Week 35.

We have now finished the introductory chapter on axiomatic geometry and started om chapter 2, where we construct and analyze an important model for the hyperbolic plane: the Poincar ¨¦ upper halfplane. The points and lines here are in exact correspondence with points and chords in an open disk in R². The correspondence goes via orthogonal and spherical projections, so we had to study the latter in some detail.  Next we want to define a "group of congruences" for this geometry, and the idea is to find a suitable group of transformations of the extended complex plane using complex function theory, and then restrict to those preserving the upper half plane.

Exercises for September 2: 2.1.2 and 2.1.4.