Comments on: A brief explanation of the "Hairy Ball Theorem" http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html Brain candy for Happy Mutants Sun, 19 May 2013 22:10:00 +0000 hourly 1 http://wordpress.org/?v=3.4.1 By: Tynam http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1288414 Tynam Tue, 06 Dec 2011 17:46:00 +0000 http://boingboing.net/?p=132878#comment-1288414 Everything you say is true - That's why I said "and related theorems". The first is, obviously, Borsuk-Ulam. The second you can get at quite a few ways, but I agree that your approach is much more natural. But seeing as we're on BB not math.stackexchange, I felt like throwing in a few extra examples while we're on this topic. They're related. Fixed point theorems are just cool in general. Full appreciation of the coolness of fixed point theorems may require several years study of analysis and topology. Everything you say is true – That’s why I said “and related theorems”.
The first is, obviously, Borsuk-Ulam. The second you can get at quite a few ways, but I agree that your approach is much more natural.

But seeing as we’re on BB not math.stackexchange, I felt like throwing in a few extra examples while we’re on this topic. They’re related. Fixed point theorems are just cool in general.

Full appreciation of the coolness of fixed point theorems may require several years study of analysis and topology.

]]> By: shuckster http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1288289 shuckster Tue, 06 Dec 2011 15:20:00 +0000 http://boingboing.net/?p=132878#comment-1288289 http://www.youtube.com/watch?v=B4UGZEjG02s Original source, better quality, no ads. http://www.youtube.com/watch?v=B4UGZEjG02s

Original source, better quality, no ads.

]]> By: Woden Kusner http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1288005 Woden Kusner Tue, 06 Dec 2011 00:47:00 +0000 http://boingboing.net/?p=132878#comment-1288005 Edit edit: Nevermind, I was just being opinionated. I really think these examples "naturally" illustrate other theorems. Edit: (so as not to be obscure) i.e. I feel the first follows most naturally from Borsuk-Ulam or IMVT depending how you state it, and the second from Banach CMT. If you consider the map and use Brouwer, it's really not as nice a result (no unique fixed point, no construction, ...). Edit edit: Nevermind, I was just being opinionated.

I really think these examples “naturally” illustrate other theorems.

Edit: (so as not to be obscure) i.e. I feel the first follows most naturally from Borsuk-Ulam or IMVT depending how you state it, and the second from Banach CMT. If you consider the map and use Brouwer, it’s really not as nice a result (no unique fixed point, no construction, …).

]]> By: Jake0748 http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287988 Jake0748 Tue, 06 Dec 2011 00:29:00 +0000 http://boingboing.net/?p=132878#comment-1287988 I remember my high school geometry teacher mentioning this back in the dark ages.  Although, I always remember him calling it the "hairy ape" theorem.  (Maybe he just thought hairy balls were a little to giggle-provoking for his 10th grade audience.  I remember my high school geometry teacher mentioning this back in the dark ages.  Although, I always remember him calling it the “hairy ape” theorem.  (Maybe he just thought hairy balls were a little to giggle-provoking for his 10th grade audience. 

]]> By: jimkirk http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287881 jimkirk Mon, 05 Dec 2011 22:39:00 +0000 http://boingboing.net/?p=132878#comment-1287881 Male Pattern Baldness for the win! Male Pattern Baldness for the win!

]]> By: Judonerd http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287860 Judonerd Mon, 05 Dec 2011 22:18:00 +0000 http://boingboing.net/?p=132878#comment-1287860 boom. boom.

]]> By: Rob Stevens http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287837 Rob Stevens Mon, 05 Dec 2011 22:07:00 +0000 http://boingboing.net/?p=132878#comment-1287837 It certainly took a while for this to start rolling downhill, but finally, here we go. Yay! It certainly took a while for this to start rolling downhill, but finally, here we go. Yay!

]]> By: Haggie http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287808 Haggie Mon, 05 Dec 2011 21:49:00 +0000 http://boingboing.net/?p=132878#comment-1287808 It would be even funnier if Pete Schweddy had proven this theorem... It would be even funnier if Pete Schweddy had proven this theorem…

]]> By: RJ http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287660 RJ Mon, 05 Dec 2011 20:22:00 +0000 http://boingboing.net/?p=132878#comment-1287660 Join us tomorrow when we discuss the Dat Ass Theorum, which deals with spherical curvatures and friction coefficients. Join us tomorrow when we discuss the Dat Ass Theorum, which deals with spherical curvatures and friction coefficients.

