that moment when u open a topology book to some weird shit like this which your brain now registers as completely plausible and reasonable from all the damage your math major has done to your mind and for a fleeting, fleeting second of lucidity u wonder to urself “wtf am I doing with my life?”
In an important field of mathematics called topology, two objects are considered to be equivalent, or “homeomorphic,” if one can be morphed into the other by simply twisting and stretching its surface; they are different if you have to cut or crease the surface of one to reshape it into the form of the other.
Consider, for example, a torus — the dougnut-shape object shown in the intro slide. If you turn it upright, widen one side and indent the top of that side, you then have a cylindrical object with a handle. Thus, a classic math joke is to say that topologists can’t tell their doughnuts from their coffee cups.
On the other hand, Moebius bands — loops with a single twist in them — are not homeomorphic with twist-free loops (cylinders), because you can’t take the twist out of a Moebius band without cutting it, flipping over one of the edges, and reattaching.
Topologists long wondered: Is a sphere homeomorphic with the inside-out version of itself? In other words, can you turn a sphere inside out? At first it seems impossible, because you aren’t allowed to poke a hole in the sphere and pull out the inside. But in fact, “sphere eversion,” as it’s called, is possible.
Incredibly, the topologist Bernard Morin, a key developer of the complex method of sphere eversion shown here, was blind.