Have you ever seen how a cat lands on its feet? Even 6 week old kittens seem to be able to do it perfectly from any position. In this lecture we will discuss how cats do it by changing their shape in a particular way. Remarkably, the mathematical structures relevant for “cat landing physics” are very similar to the ones necessary to understand quantum physics of light and the standard model of elementary particles. We will talk about the concept of emergent behavior using simple but very surprising mechanical demonstrations. Then we will discuss how various geometrical and topological concepts can emerge as a result of a conspiracy between interacting microscopic particles. We conclude by applying some of these ideas to the quantum Hall effect - one of the most fascinating discoveries of 20th century.