Talk:Hairy ball theorem

Uncomfortable

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)

Cyclone?

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

The cyclone consequence is mentioned in "The Penguin Dictionary of Curious and Interesting Geometry" by David Wells [1]. See also [2] Richard W.M. Jones 10:47, 8 Jan 2005 (UTC)

That doesn't hold true at least when there is no wind on the Earth at all. I can show another counterexample, but I can't express it with words right now... Grue 13:08, 8 Jan 2005 (UTC)

It depends on what you mean by 'cyclone'. The theorem implies the existence of an 'eye' where there is no wind. That's all. :Metterklume 04:59, 10 Jan 2005 (UTC)

I don't think the atmosphere is a 2d vector field. First, the atmosphere has depth; second, air is compressible. 209.81.57.171 28 June 2005 18:28 (UTC)

I don't understand this theorem. --Abdull 6 July 2005 15:36 (UTC)

Did you make the same initial mistake I did? I thought it was referring to a single hair at first. The way I undrestand this theorem now is the hairs on a tennis ball. Sure, you could comb them all in a bunch of directions, but to make it "smooth" in a mathematical sense, every hair has to be minisculy different in direction than it's neighbor. May seem possible at first when you start with one side. But if you try to make the whole ball smooth, there will always be a discontinuity somewhere, where two neighbors are too different.--Philosophistry 06:45, 6 August 2005 (UTC)

Added a statement to that effect. --yoshi 15:28, 12 February 2006 (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:

One consequence of the hairy ball theorem: There is always a place on the surface of Earth where the wind is still (such as in the eye of a cyclone). This applies on a scale of roughly 10 km, the height of the weather; to that approximation, the velocity of the wind is a continuous tangent vector field on the surface of Earth, and Earth is topologically equivalent to a ball.

I would suggest considering the projection of the wind vector field (on the surface of the earth, or equally, but less fascinatingly, an arbitrary ball cutting through the atmosphere) onto the tangentspace of the earthsphere. In fact assuming that air can not flow through the surface of the earth sphere the normal relative to the earth sphere surface vanishes and hence the actual wind vector automatically lives on the tangent space. Hence a true form of the statement could be that there is neccesarily at least one windless spot on the surface of the earth. If we assume the vector field to be source free (in the language of physics) the cyclone statement is in fact correct with this reading of "on the surface of earth", however the boundary wind field is not even approximately source free, as for example, convection cells would show even for incompressible air. I suggest:

"It follows from the theorem that there is always a windless spot (e.g. the eye of a cyclone) on the surface of the Earth."

-- 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 text
One 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 that
there is always a place where the air is perfectly still.

-- postglock 09:05, 2 August 2005 (UTC)

I don't see any problem at all here, except one of interpretation. The statement is quite clear if you read it carefully: wind can be considered a vector at each point on the Earth's surface. It specifically says "on the surface", thus the height of the atmosphere is irrelevant - we are only concerned with the direction of the vector on the surface. It could be clarified a little be stressing that point, and perhaps by specifying that we are only considering the component of wind parallel to the surface, but that's all that's really needed. -dmmaus 11:13, 11 August 2005 (UTC)

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)

You have reverted my inclusion of this example: which is not 'erroneous' as you seem to think - it has been set as an examination question to final year undergraduates. I don't feel your analysis of it as 'trivial' is really a good enough reason to do that. Certainly, it is in a sense a rephrasing of the theorem, and that's all. But that's exactly why it is a good, casual illustration, and different from the 'hair' metaphor. Charles Matthews 16:44, 8 December 2005 (UTC)

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)

Not necessary - it's being pedantic about what is meant to be an example of what the theorem can mean, not the formal statement of a piece of fluid dynamics. As I recall, it was Southampton University. While we're not being offensive, infinitesimally close is ruled out of court because it has no definite meaning. Charles Matthews 23:04, 8 December 2005 (UTC)

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)

Look, all we need for this is basic thought-experiment stuff, with a notional weathervane and anemometer set up at ground level. I'm sure I've seen TV weather maps with arrows, isobars and so on and the presenter doesn't have to go on saying that this is at ground level not up with the cirrus clouds. Modelling the Earth as a topological sphere is also a bit approximate, but again as a thought experiment it is perfectly fine. Mathematicians are often enough called pedants, but sheesh! Charles Matthews 07:49, 9 December 2005 (UTC)

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)

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)

The "one pole" example is a dipole, not a "cyclone" as defined in the article. Keith McClary (talk) 03:05, 27 July 2017 (UTC)

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)

Theorem linked to from humour site

"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?

