy seeing it.

The theorem states that a continuous vector field tangent to the sphere must contain a zero vector. In less abstract terms, it states that you cannot "comb a hairy ball without creating a cowlick".

A picture is worth a thousand words :



For instace, this is not true of a torus :



This is related to the fact that there is no three-dimensional equivalent of the complex numbers. The 3-sphere (the surface of a four-dimensional ball) does admit continous, nonvanishing tangent vector fields, and this is in turn related to the fact that there is a four-dimensional equivalent of the complex numbers (the quaternions).


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InvisibleChespirito
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Registered: 02/13/09
Posts: 3,259
Re: The Hairy Ball Theorem [Re: isaacein]
    #11194785 - 10/06/09 05:58 PM (8 years, 13 days ago)

Ha I dont know much about topology, however I was at a party once and my friends topology professor was there for some reason.  My friends and I were trashed and we made fun of him and his field for having a theory called the Hairy Ball theorem.  Seemed amusing at the time though we were plowed under man, I think I sexually harassed him afterwards just to see how obnoxious I could be.  Ah undergrad


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Invisiblecortex
[ H ] ψ = [ E ] ψ
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Re: The Hairy Ball Theorem [Re: Chespirito]
    #11194815 - 10/06/09 06:03 PM (8 years, 13 days ago)

:lol:


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Offlineisaacein
exp(ix) = cosx + isinx


Registered: 05/21/08
Posts: 1,141
Last seen: 6 years, 7 months
Re: The Hairy Ball Theorem [Re: Chespirito]
    #11195032 - 10/06/09 06:34 PM (8 years, 13 days ago)

Quote:

Chespirito said:
Ha I dont know much about topology, however I was at a party once and my friends topology professor was there for some reason.  My friends and I were trashed and we made fun of him and his field for having a theory called the Hairy Ball theorem.  Seemed amusing at the time though we were plowed under man, I think I sexually harassed him afterwards just to see how obnoxious I could be.  Ah undergrad




Haha! Good story.


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OfflineChuangTzu
starvingphysicist
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Folding@home Statistics
Registered: 09/04/02
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Re: The Hairy Ball Theorem [Re: Chespirito]
    #11195683 - 10/06/09 08:13 PM (8 years, 13 days ago)

Quote:

Chespirito said:
I think I sexually harassed him afterwards just to see how obnoxious I could be.




Did you tell a lewd version of the coffee cup and donut joke?


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InvisibleAnnapurna1
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Registered: 05/21/02
Posts: 5,646
Loc: innsmouth..MA
Re: The Hairy Ball Theorem [Re: isaacein]
    #11197978 - 10/07/09 01:54 AM (8 years, 13 days ago)

heres a couple of proofs...but dont ask me to try to make head or tail out of them ..

http://www.math.byu.edu/~jarvis/sperner.pdf
http://topologicalmusings.wordpress.com/2008/07/22/analyzing-the-hairy-ball-theorem/


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