I'm looking for references to applications of the Hairy ball theorem.

This is a result of mathematics (topology), but I am interested in applications.

I already visited wikipedia and cited references, but I need a little more explanation in magnetic fields, meteorology and applications in computer graphics. If any other application would be excellent.

I've also read some questions on this site and some articles uploaded (an article of John Milnor for example, in math.stackexchange), but none deals specifically with applications.

Maybe it's a bit difficult because I seek not know much about magnetic fields or computer graphics, but I'm just interested

I would like to know if this result is really effective to model these situations, it is mentioned briefly in some texts, but not how they actually work.

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    $\begingroup$ I'm too lazy to go to the Wiki-page and see what was listed, but an obvious example is that there always exists a point on the earth where the wind is not blowing. (To be rigorous we must add the qualifier that vertical winds are allowed to exist at such points, i.e. tornadoes or whatever upward-drift you can think of) $\endgroup$ Jun 12, 2012 at 21:11
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    $\begingroup$ Good way to confuse electromagnetism students: Ask them to describe the instantaneous electric field created by an isotropic light source (like the sun or a red-hot particle). If you don't understand incoherent light, the question seems impossible to answer thanks to the hairy ball theorem. I wouldn't quite call that an "application" though. $\endgroup$ Jun 13, 2012 at 3:55

2 Answers 2


One recent example is an application to magnetic flux lines in type-II superconductors (link: Nature). See also this link.


You could also this an application in this answer, which shows that there is always at least two points on Earth where the geomagnetic field is vertical.


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