Talk:Hairy ball theorem
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In the main picture for this article why is the hairy sphere described as having 'uncomfortable' tufts.......
Imagine if you were combing your hair and you have a cowlick. It's not that the tuft is actually uncomfortable, it's that the sight of them makes you uncomfortable. Nobody likes looking at them anyways, right? 24.23.13.3 (talk) 21:47, 22 January 2015 (UTC)
I think the last sentence "It follows from the theorem that there is always a cyclone somewhere on the Earth's surface." is incorrect. It only follows that there is a place where is no wind. Metterklume
I don't understand this theorem. --Abdull 6 July 2005 15:36 (UTC)
I like the idea of the last sentence, but wind has three dimensions not two, so it's not true! Should we delete the sentence or say "if we imagined that wind has only two dimensions instead of three..."? --Michael
The last sentence needs to be removed as the HB theorem does not apply to 3d atmospheric circulation. -- Cord
The weather is approximately 2D; the circumference of Earth is much larger than the height of Earth's troposphere (which is where weather as we know it takes place): on the order of 10 km. The article is currently protected from Farking fallout (see below), but this is how I would write it:
-- Toby Bartels 23:13, 2005 August 1 (UTC)
I've removed this section unless anyone can word it with sufficient clarity without making it trivial (which I cannot imagine). While the height of the troposphere may be considered irrelevant in comparative scale, its existence nevertheless allows two (or more) opposing vectors directly above any point on the earth. In fact, this does happen, see the picture (right) from atmospheric circulation.
Original textOne surprising consequence of the hairy ball theorem: The Earth is approximately a ball,and at each point on the surface, wind has a direction. It follows from the theorem thatthere is always a place where the air is perfectly still.
-- postglock 09:05, 2 August 2005 (UTC)[reply]
That is a fair point, and makes sense topologically, but it now seems a little trivial to me. With this explanation, the rephrasing should include factually: "considering all points of the atmosphere infinitesimally close to the surface of the earth, there must be at least one where the wind has no component parallel to the surface." That makes sense to me, and appears to be true, but I feel that the "real-world" and surprising aspect of the original statement has disappeared. -- postglock 13:43, 16 August 2005 (UTC)[reply]
Hi Charles, just wanted to mention that I didn't mean to offend with the revert, the edit summary can some off a little terse some times (and I certainly appreciate your Sensei's Library contributions as well)! Nevertheless, I still do not feel there has been a thorough refutation to the arguments posted above. I am also a little confused as to your comments; do you agree or disagree that my rephrasing above is necessary? Do you feel that any of the arguments presented above regarding the height of the surface providing "multiple" vectors are fallacious? Also, without being offensive, I don't feel that the inclusion of the example as a university examination question justifies anything, at least not from my experience of university! :-) -postglock 22:46, 8 December 2005 (UTC)[reply]
Sorry, I can't seem to agree with you on this one... I understand what you mean by using it as an example, but I still think that while it does not have to be a formal statement, it should nevertheless be implicitly valid. I think that an example like that aims to make the reader think "oh yeah, that is an interesting (and perhaps surprising) implication of the theorem." That is a fair aim, but if upon further reflection the implication is incorrect, well, then there appears to be no point making the initial link. Also, I admit my "infinitesimally close" phrase was perhaps not the best to use. I imagine you understand the usage was to imply the restriction of wind vectors to a topological sphere (in the sense that the "surface" is two-dimensional), the actual (constant or variable) height of the surface of the "sphere" above ground is irrelevant. Obviously to speak of a wind vector at a specified latitude and longitude is meaningless without altitude specified. -postglock 05:30, 9 December 2005 (UTC)[reply]
Sorry for the delay in answering, I've had some wikipedia-edit trouble. I understand what you are saying – if we restrict the meaning of "wind" to something whose presence can be ascertained purely by equipment such as weathervanes and anemometers you are absolutely correct. On the other hand, if we consider an even simpler thought-experiment using (for example) flags on flagpoles, the example is again invalid. It appears to me that the validity of the example depends on restricting the definition of wind to an unintuitive level. I don't think the average man would consider a weathervane or anemometer when thinking of wind, but rather something as simple as air on his face, or leaves moving in the air. -postglock 23:31, 21 December 2005 (UTC)[reply]
I think that as long as you clarify that you are making a few assumptions, the corollary can be returned to what it was originally. "If we make a few basic assumptions about the nature of wind on the surface of the earth, such as that wind can be represented as a two dimensional vector ...". Another option would be to construct a non-earth body for which this would be true. "If we consider a very large perfect sphere with radius comparable to that of earth's that may or may not be in outer space and possibly could be orbiting a star that theoretically could be quite similar to our sun, and there is wind blowing along a two dimensional vector field on its surface, then ..."
