The ‘Hairy Ball’ Theorem

Salons and barbershops have closed due to the Covid-19 outbreak. Since then, haircut has become a recurrent topic of conversation.

A reason is that attending a remote meeting with messy hair may be awkward (for me, just good memories …). However, it should not be, as mathematics offers an excellent excuse: the ‘hairy ball’ theorem. This theorem states that if a sphere is covered by hairs, it is impossible to smoothly comb them down so that there are no whorls. In more rigorous term, there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres.
It is plain that this excuse relies on a ‘spherical cow assumption’, namely the head is topologically equivalent to a sphere and there are not hairless part …

Remarkably, the ‘hairy ball’ theorem has several fluid dynamics applications. The most striking one is connected to the Earth’s atmosphere. The height of the atmosphere is negligible compared to the diameter of the Earth. Accordingly, the wind may be regarded as a two-dimensional vector field that is defined continuously everywhere on the surface of the Earth. If the wind blows somewhere, then there is another place where it doesn’t (this is the eye of a cyclone/anticyclone).

Another interesting application is related to the Bénard–Marangoni convection cells. These develop in a plane horizontal fluid layer heated from below (see the video). As discussed in this paper, the ‘hairy ball theorem’ predicts the existence of at least one zero velocity point and this explains why a sole convection cell cannot be observed.