User talk:Zhw

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greetings
Hi Zhw,

I notice you have made a few additions to mathematics articles. You might consider adding yourself to WikiProject Mathematics/Participants. You might also want to put Wikipedia Talk:WikiProject Mathematics on your watchlist; that's where people discuss various things about mathematics articles. Dmharvey 03:45, 2 April 2006 (UTC)

Calling programmers
We need coders for the WikiProject Disambigation fixer. We need to make a program to make faster and easier the fixing of links. We will be happy if you could check the project. You can Help! --Neo139 09:21, 5 August 2006 (UTC)

Image tagging for Image:Kbattleship.jpg
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Hairy ball theorem removal from quaternions and spatial rotation
Hi - can you tell me why the hairy ball theorem does not explain the degeneracy of a two-parameter description of the points on a sphere? (just learning this stuff) PAR (talk) 22:26, 4 April 2010 (UTC)
 * Well, I guess it depends on what exactly is meant by non-degenerate. If it means that the parameterization is a diffeomorphism, then it's obviously impossible for topological reasons. If it just means that it's a local diffeomorphism, then sure, the hairy ball theorem implies that (such a local diffeomorphism would map a constant vector field on a square to a non-vanishing vector field on the sphere), though there might be a more elementary way to prove it. I removed that remark, because it seemed somewhat out of place: for someone not familiar with topology it might not be evident what the hairy ball theorem has to do with anything, and its page does not list this fact among the consequences. Feel free to revert the edit, if you feel otherwise. Zhw (talk) 00:13, 5 April 2010 (UTC)


 * I am approaching this as someone trying to understand why and when a given parameterization of the points on an n-sphere yields certain points that are not differentiable, so I don't feel qualified to revert the edit. I was trying to understand the application of the hairy ball theorem to that problem, when suddenly you removed it, so I figure you are a good person to talk to. As the article stands now, it simply states that a parameterization using two coordinates on a 2-sphere is impossible, with the only link to an explanation (correct or not) removed. I mean you say "obviously impossible for topological reasons" and I'm trying to get into a position where I can say "yes, obviously". May I ask you to edit the article and provide a short explanation, and a link or two, so that someone who is not too familiar with topology (like me) can pursue the question a little further? I mean, hopefully, a link or two that points to an article or article section that more or less directly addresses the problem rather than a link to the topology article.


 * Perhaps you have some insight into the particular problem I am struggling with. I am trying to understand why a total of two unit quaternions map to a single rotation in 3-space. I visualize a unique rotation in 3-space as 1-to-1 (bijective?) with a unique line through the origin of a 4-dimensional Euclidean space. The space in this case is parameterized by a general quaternion. When you constrain the quaternion describing a rotation to be a unit quaternion, you wind up with two points on the line describing the rotation, and any attempt to reduce that description to a single point will give a discontinuity that doesn't "really exist" but is rather a consequence of the parameterization. This was the point that was confusing me, I was thinking the "double cover" was maybe reflecting some wierd topology of the space of rotations. Does this sound like a valid train of thought?


 * With regard to the hairy ball theorem, it states that you can't comb the hair on an n-ball where n is odd, without creating a discontinuity. In the case of 3-D rotations, n is even. Can you comb the hair on an n-ball where n is even? Specifically, can you comb the hair on an a 4-ball? What does this imply about the possibility of a 3-parameter parameterization of the space of rotations? Thanks for any help you can provide. PAR (talk) 09:37, 5 April 2010 (UTC)


 * As I said, the hairy ball theorem does imply the impossibility of such a parameterization, though, depending on the exact definition of "degenerate", there might be easier ways to prove it. I'm sorry, I am not aware of any source where this fact is explained in more detail. Your insights about the space of rotations are generally correct. I'm not going to delve into a long explanation here, but you can contact me by email or jabber (p.capriotti at gmail), if you wish, and I can try to help you make sense of it. Zhw (talk) 09:56, 5 April 2010 (UTC)


 * Thinking about this some more, I have now a very simple explanation why there is no such parameterization. The idea is simply that a global non-degenerate parameterization is a covering map, and n-spheres (with n > 1) are simply connected, so they have no non-trivial covering space. This answers your last question: the 3-sphere admits a non-vanishing vector field, but no global parameterization. Again, feel free to contact me if you are unfamiliar with the terminology, I'm not an expert myself, but maybe I can shed some light on some of this stuff. Zhw (talk) 10:16, 5 April 2010 (UTC)