Möbius Transformations Revealed [HD]

Möbius Transformations Revealed is a short video by Douglas Arnold and Jonathan Rogness which depicts the beauty of Möbius transformations and shows how moving to a higher dimension reveals their essential unity. It was one of the winners in the 2007 Science and Visualization Challenge and was featured along with the other winning entries in the September 28, 2007 issue of journal Science. The video, which was first released on YouTube in June 2007, has been watched there by more than a million and a half viewers and classified as a Top Favorite of All Time first in the Film & Animation category and later in the Education category. It was selected for inclusion in the MathFilm 2008 DVD, published by Springer.

credit: Möbius Transformations Revealed was built primarily using POV-Ray with some help from Mathematica. The soundtrack is a performance of "Von fremden Ländern und Menschen" ("Of Foreign Lands and Peoples") from Robert Schumann's Kinderszenen, Op. 15. The pianist is Donald Betts.

[timgoppelsroeder]

The connection between the riemann sphere and mobius transformations just became alot more clear thanks

[AsiaCrasie]

cool video, but I still cannot solve my complex analysis problems

[EyeoftheAbyss]

10 years old yet the best video on this topic :/

[MrCavitysChessCorner]

Insanely awesome. Projects like these are why I get up in the morning. I'm getting trained (lectured thrice a week) to work in a hyperbolic geometry lab. Needless to say I'm excited to do this kind of work.

Geodesics

[triton]

Simply amazing, that sphere visualisation blew me away.

[jeppejabob]

A beautiful way to connect the Riemann sphere to Möbius trasformation. Very intuitive.

[chrisvolk]

Beautiful, simple and elegant. Thank you.

[Cyberautist]

I saw these inversions for like 5 years in some videos about complex math. I never understood what they actually doing and why. Even in videos about the riemann-hypothesis these inversions were used, but I never saw or understand, that these graphics were meant to be understood as inversions; or what inversion on the complex plain could possible mean. Thank you so much for making that clear to me.

[kockarthur]

A rotation of the Riemann sphere about a horizontal axis does not represent an inversion of the complex plane. It represents an inversion and a rotation (combined). The true sphere representation of an inversion corresponds to reaching through the sphere, pinching the other side, and pulling it through, i.e. literally turning it inside out (inverting it).

[leandersmainchannel]

THANKS FOR RECOMMENDING THIS TO ME AFTER I GOT STUCK AFTER TRYING TO SEARCH IT

Thank you so much.

[dylanwilliams]

Got my complex analysis final coming up, thanks :)

[shashwatbhatnagar]

I am really speechless. What a wonderful educational content and the way you explained it is totally amazing

[Lhirstev]

This is one of the first videos I have ever seen on YouTube. I am pretty sure that I remember I had looked this up under recently uploaded on YouTube when my teacher in high school had shown me what YouTube was. Back then, there was not that many videos on YouTube and I think I had watched every single video uploaded around the same time as I saw this? But I had seen the original video by whomever had uploaded that in September of 2005. Lol. I bet against YouTube Gaining Success.. :/ I remember talking about it in class.

[jacifafernandes]

Wow ! This was really helpful. Thank you so much.

[resonarc]

terence tao mentioned this video in his 2009 induction ceremony speech, which is why i came here. i like how simply this is explained.

[orktv]

Wish there was a video like this showcasing how the conjugacy classes preserve certain fixed points. From what I understand this arises due to a specific combination of translation and rotation of the Riemann sphere.

[bsharpmajorscale]

When it inverts, the empty space almost looks like a silhouette of Tweety Bird. :P

[arthursgarage]

Learning these for smith charts, neat

[nickjiang]

cant believe this is a video made 10 yrs ago. eggcellent video

[PROcrastiDRIVESVofficial]

Awesome video, thanks! It's amazing how simple it becomes.