]]> By: GlenBlank http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287661 GlenBlank Mon, 05 Dec 2011 20:22:00 +0000 http://boingboing.net/?p=132878#comment-1287661 <blockquote>i don't like the "it's math. so don't bother questioning it" assertion.</blockquote> But that's not what he said.  He said don't go wasting your time <i>playing around with a hairy ball</i> trying to prove the theorem wrong, because the 'Hairy Ball Theorem' is math (i.e., it's not a prediction of the physical behavior of actual hairy balls, it's about vector fields in algebraic topology) - and it's a theorem that's already been proven. So nothing you observe while playing around with an actual hairy ball is going to disprove it. So you'd be wasting your time doing that. :-)

i don’t like the “it’s math. so don’t bother questioning it” assertion.

But that’s not what he said.

 He said don’t go wasting your time playing around with a hairy ball trying to prove the theorem wrong, because the ‘Hairy Ball Theorem’ is math (i.e., it’s not a prediction of the physical behavior of actual hairy balls, it’s about vector fields in algebraic topology) – and it’s a theorem that’s already been proven.

So nothing you observe while playing around with an actual hairy ball is going to disprove it.

So you’d be wasting your time doing that. :-)

]]> By: squidfood http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287624 squidfood Mon, 05 Dec 2011 20:00:00 +0000 http://boingboing.net/?p=132878#comment-1287624 Yeah but... your cat is actually the donut. Yeah but… your cat is actually the donut.

]]> By: querent http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287614 querent Mon, 05 Dec 2011 19:50:00 +0000 http://boingboing.net/?p=132878#comment-1287614 Just make sure your v-field isn't Lipschitz!  Cause that would be gross. Just make sure your v-field isn’t Lipschitz!  Cause that would be gross.

]]> By: Marktech http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287599 Marktech Mon, 05 Dec 2011 19:40:00 +0000 http://boingboing.net/?p=132878#comment-1287599 And just look at that hairy banana.  Just look at it. And just look at that hairy banana.  Just look at it.

]]> By: SamSam http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287600 SamSam Mon, 05 Dec 2011 19:40:00 +0000 http://boingboing.net/?p=132878#comment-1287600 Yes, that's described in the video. Balls are the same as banana-shaped objects, which are the same as rabbit-shaped objects. Unless you count the hole in the rabbit, in which case a rabbit is a donut, not a ball. Yes, that’s described in the video. Balls are the same as banana-shaped objects, which are the same as rabbit-shaped objects. Unless you count the hole in the rabbit, in which case a rabbit is a donut, not a ball.

]]> By: Tynam http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287588 Tynam Mon, 05 Dec 2011 19:35:00 +0000 http://boingboing.net/?p=132878#comment-1287588 Technically, the theorem doesn't apply to bananas, just spheres.  But bananas are topologically equivalent, I suppose. Technically, the theorem doesn’t apply to bananas, just spheres.  But bananas are topologically equivalent, I suppose.

]]> By: Tynam http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287579 Tynam Mon, 05 Dec 2011 19:29:00 +0000 http://boingboing.net/?p=132878#comment-1287579 Other fun consequences of Brouwer's Fixed Point Theorem and its related theorems: Somewhere on the equator there are two antipodal points with exactly the same temperature.  (Or barometric pressure, if you'd rather stick with air movement.) Take two identical pieces of paper. Crumple one up and put it on the other. There is at least one point on the crumpled piece that is directly over the corresponding point on the flat one. (You'd have to use transparent film with a numbered grid, or something, to see this for yourself.) (Alternatively, take a map that includes your current location and drop it on the floor. At least one point on the map is directly over the point it represents.) Other fun consequences of Brouwer’s Fixed Point Theorem and its related theorems:

Somewhere on the equator there are two antipodal points with exactly the same temperature.  (Or barometric pressure, if you’d rather stick with air movement.)

Take two identical pieces of paper. Crumple one up and put it on the other. There is at least one point on the crumpled piece that is directly over the corresponding point on the flat one.
(You’d have to use transparent film with a numbered grid, or something, to see this for yourself.)

(Alternatively, take a map that includes your current location and drop it on the floor. At least one point on the map is directly over the point it represents.)

]]> By: bo1n6bo1n6 http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287575 bo1n6bo1n6 Mon, 05 Dec 2011 19:25:00 +0000 http://boingboing.net/?p=132878#comment-1287575 Hairy balls need combing? Hairy balls need combing?

]]> By: Robert Sanders http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287574 Robert Sanders Mon, 05 Dec 2011 19:22:00 +0000 http://boingboing.net/?p=132878#comment-1287574 "Most sets of values would give rise to universes that, although-" Shut the fuck up Stephen Hawkins and come my hairy balls!   “Most sets of values would give rise to universes that, although-” Shut the fuck up Stephen Hawkins and come my hairy balls!  

