Mathematics may be the most pure of the sciences. The most abstract of disciplines. And being the most abstract it allows for many problems that seem impossible or just plain flights of fancy.
One such problem is how to turn a sphere inside out.
Now we're talking about math and we're talking about the concept of a sphere. Don't think about trying to take a balloon and turning it inside out. It's not a real object mathematicians are talking about. A physical object can't be pushed through itself. A mathematical entity can. A real sphere can be torn and creased. A mathematical entity can't.
Even with a set of self imposed rules it turns out you can turn a sphere inside out (video).