okay so what's the Hairy ball theorem?

at the simplest level, it basically says that;

You can't comb a ball of hair without having some hair sticking up. Given a ball of hairs all over it, it is impossible to comb the hairs continuously and have all the hairs lay flat.

Imagine a ball of hair, now to mathematicians it will look like a continuous 'vector field'.

What on earth does that even mean? A vector is nothing but a quantity that has both magnitude and direction (Follows the triangular law of vector addition)

A continuous vector field is like the flow of wind. A breeze of wind, all flowing in one direction like a bunch of arrows.

The statement of this theorem goes like this;

If you've a ball covered entirely with hair, it’s impossible to comb it in a way that every single hair will lie flat.

If you're getting ready for a party and can't comb your hair properly because there's this one cowlick that just doesn't sit down, then don't worry it will somehow come down because our head is not perfectly spherical! Hence the theorem doesn't apply to us lol

This theorem has a detailed formulation in the field of algebraic topology with n number of dimensions for each sphere but right now we won't go into the details, that's all for the next time.

Let me leave you with a perspective; The Hairy Ball Theorem implies that it can't be windy everywhere: there is always at least one point on Earth where there is no wind. Because as I said wind acts as a continuous vector field, there will a point of earth

Where either there will no wind at all or it will not be along the curvature of the earth. Nature doesn't allow perfection and that truly reflects into the elegant equations of mathematics, utterly remarkable things come out of it, all you have to do is build a perspective ;)

I can predict a wave of jokes incoming for this thread, y'all making fun of multivariable calculus and topology lmao