Abstraction lies at the heart of mathematics. It makes math powerful, but at the same time, it can make math hard to understand. Abstraction makes math simultaneously beautiful and austere, useful and esoteric.
But a picture can tame the mad monster of abstraction, and sometimes, a video can do so even better. Now, a pair of mathematicians has created a video (see http://www.youtube.com/watch?v=JX3VmDgiFnY) that shows how to visualize and understand Möbius transformations, which are a fundamental and highly abstract mathematical tool. The new video, “Möbius Transformations Revealed,” has become an Internet sensation, with 60,000 hits on YouTube so far. It also won honorable mention in the Science 2007 Science and Engineering Visualization challenge.