Dynkin diagrams and regular polyhedra!!! And the list of all possible finite simple groups!!!

I mean, cute applications listed here are cute, but you should also put a tangible, direct representation of human achievement in mathematics. It might be a bit hard to properly explain them, but still ...

As a person with background in elementary particle physics, the fact that human beings have classified possible symmetries themselves (not just the symmetry realized) strikes my inner cord quite strongly.