Thackeray 703

Thackeray 427

### Abstract or Additional Information

This is the third in a series of talks establishing the basic "synthetic" properties of hyperbolic space: the existence of geodesics, what angles and circles look like, etc. In the first talk I introduced the Poincare ball model of hyperbolic space and explicitly described its geodesic arcs with one endpoint at the origin. In the second, I characterized the isometries of this model and showed that they act transitively on its orthonormal frame bundle. In this third talk, after briefly revisiting the first two, I will bring their descriptions together to completely describe all hyperbolic geodesics, then establish Euclid's first four postulates and the failure of the fifth. Time permitting, I will go on to prove the "Gauss--Bonnet formula" for the area of hyperbolic triangles.