A brief explanation of the "Hairy Ball Theorem"

Why hairy balls can't be combed to lie perfectly flat and what that has to do with wind currents on planet Earth. Surprisingly, this is all totally safe for work.

Via New Scientist and hectocotyli


  1. This explains some of the problems I have with my cat.  BTW, how big is this guaranteed area of no wind on the earth?

    1. 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.

      1. 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.

    2. 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.

    1. Technically, the theorem doesn’t apply to bananas, just spheres.  But bananas are topologically equivalent, I suppose.

      1. 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.

  2. 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.

    1. 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. :-)

  3. 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.

  4. 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.)

    1. 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, …).

      1. 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.

  5. Join us tomorrow when we discuss the Dat Ass Theorum, which deals with spherical curvatures and friction coefficients.

  6. 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. 

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