The beauty of mathematics, part 1

The short video (also available on YouTube) demonstrates effects of linear fractional transformations, also known as Möbius transformations, on a square segment of the plane. The subject is a key point in Complex Analysis, one of my favorite classes to teach due to the sheer number of beautiful visuzalizations like this one that are possible. The video was made by Douglas N. Arnold and Jonathan Rogness of the University of Minnesota and has become widely celebrated; it recently was a winner in the 2007 Science and Engineering Visualization Challenge.