Via The Art of Problem-Solving, a great video on Mobius transformations. I never really got how the inversion transformation fit in with the others before seeing this!
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Mark Chu-Carroll (aka MarkCC) is a PhD Computer Scientist, who works for Google as a Software Engineer. My professional interests center on programming languages and tools, and how to improve the languages and tools that are used for building complex software systems.
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Wonderful Mobius Transformation Video
Category: topology
Posted on: June 25, 2007 8:43 PM, by Mark C. Chu-Carroll
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That's awesome, Mark! It's really interesting how geometry all fits together.
I wonder if the sphere analogy was intended originally, or if it was merely realized later that the Mobius transformations correspond to simple movements on a sphere?
Posted by: Xanthir, FCD | June 25, 2007 9:23 PM
Very, very cool. This is the first time I "got it."
Posted by: revere | June 25, 2007 10:13 PM
Wow. Just wow. Cam we have the math behind the video?
Posted by: Canuckistani | June 25, 2007 10:30 PM
Krantz, S. G. "Möbius Transformations." §6.2.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 81, 1999.
Needham, T. "Möbius Transformations and Inversion." Ch. 3 in Visual Complex Analysis. New York: Clarendon Press, pp. 122-188, 2000.
Weisstein, Eric W. "Möbius Transformation." From MathWorld--A Wolfram Web Resource.
Posted by: Jonathan Vos Post | June 25, 2007 11:26 PM
Superb use of visuals. As it was for you, this generated an "aha!" moment for me. I love "aha!" moments.
Posted by: Duae Quartunciae | June 25, 2007 11:38 PM
That is a great find.
Posted by: Josiah Carlson | June 26, 2007 1:21 AM
Wonderful. This is one of the best pieces of mathematical educational material I've ever seen. But more than that, this is one of those "isn't the universe we live in cool" moments for me. When you can combine education with that kind of wonder and delight, you've really got something good.
Huge kudos to the authors.
Posted by: Crow | June 26, 2007 8:54 AM
Overwhelming! I feel burning interest to complex analysis after your movie. A great moment of enlightment (both literally and allegorically). Thanks.
Posted by: Vlad Shcherbina | June 26, 2007 11:42 AM
I don't know from math, but this was really beautiful on a purely visual level. Too bad about the sedate piano accompaniment...those colors seem more suited to an Anthem of the Sun-era Grateful Dead soundtrack, man.
Posted by: CCP | June 26, 2007 12:29 PM
There's a YouTube account for visuals used in a topology seminar: http://www.youtube.com/user/bothmer
They're pretty cool too. Since it's for a seminar, they also provide a more rigorous explanation in the description box.
Posted by: Agnostic | June 26, 2007 12:50 PM
Two minutes ago I didn't even know what a Mobius transformation was. Seeing that inversion in the plane corresponds to rotating the projective sphere- that was truly enlightening. Does this generalise to higher dimensions?- e.g. can I think of an inversion in 3-D space as corresponding to rotation of a projective hypersphere?
Posted by: Stephen Wells | June 26, 2007 1:47 PM
Very nice indeed. Some years ago, I tried to do similar things for the Hyperbolic Plane, but while I came up short, these guys succeeded brilliantly. The Schumann was good accompaniment, but I too would have preferred something a little jumpier. Maybe Feltsman doing a fugue from the WTC.
Posted by: Jonathan Lubin | June 26, 2007 2:39 PM
That is the type of music that all maths should be set to, ever. From now on, I'm not learning anything without a piano accompaniment.
Posted by: Paul Carpenter | June 26, 2007 2:48 PM
I second the Jonathan Vos Post #4 reference to Tristan Needham, 'Visual Complex Analysis' and his use of "amplitwisting".
There is a website with PDF extracts:
http://www.usfca.edu/vca/
Posted by: Doug | June 26, 2007 4:33 PM
Paul Carpenter:
I dunno — I think this video, in particular, would have gone very well with a Kraftwerk track like "Neon Lights".
By the way, those who want more details should check out the Secret Blogging Seminar's comments on this video.
Posted by: Blake Stacey, OM | June 29, 2007 1:45 PM