Goal. Exploit the geometry of the translation surface and the way matrices act on the saddle connection vectors. We'll be working through this in the context of the torus and saddle connections in that setting. For the torus one intuitive way to restrict the set of slopes we're considering is to consider only those saddle connections who have slope 1 and horizontal component bounded by Q. If you know about Farey fractions there's a relationship here. Exercise 2.1. Once we've restricted the set of saddle connections to those with slope 1 and horizontal component Exercise 2.3. Show that these two sets are equal 1. Given a point (a,b) Ef show that it represents a translation surface with a horizontal saddle connection of length 1. 2. Harder! Given a translation surface with a horizontal saddle connection of length