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Möbius maps animation

Post Icon Posted: Submitted by Daniel Wood on 21 November 2007 - 9:29pm.

Joined: 2006-10-10
Posts: 64

Hey everyone,

I stumbled across this on YouTube. It's rather good.

http://www.youtube.com/watch?v=JX3VmDgiFnY&NR=1

Daniel

‹ Quantum Chemistry DG Countable Ordinals ›
Post Icon Posted: 21 November 2007 - 9:44pm

Joined: 2006-08-31
Posts: 694

Very, very cool.
Post Icon Posted: 21 November 2007 - 10:12pm

Joined: 2007-10-03
Posts: 382

Wow, very nicely done, the animations are all so smooth and everything. Really good find !
Post Icon Posted: 22 November 2007 - 3:38am

Joined: 2006-11-02
Posts: 1017

Awesome!
Post Icon Posted: 22 November 2007 - 1:46pm

Joined: 2006-10-01
Posts: 427

Turbo!

Intriguing question: "can you use the description of mobius transformations as euclidian motions of a sphere to justify the existence of the canonical isomorphism $ \phi:M\"{o}b\rightarrow PSL(2,\matbb{C}) $ proved in discussion groups?"
Post Icon Posted: 23 November 2007 - 1:08am

Joined: 2006-10-10
Posts: 64

I don't know...

Daniel
Post Icon Posted: 30 November 2007 - 4:36am

Joined: 2006-10-01
Posts: 427

Hint: (if anyone really cares) $ PSL(2,\bbold{C}) $ has faithful action on (that is to say "the natural space in which our group acts is"- try it yourself applying the matrix before and after division) $ \bbold{P}^1\bbold{C}\cong $Riemann Sphere $ \cong S^2 $ (to see the first homeomorphism, observe the decomposition of projective space below and remember that the riemann sphere is $ \bbold{c} $ together with a point at infinity, the second is obvious if you know the riemann sphere- try wikipedia).

We also have $ \bbold{P}^1\bbold{C}\cong (1,z): z\in \bbold{C}\bigcup (0,1) $ (since we are working in projective space, a non-zero first coordinate can be divided to equal 1, the only other case {since both coordinates cannot be zero} is that the first is zero and the second can be divided to 1). The question then becomes where do we move those points and how....

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