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."

That is one of the coolest things I’ve ever seen in my life.

None of this made any sense when I was a senior-year undergraduate in complex analysis, and it’s completely sensible and tractably presented here. Jumping into the theorems would make a lot more sense now, given this context. Thanks.

This is an outstanding way for mathematicians to convey the elegance and importance of their work to the public, with the added bonus of being a palatable introduction to a tricky subject for mathematicians-in-training.

I hope the subtext got across — the connection of the sphere to Mobius transformations of the plane is important because of its elegance and its practical utility as a tool for proving other concepts.

None of this made any sense when I was a senior-year undergraduate in complex analysis, and it’s completely sensible and tractably presented here. Jumping into the theorems would make a lot more sense now, given this context. Thanks.

This is an outstanding way for mathematicians to convey the elegance and importance of their work to the public, with the added bonus of being a palatable introduction to a tricky subject for mathematicians-in-training.

I hope the subtext got across — the connection of the sphere to Mobius transformations of the plane is important because of its elegance and its practical utility as a tool for proving other concepts.

Just

lovely.My precalculus students and I have just finished a unit on transformations of functions in two dimensions, and though the mathematics at work here is well beyond their grasp at the moment, I am definitely going to show this to my kids tomorrow as a taste of what lies ahead.

Thank you.

You can interactively play with this kind of transformation here: it’s fun and kind of trippy:

http://www.rotorbrain.com/foote/interactive/hacks/ConformalMapping.html

Jonathan Foote