For your viewing pleasure, a fantastically beautiful video about Möbius transformations, which are functions of the form
where z, a, b, c, and d are complex numbers, and . For example, is a Möbius transformation with b=2, c=1, and a=d=0. is also a Möbius transformation. However, isn’t, because . Since any Möbius transformation sends a complex number z to another complex number , it can be thought of as a transformation on the complex plane. The question is, what sorts of transformations are possible? That’s what the video is about.
The video was made by two mathematicians at the University of Minnesota, Douglas Arnold and Jonathan Rogness. There’s more information about the video here.