University of Minnesota researchers Douglas Arnold and Jonathan Rogness created an engaging and (relatively) simple video explaining the abstract and mind-bending mathematics of Möbius transformations. From Science News:
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 multicolored square into new shapes using Möbius transformations...Link to YouTube video, Link to Science News
"You need some pretty heavy mathematical machinery that people usually don't do until their first year of grad school to prove the stuff in the video," Rogness says, "but we've been showing this to high school students and they seem to get it."