The hairy ball theorem states that on a 2-dimensional sphere, any continuous tangent vector field must vanish at least once. In dimension one, it is easy to create a tangent vector field on the circle that does not vanish. It is less easy, however, to find, on a three-dimensional sphere, not one but three continuous tangent vector fields that do not vanish and are even linearly independent. I propose to explore these results and their generalizations in higher dimensions.