I've noted that it's merely a curiosity, and that the cyclone definition is very weak. Doktor 16:40, 24 October 2006 (UTC)[reply]
I don't think it is assumed that the wind "cannot flow into or out of the point". The field just needs to be continuous, not divergence free. Keith McClary (talk) 03:05, 27 July 2017 (UTC)[reply]
"And the award for funniest name for a mathematical theorem goes to:"http://forums.fark.com/cgi/fark/comments.pl?IDLink=1600618
-You shouldn't lock this from being edited. I'm sure more people are getting use out of it for humorous reasons than there are people getting use out of it for mathematical reasons. Power to the people; isn't that the point of wikipedia?
---I'd have to point out that if you had a comb that was curved instead of straight, it may in fact be possible to comb a hairy sphere, or even a hairy ball (we all know that not all balls are spherical - rugby, american "football" etc, etc). --81.77.200.248 12:04, 6 August 2005 (UTC)[reply]
The cylinder and the sphere are not homeomorphic. They are distinguished by their fundamental groups. The fundamental group of the cylinder is the group of additive integers and the fundamental group of the sphere is trivial. The Mercator projection is only homeomorphic on the complement of the poles of the 2-sphere. — Preceding unsigned comment added by Okieinexile (talk • contribs) 13:56, 25 April 2012 (UTC)[reply]
The sphere is not equivalent to the mathematical object known as a cylinder, and in fact a cylinder can easily be combed. I think Quentin's original statement might have meant that the sphere is topologically equivalent to a capped cylinder, i.e., a cylinder together with a disk at each end -- which is often what laypersons mean when they refer to a cylinder shape. Joule36e5 (talk) 00:28, 4 August 2012 (UTC)[reply]
This is number 32 in the most popular articles on Wikipedia. Getting people to find out about it is (for most wikipedians) a good thing. The article is interesting also!
POOOOOOOOOOOOOOOOOOOOOOOOOOOOP —Preceding unsigned comment added by 128.174.114.66 (talk) 22:27, 31 March 2010 (UTC)[reply]
Why is this called 'hairy ball theorem' if it is about spheres? A ball is solid whereas a sphere is hollow, and here it seems that the vector field is only defined on the surface (i.e. the sphere), not on the interior... --Army1987 14:43, 9 April 2006 (UTC)[reply]
Meh, It doesn't really matter, since as you say we are onlt concerned with the outside. This theorem is commonly referred to as the hair(y) ball theorem, and it was proabbly consieved that way due to the fact that a sphere is the 3D analog of a disc. Topologicaly speaking they are both contractable and are commonly considered in topology. --68.214.117.9 11:40, 5 July 2006 (UTC)[reply]
A potential candidate can be found here --146.145.37.154 21:18, 26 June 2006 (UTC)[reply]
The hairy donut looks funny, but we need a combed hairy donut. --Pjacobi 10:33, 5 July 2006 (UTC)[reply]
Hey guys, this was the best I could do. I know the sphere is nothing close to what we want but that is all I can do without spending many many hours on it. As for the donut, I think it looks nice, and it looks like it definitely needs to be combed. I would like that those stay up till anything better comes up. --Polfbroekstraat 11:23, 5 July 2006 (UTC)[reply]
What I could do in theory is create an image of a circle in a spiraling vector field. --Polfbroekstraat 11:46, 5 July 2006 (UTC)[reply]
A google translation of this gives the name "Sentence of the hedgehog". [3] violet/riga (t) 13:23, 28 August 2006 (UTC)[reply]
It would be helpful if someone could add an outline of a proof of the theorem to the article. As is the article just begs the question of how to prove it. Dugwiki 17:24, 7 November 2006 (UTC)[reply]
The article states that you need to have "is at least one p such that f(p) = 0". Is it possible to generate an f(p) s.t. there is only one p s.t. f(p)=0, or do you need at least two? I can't visualise a situation in which you only have one problem point, but of course that doesn't mean it's not possible. Perhaps, if it is possible to construct a vector field with only one problem point, we should include an explanation of it? AdinaBob 19:00, 23 February 2007 (UTC)[reply]
As has been mentioned, this is one of wikipedia's most popular articles. A brief, yet encyclopedic note on the fact that people find this funny should be in there.--Loodog 04:50, 24 February 2007 (UTC)[reply]
gluLookAt(eyeX, eyeY, eyeZ, centerX, centerY, centerZ, upX, upY, upZ) actually takes two vectors for camera orientation: f = center - eye and up = (upX, upY, upZ). The third vector, which is computed from the function, is simply the dot product of the former two. Therefore gluLookAt is continuous and has nothing to do with the Hairy ball theorem. Hairy ball theorem states that there cannot be continuous function f(v) which returns v' orthogonal to v.