The point is to write an encyclopedia, rather than anything else. Charles Matthews 06:52, 2 August 2005 (UTC)

--- 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)

Topologically, a sphere is the same as a football, or a cylinder. And the shape of the comb doesn't matter. This isn't to say you can't comb a sphere; you certainly can, but you can't make the end result in a particular way. Think of it if you had control of all the hairs on a sphere, you couldn't make them lie down flat without having two next to each other have drastically different directions. Quentin mcalmott 05:29, 11 August 2005 (UTC)

I don't think it's true that a cylinder is topologically the same as a sphere. Don't the edges of the cylinder prevent any attempts at creating a diffeomorphism? Threepounds 04:11, 8 December 2005 (UTC)

(1) It is, because in Top only homeomorphisms count and you can easily construct a homeomorphism from $S^{2}$ to the cylinder. (2) Surprisingly, if a space X admits a differentiable structure, any space X' homeomorphic to X also admits a differentiable structure. Therefore, the cylinder admits a differentiable structure and is actually diffeomorphic to the sphere. For a differentiable structure on a topological manifold, you don't actually need diffeomorphisms from the manifold to the model space in the atlas; you just need homeomorphisms which, when composed on overlaps of chart neighborhoods, are differentiable maps from (subsets of) the model space to (subsets of) the model space. Heuristically, then, you just need the kinks to composition on the overlapping maps to ``smooth out the kinks. (You can actually work out an explicit formula for stereographic projection from the cube to $\mathbb {R} ^{2}$ to turn the cube into a differentiable manifold.) (I know I am replying to a five-year-old question. Who cares? It's interesting material.) 99.186.238.191 (talk) 03:44, 12 December 2010 (UTC)

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)

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)

humour has a place

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!

Hairy balls... I get it. LOL and Happy Holidays from The Doctahedron, 16:01, 25 December 2011 (UTC)

Ball or sphere?

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)

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)

Not quite.

• In 2D: disc is solid, circle is hollow
• In 3D: ball is solid, sphere is hollow

I think the reason for the name is that most spherical objects in real life, which is where they are most likely to be "hairy", are known as balls rather than spheres. -- Smjg 13:36, 8 November 2006 (UTC)

Image of a ball in a vector field

A potential candidate can be found here --146.145.37.154 21:18, 26 June 2006 (UTC)

As for the image added recently, it is not very appropriate because vectors are not tangent to the sphere. --Army1987 09:28, 5 July 2006 (UTC)

The hairy donut looks funny, but we need a combed hairy donut. --Pjacobi 10:33, 5 July 2006 (UTC)

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)

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)

Google translation of German article

A google translation of this gives the name "Sentence of the hedgehog". [3] violet/riga (t) 13:23, 28 August 2006 (UTC)

Satz is German for theorem, does mean sentence/proposition. Charles Matthews 14:03, 8 November 2006 (UTC)

Possible to provide proof outline?

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)

+1, as with any theorem, I'd like to see proof.--Jasper Deng (talk) 22:15, 21 July 2014 (UTC)

How many problem points on a sphere?

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)

To answer my own question, it seems that by the Euler characteristic, the sphere must have at least two "problem points". I think there should be some mention of this in the hairy ball theorem. I'll put it in when I find some time.AdinaBob 19:59, 23 February 2007 (UTC)

You can put the two points together, to get what's called an index 2 zero. What's it look like? Pick a point p on the sphere and a great arc A going through it. Now consider all circles on the sphere that go through p and have "centers" on A. Then let the vector field flow consistently around the circles but always die at p. This is the vector field with just one zero. 24.59.106.118 16:37, 23 September 2007 (UTC)

Encyclopedic note on humor

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)

Error in paragraph Application to Computer Graphics

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.

