The Hairy Ball Theorem

Theorem of the week:

The Hairy Ball Theorem

This says you can't comb a hairy ball without introducing discontinuities such as partings or whorls, unless there's a bald spot.  (There's a more mathematical statement below under the heading "theorem".)  The proof is from An Extremely Short Proof of the Hairy Ball Theorem, by P McGrath, but I've put it into my own words, completely removed all maths notation, and added pictures to make it as accessible as possible.  In addition to being extremely short, it's extremely elegant, and somewhat reminiscent of the Ham Sandwich Theorem.

Theorem

It is not possible to impose a continuous vector field onto a sphere, such that the vectors are all tangential to the surface, unless the field is zero somewhere

Proof

Let's assume the sphere does have a continuous, tangential, everywhere non-zero vector field, and attempt to derive a contradiction.

Draw a small circle around a point p.  Do one lap around the circle, and count how many full rotations the vector field makes, relative to your direction of motion.  Each clockwise rotation counts as +1 and each anticlockwise rotation as -1.  By continuity the count must be an integer.  Since the circle is small the field will be more or less constant, and the count must therefore equal +1 or -1.  For simplicity lets assume +1.

Continuously grow the circle, keeping it centred on p, until it becomes a great circle.  The count does not change, because that would require a discontinuous jump in it's value.  Now fix two opposite points on the circle and continuously rotate the rest of the circle through 180 degrees, like a hula hoop.  Again the count cannot change, and so must still be +1.  However, we are now going the opposite way round the same circle so it should now be -1.

Reductio ad absurdum!