Turning a sphere inside-out was thought to be impossible for decades. It was officially declared possible in 1958, but remained impossible to picture. It was only until the 1970s, when a graphics expert got involved, that people could finally see the craziest bubbling of a beach ball ever mathematically conceived of.
Mathematicians have a certain protocol for 'everting a sphere.' For one thing, this isn't a sphere made of a material that we have any access to. It is a flexible membrane that can actually pass through itself. The sphere isn't pierced and drawn through a hole. It's just squished down in a certain formation until it everts.
But there are rules. The sphere can't be torn or ripped. It also can't be creased, since in math any crease gets pushed down to an infinitely thin edge, breaking it. The sphere had to be somehow drawn through itself in easy rounded ways, that left no creases, so it could be completely turned inside out. This was thought to be impossible to do, until the 1950s. Then it was thought to be only impossible to see. It was not until 1977, computer graphics, and the film Turning a Sphere Inside Out, that people could picture it. Take a look at an everting sphere.
Interestingly, it is impossible to turn a rubber-band-like circle inside out this way. You can turn a sphere inside-out according to these rules, but not a two-dimensional circle. Living in three dimensions has its positive aspects, in theory, and in computer animation.
Top Image: Soylent Green
Via SUNY Stony Brook