## Wednesday, November 21, 2007

### Möbius transformations

This video is not quite a theory of everything but - as its creators forgot to tell you - it is about the tree level approximation of a theory of everything. The Möbius transformations are conformal i.e. angle-preserving, one-to-one maps from a sphere onto a sphere (or from a complex plane onto itself, or from anything of the same topology such as the Stanford bunny) which is why they are essential in perturbative string theory to bring a sphere diagram into a standard form.

Incidentally, the sphere is conformally equivalent to the plane because of Riemann's stereographic projection.

I didn't want to have too many birthday postings at the same moment but August Ferdinand Möbius was born on November 17th, 1790. This early 19th century string theorist is famous for his work on projective geometry, number theory, theoretical foundations of astronomy, and - of course - the Möbius strip, the characteristic one-loop Feynman diagram that only appears in vacua that contain both open and unoriented strings.

Benoît Mandelbrot celebrated his 83th birthday yesterday: congratulations.