]]> By: lknope http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287572 lknope Mon, 05 Dec 2011 19:21:00 +0000 http://boingboing.net/?p=132878#comment-1287572 This may be the only time I've ever wished I had a hairy ball to play with.  This may be the only time I’ve ever wished I had a hairy ball to play with. 

]]> By: Tynam http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287562 Tynam Mon, 05 Dec 2011 19:14:00 +0000 http://boingboing.net/?p=132878#comment-1287562 One could argue that, but it seems over-precise for this application. Granted that gases are discontinuous, it only matters for applications where that actually affects their behaviour. If you're modelling diffusion, sure.  If you're modelling wind, a vector field still works pretty well. One could argue that, but it seems over-precise for this application. Granted that gases are discontinuous, it only matters for applications where that actually affects their behaviour. If you’re modelling diffusion, sure.  If you’re modelling wind, a vector field still works pretty well.

]]> By: Guest http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287558 Guest Mon, 05 Dec 2011 19:11:00 +0000 http://boingboing.net/?p=132878#comment-1287558 It's a sad state of affairs for me when I can't think of a joke to make about this post. I was going to add 'in my defence I am drunk'... but that makes my predicament even more shameful. It’s a sad state of affairs for me when I can’t think of a joke to make about this post.
I was going to add ‘in my defence I am drunk’… but that makes my predicament even more shameful.

]]> By: Tynam http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287554 Tynam Mon, 05 Dec 2011 19:09:00 +0000 http://boingboing.net/?p=132878#comment-1287554 Technically according to the theorem, the guaranteed area isn't an area at all, it's a point... in theory it could be infinitesimally small. (Although in practice in this example, it needn't be.) One of the joys of pure maths is that once you've proven something <i>exists</i>, you mostly stop worrying about it. Technically according to the theorem, the guaranteed area isn’t an area at all, it’s a point… in theory it could be infinitesimally small. (Although in practice in this example, it needn’t be.)

One of the joys of pure maths is that once you’ve proven something exists, you mostly stop worrying about it.

]]> By: theophrastvs http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287551 theophrastvs Mon, 05 Dec 2011 19:07:00 +0000 http://boingboing.net/?p=132878#comment-1287551 i don't like the "it's math. so don't bother questioning it" assertion.  but still, it's a very nice (unnecessarily terse) explanation. i don’t like the “it’s math. so don’t bother questioning it” assertion.  but still, it’s a very nice (unnecessarily terse) explanation.

]]> By: Stephan Zielinski http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287545 Stephan Zielinski Mon, 05 Dec 2011 19:03:00 +0000 http://boingboing.net/?p=132878#comment-1287545 A single point minimum, but there may be more. However, the guarantee is of no HORIZONTAL wind; there could still be an updraft or a downdraft.  And one could argue that the movement of wind on the real world is not particularly well modeled by a continuous vector field in the first place; ultimately, the atmosphere is highly discontinuous, being made of various teeny molecules with big gaps of vacuum between them. A single point minimum, but there may be more.

However, the guarantee is of no HORIZONTAL wind; there could still be an updraft or a downdraft.  And one could argue that the movement of wind on the real world is not particularly well modeled by a continuous vector field in the first place; ultimately, the atmosphere is highly discontinuous, being made of various teeny molecules with big gaps of vacuum between them.

]]> By: pencilbox http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287524 pencilbox Mon, 05 Dec 2011 18:54:00 +0000 http://boingboing.net/?p=132878#comment-1287524 Look at that hairy banana. Don't comb it. Just look at it. Look at that hairy banana.

Don’t comb it. Just look at it.

]]> By: Zero Sonico http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287504 Zero Sonico Mon, 05 Dec 2011 18:45:00 +0000 http://boingboing.net/?p=132878#comment-1287504 That's why I shave. ... I'm really sorry, but someone had to. That’s why I shave.

… I’m really sorry, but someone had to.

]]> By: ROSSINDETROIT http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287495 ROSSINDETROIT Mon, 05 Dec 2011 18:40:00 +0000 http://boingboing.net/?p=132878#comment-1287495 Cat butts and hairy balls.  Happy Monday, bOINGbOING! Cat butts and hairy balls.  Happy Monday, bOINGbOING!

]]> By: flowergardenslayer http://boingboing.net/2011/12/05/a-brief-explanation-of-the-h.html#comment-1287462 flowergardenslayer Mon, 05 Dec 2011 18:23:00 +0000 http://boingboing.net/?p=132878#comment-1287462 This explains some of the problems I have with my cat.  BTW, how big is this guaranteed area of no wind on the earth? This explains some of the problems I have with my cat.  BTW, how big is this guaranteed area of no wind on the earth?

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