# MÃ¶bius Transformations

This short video is simply lovely. It illustrates (with perfectly chosen music) an important set of mathematical transformations dear to many of us, the MÃ¶bius transformations. It is by Douglas Arnold and Jonathan Rogness of the University of Minnesota. It’s a pleasure to watch, whether you work in a mathematically related field or not. Try it:

Also worth looking at (for a host of other reasons) are the comments on the video from random people, for which you’ll have to go to the YouTube site itself. Be sure you’re not drinking any very hot liquids.

-cvj

### 12 Responses to MÃ¶bius Transformations

1. See this ancient post for a very nice artistic application of the same idea.

2. Clifford says:

-cvj

3. Elliot says:

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.

4. Clifford says:

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

5. Elliot says:

thanks for the reply. I believe i get the general concept and as always appreciate the thoughtful response.

e

6. Mary Cole says:

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.

7. Clifford says:

Wonderful! What was the book? I love books like that.

-cvj

8. D says:

Nice! What DO the lines and circles on the plane correspond to on the sphere?

9. Clifford says:

Hi,

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

Best,

-cvj

10. Mary Cole says:

The book is ‘Make Shapes’ by Gerald Jenkins and Anne Wild.