Hairy Ball
Imagine a big lump of protein grows hair all around itself, and asks you in a shrieking voice to comb its hairs so that it can be a pretty and smooth hairy ball, with no “parting” to be found.
This is a topology problem about “vector fields” on a sphere. The best one can do is to comb the hairs to make everywhere smooth except one point on the ball. Remarkably, the meteorologist then deduces from the hairy ball that there must be always a cyclone somewhere on earth.
Concepts of Modern Mathematics has a lively discussion on the hairy ball.