The hairy ball theorem
Vladimir Cuesta †
Instituto de Ciencias Nucleares, Universidad Nacional Aut´ noma de M´ xico, 70-543, Ciudad de
M´ xico, M´ xico
Abstract. I show that for the sphere S 2 , if I choose a ﬁnite number of points then I can deﬁne a tangent
vector for each point. If I consider all the points in the sphere S 2 , I can not deﬁne a tangent vector for
1. The hairy ball theorem
If I take one point p1 of S 2 I can deﬁne a vector on it independent of everything. Now, if I take two
different points p1 and p2 on S 2 , I can put p1 on a ﬁrst sphere S 2 and p2 on a second sphere S 2 , in
this case a tangent vector can exist on the ﬁrst sphere and the point p1 and a tangent vector can exist
(locally) for the second sphere and the point p2 in an independent way with a given direction, ﬁnally
I can put it together without problems because p1 and p2 are different. Like third case, I take the
different points p1 , p2 and p3 and I can put it on three different spheres S 2 and I can construct tangent
vector (locally) for the three points with a given direction and ﬁnally put all together on the same
For the case on n different points I can follow the previous procedure without problems.
However, for the case of all the points on S 2 , the previous algorithm will fail because I can not
separate tangent vectors at the ﬁnal step. I mean, It does not exist space for separating different
Sometimes, you can describe the theorem like: You can not comb the hairs without the presence
of cowlicks for a ball with hairs all over it. In fact, all the hairs will be vertical.
2. Conclusions and remarks
The previous argument can be applied on different 2-surfaces, 3-surfaces, 4-surfaces and so on,
making the difference for the case of ﬁnite points or the case of complete surfaces.
The hairy ball theorem 2
The reader can note the importance for computer studies on wind, storms, ﬂuids, ﬁelds and so
I present the following example, you can think currents of wind on a portion of the plane,
probably all the wind on the same direction (in this case I have a ﬁnite number of points and I can
deﬁne a vector for all the points on the portion of the plane). For an unknown reason, the wind can be
accumulated on a little 2-surface and according to the hairy ball theorem, there is not enough space
for doing that and the wind will leave the plane and the air will transform into a twister. I mean, the
hair will be vertical for complete 2-surfaces. For higher dimensions, the reasoning is similar.
I appreciate very fruitful discussions with my children and their mothers, my brothers, friends and
some members of the Department of Mathematics of the Centro de Investigaci´ n y de Estudios
Avanzados del Instituto Polit´ cnico Nacional (in particular some people of the football team).