Comments on: Möbius Transformations http://asymptotia.com/2007/11/28/mobius-transformations/ Sat, 30 Aug 2008 11:22:42 +0000 http://wordpress.org/?v=2.5.1 By: Anti Memoirs » Möbius Transformation http://asymptotia.com/2007/11/28/mobius-transformations/#comment-95259 Anti Memoirs » Möbius Transformation Mon, 03 Dec 2007 07:08:31 +0000 http://asymptotia.com/2007/11/28/mobius-transformations/#comment-95259 [...] تولید کنندگان: Douglas Arnold and Jonathan Rogness منبع اولیه‌ی من: Asymptotica [...] [...] تولید کنندگان: Douglas Arnold and Jonathan Rogness منبع اولیه‌ی من: Asymptotica [...]

]]> By: Mary Cole http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94692 Mary Cole Fri, 30 Nov 2007 11:01:16 +0000 http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94692 The book is 'Make Shapes' by Gerald Jenkins and Anne Wild. The book is ‘Make Shapes’ by Gerald Jenkins and Anne Wild.

]]> By: Clifford http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94564 Clifford Fri, 30 Nov 2007 02:02:54 +0000 http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94564 Hi, Have another look at the video. You can see the shape on the actual sphere itself. Best, -cvj Hi,

Have another look at the video. You can see the shape on the actual sphere itself.

Best,

-cvj

]]> By: D http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94536 D Thu, 29 Nov 2007 23:20:19 +0000 http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94536 Nice! What DO the lines and circles on the plane correspond to on the sphere? Nice! What DO the lines and circles on the plane correspond to on the sphere?

]]> By: Clifford http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94455 Clifford Thu, 29 Nov 2007 16:12:47 +0000 http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94455 Wonderful! What was the book? I love books like that. -cvj Wonderful! What was the book? I love books like that.

-cvj

]]> By: Mary Cole http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94453 Mary Cole Thu, 29 Nov 2007 16:03:05 +0000 http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94453 Thank you for this wonderful video clip! I visited the science museum in London recently and was completely enthralled by the section on mathematical shapes. On the strength of this, I bourght a book on making mathematical models ('to cut out, glue and decorate') as I thought it would make for some interesting Christmas decorations this year! I'm probably being wildly optimistic about finding time to do this, but I live in hope. Thank you for this wonderful video clip! I visited the science museum in London recently and was completely enthralled by the section on mathematical shapes. On the strength of this, I bourght a book on making mathematical models (’to cut out, glue and decorate’) as I thought it would make for some interesting Christmas decorations this year! I’m probably being wildly optimistic about finding time to do this, but I live in hope.

]]> By: Elliot http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94247 Elliot Thu, 29 Nov 2007 01:17:40 +0000 http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94247 thanks for the reply. I believe i get the general concept and as always appreciate the thoughtful response. e thanks for the reply. I believe i get the general concept and as always appreciate the thoughtful response.

e

]]> By: Clifford http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94211 Clifford Wed, 28 Nov 2007 22:54:53 +0000 http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94211 Indeed, these form the conformal group in two dimensions., with six parameters, acting on the complex plane with coordinate   [tex]z[/tex]: [tex] z\to\frac{az+b}{cz+d}\ ,\quad ad-bc=1\ . [/tex] It has its generalizations, off in conformal geometry, algebraic geometry, etc, taking you to projective spaces and so forth. There are then natural generalizations of the two-sphere in the video and the actions on it which generate the inversion, special conformal transformation and so forth. I know very little about that, and so will defer to others. As for the AdS connection, the transformation naturally acts on [tex]AdS_3[/tex] as a symmetry (I'm going Lorentzian henceforth). Then the whole thing generalizes to other AdS spaces, yes, in the sense that the conformal group is generally [tex]SO(D,2)[/tex], the conformal group in [tex]D[/tex] dimensions, and this acts naturally on [tex]AdS_{D+1}[/tex], one dimension higher. There is a natural geometrical object in this presentation too, as one can write [tex]AdS_{D+1}[/tex] as an hyperboloid in a space again one dimension higher with signature (D,2), where SO(D,2) acts as a simple isometry. So in the general sense that AdS shows up on one side of the AdS/CFT correspondence, it is related..... but there's of course a lot more to the correspondence than the AdS part. Cheers, -cvj Indeed, these form the conformal group in two dimensions., with six parameters, acting on the complex plane with coordinate   z:

z\to\frac{az+b}{cz+d}\ ,\quad ad-bc=1\ .

It has its generalizations, off in conformal geometry, algebraic geometry, etc, taking you to projective spaces and so forth. There are then natural generalizations of the two-sphere in the video and the actions on it which generate the inversion, special conformal transformation and so forth. I know very little about that, and so will defer to others.

As for the AdS connection, the transformation naturally acts on AdS_3 as a symmetry (I’m going Lorentzian henceforth).

Then the whole thing generalizes to other AdS spaces, yes, in the sense that the conformal group is generally SO(D,2), the conformal group in D dimensions, and this acts naturally on AdS_{D+1}, one dimension higher. There is a natural geometrical object in this presentation too, as one can write AdS_{D+1} as an hyperboloid in a space again one dimension higher with signature (D,2), where SO(D,2) acts as a simple isometry.

So in the general sense that AdS shows up on one side of the AdS/CFT correspondence, it is related….. but there’s of course a lot more to the correspondence than the AdS part.

Cheers,

-cvj

]]> By: Elliot http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94162 Elliot Wed, 28 Nov 2007 18:57:55 +0000 http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94162 fabulous. so is there a similar dimensional heirarchy for these transformations in higher dimensions? If so and I realize I may be way out in left field here, does this relate mathematically to ADS/CFT correspondence? thanks e. fabulous. so is there a similar dimensional heirarchy for these transformations in higher dimensions? If so and I realize I may be way out in left field here, does this relate mathematically to ADS/CFT correspondence?

thanks

e.

]]> By: Clifford http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94159 Clifford Wed, 28 Nov 2007 18:47:34 +0000 http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94159 Thanks Jacques! Nice link... -cvj Thanks Jacques! Nice link…

-cvj

]]> By: links for 2007-11-28 « Chatquah and Galoshes http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94103 links for 2007-11-28 « Chatquah and Galoshes Wed, 28 Nov 2007 15:21:43 +0000 http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94103 [...] Mobius Transformations on the mathematics behind the mobius transform via Asymptotia (tags: math geometry transformations) [...] [...] Mobius Transformations on the mathematics behind the mobius transform via Asymptotia (tags: math geometry transformations) [...]

]]> By: Jacques Distler http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94089 Jacques Distler Wed, 28 Nov 2007 14:38:57 +0000 http://asymptotia.com/2007/11/28/mobius-transformations/#comment-94089 See <a href='http://golem.ph.utexas.edu/~distler/blog/archives/000011.html' rel="nofollow">this ancient post</a> for a very nice artistic application of the same idea. See this ancient post for a very nice artistic application of the same idea.

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