Firstly you mean 'cross product' not 'dot product'. Secondly, it's a standard problem in computer graphics to generate a vector orthogonal to another. For example if we want a camera at A to look at an object at B we want to generate a normalised 'up' vector for the camera that is orthogonal to the line AB. Animators often want such a function to be continuous. No such function exists because of the hairy ball theorem. This is a classic example of the hairy ball theorem at work. gluLookAt is the compromise - you have to sacrifice continuity and instead supply gluLookAt with an up vector. As a result we end up with a function that, as you rightly point out, is continuous, but unfortunately sometimes goes to zero, as the hairy ball theorem says it must. This is a real world consequence of the hairy ball theorem that computer graphics people have to work around every day and to me it seems relevant to the wikipedia entry. I might not have worded it very well however. Sigfpe 22:13, 27 February 2007 (UTC)

This is in fact wrong. Given a non zero continous vector field there is always a continuous non-zero tangent vector field.

I don't get that last statement. There is no everywhere non-zero continuous tangent vector field on the sphere so how can it be true that "Given a non zero continous vector field there is always a continuous non-zero tangent vector field.". I must be misreading it. Sigfpe 16:53, 14 March 2007 (UTC)

There is always an orthogonal* non-zero vector field. Sorry.

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)

I have made some sense of it! -jsnx (talk) 07:12, 12 February 2008 (UTC)

Hair combing

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)

billiard ball???

"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)

"you can't comb the hair on a billiard ball": This statement simply isn't true. It's trivially possible to comb all the hair on a billiard ball in such a way that no tufts, cusps or swirls are generated. This case corresponds with a vanishing tangent vector field. Shinobu (talk) 14:16, 13 January 2008 (UTC)

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)

New images

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)

Now there are also coninuous tangential vector fields on a torus surface:

• 
• 

--RokerHRO (talk) 11:43, 26 October 2009 (UTC)

Nice name

Now what was I thinking about just then? ResMar 04:30, 28 December 2009 (UTC)

What about a hairy even-ball?

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)

It is indeed possible to construct non-vanishing vector fields on odd dimensional spheres. Here is an example

$v(x_{1},x_{2},\cdots x_{2n-1},x_{2n})=(-x_{2},x_{1}\cdots -x_{2n},x_{2n-1})\quad {\text{where}}\quad x_{1}^{2}+x_{2}^{2}\cdots x_{2n}^{2}=1$

We just rotate pairs of coordinates by ${\frac {\pi }{2}}$. It is easy to check that this assigns a tangent vector smoothly at every point on the sphere $S^{2n-1}$. Metterklume (talk) 18:34, 9 October 2010 (UTC)

In other words, thinking of the sphere as the unit sphere in a complex vector space, we multiply the POSITION vector of a point by i to get a nonzero tangent vector at that point. Tkuvho (talk) 15:48, 10 October 2010 (UTC)

I read somewhere that there is no 3D complex-number system between "normal" complex numbers and quaternions, and it occurred to me that this theorem explains why. Assume the existence of a 3D complex-number system that obeys the usual multiplication rules for complex numbers and quaternions, i.e. that multiplying x by y (where y is of unit magnitude) is equivalent to a rotation of x by a constant angle that also carries it a distance proportional to its magnitude, i.e. |xy - x| = k|x|. If there are poles in the rotation, teh constant-distance condition cannot hold, so this number system cannot work. I haven't written this into the article as I feel out of my depth in stating it precisely, but it seems clear to me and I feel it might be worth pointing out that connection. Can anybody phrase it into the article? 213.205.224.243 (talk) 02:54, 17 February 2013 (UTC)

Tokamak

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)

The tokamak is one "solution" to the problem imposed by the Hairy Ball Theorem in magnetic confinement fusion. We cannot hold high energy plasmas in any chemical material, because no material can withstand the heat loading and the plasma will simply extinguish itself on physical contact. Instead, the plasma is held in place by a magnetic field. A plasma with no resistivity will follow magnetic field lines, and only slowly drift across them. Thus we need some 2D surface of tangent vectors (the B field lines) enclosing a solid volume of plasma. We need the magnetic field vectors to be everywhere non-zero, else the magnetised plasma, which is under high gas and magnetic pressure, will be squeezed out of the surface at these points.

So we cannot hold the plasma in a sphere (or any topological equivalent), since by the theorem, there will always exist points where the field must vanish. The next alternative is the torus. However, this has other confinement drawbacks. (The particles will drift across the magnetic field density gradient: there are more field lines closer to the centre of the doughnut than the outside, so the particles will be pushed outwards.) The Russians invented the tokamak which twists the field lines back into the middle, so that any particles drifting outwards are brought back inside as they move around the torus. Roberdin (talk) 16:27, 13 February 2012 (UTC)

Elaboration on combing statement

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)

Green Theorem Proof

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)