This is in fact wrong. Given a non zero continous vector field there is always a continuous non-zero tangent vector field.
Someone please make sense of the computer graphics application. It makes a claim about a vector. I can understand that given a continuous vector field then there is not always a continuous orthogonal vector field (for example take the vectors orthogonal to the surface of a sphere). But for one vector? —Preceding unsigned comment added by Polfbroekstraat (talk • contribs) 08:48, 2 November 2007 (UTC)[reply]
When combing physical hair, I think the vector field representing the direction of the hair does not have to be continuous: there may be lines or spots from which the hair is "combed away". In that sense, "you can't comb a hairy ball flat" is not the precise meaning of this theorem. In fact, I guess I can imagine how a hairy ball could be "combed flat" in an intuitive sense with hair combed away from the poles (with the hair exactly at the pole combed in arbitrary direction) and combed along the equator at the equator, so that the vector field of the hair would have two discontinuities (at the poles). Maybe there should be a short note on that in the article.81.20.159.197 13:57, 2 September 2007 (UTC)[reply]
"you can't comb the hair on a billiard ball"
Um, that's trivially true by virtue of the fact that billiard balls don't have hair...
The way I always heard it was "you can't comb a tennis ball without creating a cowlick" —Preceding unsigned comment added by 71.199.159.232 (talk) 15:00, 14 October 2007 (UTC)[reply]
The "billiard ball" version of the aphorism is clearly a popular way of stating the theorem, a simple quoted search on google finds 102 "can't comb a hairy ball flat" references, 48 for "can't comb the hair on a billiard ball", and only 2 "can't comb a tennis ball". These sayings aren't supposed to be formal restatements of the theorem, or even necessarily accurate, they should be on the page if they're well-known and refer to the theorem. 67.180.76.159 (talk) 05:10, 22 April 2008 (UTC)[reply]
What do you think about this image? It shows that one "pole" is sufficient. There is an animated version of this ball, too: File:Hairy ball one pole animated.gif. --RokerHRO (talk) 10:35, 26 October 2009 (UTC)[reply]
Now there are also coninuous tangential vector fields on a torus surface:
--RokerHRO (talk) 11:43, 26 October 2009 (UTC)[reply]
Now what was I thinking about just then? ResMar 04:30, 28 December 2009 (UTC)[reply]
Ok, you can't comb a hairy odd-ball flat, but is there any theorems concerning even dimensioned balls (i.e. odd dimension spheres)? Does an odd-dimension sphere always allow a continuous tangent vector field? Or is it variable? Or is it unknown? In particular, I would like to know if a 4-ball (3-sphere) can be combed flat. PAR (talk) 19:36, 4 April 2010 (UTC)[reply]
I can't find the connection with the Tokamak, except that it uses some torus image. If this is sufficient, I suggest to add "see also : doughnut". Jick01 (talk) 11:04, 4 February 2011 (UTC)[reply]
To make the statement most useful, the article should connect the technical and nontechnical language. I shall attempt to do so, but am unsure, so am posting my attempt here: The flat hairs are the tangent vectors. To comb them is to make the field continuous. The cowlick is a pole, at which the function vanishes. ᛭ LokiClock (talk) 06:38, 9 July 2011 (UTC)[reply]
I heard there was a way to prove the Hairy Ball Theorem using Green's Theorem. Does anyone know how this would be done? — Preceding unsigned comment added by 2607:F470:6:4001:B852:5D3F:AF01:405 (talk) 15:33, 26 April 2022 (UTC)[reply]