You might be surprised to learn that you can't comb the hairs flat on a coconut without creating a cowlick. Perhaps even more surprising, this silly claim with an even sillier name, "the hairy ball theorem," is a proud discovery from a branch of math called topology. Juvenile humor aside, the theorem has far-reaching consequences in meteorology, radio transmission and nuclear power.
Here, "cowlick" can mean either a bald spot or a tuft of hair sticking straight up, like the one the character Alfalfa sports in "The Little Rascals." Of course, mathematicians don't refer to coconuts or cowlicks in their framing of the problem. In more technical language, think of the coconut as a sphere and the hairs as vectors. A vector, often depicted as an arrow, is just something with a magnitude (or length) and a direction. Combing the hair flat against the sides of the coconut would form the equivalent of tangent vectors—those that touch the sphere at exactly one point along their length. Also, we want a smooth comb, so we don't allow the hair to be parted anywhere. In other words, the arrangement of vectors on the sphere must be continuous, meaning that nearby hairs should change direction only gradually, not sharply. If we stitch these criteria together, the theorem says that any way you try to assign vectors to each point on a sphere, something ugly is bound to happen: there will be a discontinuity (a part), a vector with zero length (a bald spot) or a vector that fails to be tangent to the sphere (Alfalfa). In full jargon: a continuous nonvanishing tangent vector field on a sphere can't exist.
