Brouwer fixed-point theorem

The Brouwer Fixed-Point Theorem is about shapes that can fit inside of a circle. It says that for any shape that can fit inside a circle, there exists a point inside the circle where the shape looks like it has been pushed inward; this is called the "fixed-point". You can think of it like this: if you push a piece of paper into a cup, no matter how much you push it, you will always have one point inside the cup where the paper looks like it has been pinned down. That's the fixed-point.