Web edition: November 13, 2007 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. A Möbius
transformation begins with a plane and moves each point to a new
location according to certain rules. In their video, Douglas N.
Arnold and Jonathan Rogness of the University of Minnesota in
Minneapolis transform a multicolo