# What can I do to make hyperbolic geometry seem less hard?

Unfortunately, as I have said on another page, you can't understand it if you are not prepared to put in some work. Nevertheless, here are a few tips that may help you to work more efficiently. If I can think of any more, I will add them.

1. A very good introduction to hyperbolic geometry can be found in the book Flavors of Geometry , edited by Silvio Levy. If you read the relevant essay in this book, skipping the parts you don't understand, you will find answers to all sorts of questions, such as why hyperbolic geometry is called hyperbolic and why it is more than just a mathematical curiosity. You will also find helpful pictures of the relationships between the various models. I strongly recommend that you read this before we get on to the topic in lectures. Then you will have some idea of where I am going. Another strongly recommended book is Morris Kline's History of Mathematical Thought , which contains very interesting discussions of how, why and by whom non-Euclidean geometry was developed. Non-Euclidean geometry is one of the few areas of mathematics where a historical perspective is not just interesting but extremely helpful if you want to understand it.

2. Although calculations are quite complicated in hyperbolic geometry, involving various integrals, they can often be simplified by (i) making a good choice of model and (ii) exploiting symmetries. For example, to calculate the distance between two points in the disc model is not all that hard if one of the points is at the centre of the disc. But you can arrange for that by using an appropriate Möbius transformation. This will preserve distances and move one of your points to the centre. The half-plane model is useful because calculations with vertical lines are simpler, and one can use symmetries to make some lines vertical. This trick is exploited in the calculation of the area of a hyperbolic triangle. In general, it is a very good idea to become fluent in transferring between models, and in applying distance-preserving transformations (isometries) within models. This is one reason for the strong emphasis on symmetry groups in the course. (Another reason can be found here .)

3. One way to get a feel for the hyperbolic plane is to imagine a tessellation of the disc model by hyperbolic triangles, and an ant crawling around on it, with the disc continually "rotating" in order to keep the ant at the (Euclidean) centre, just as one might rotate a football with an ant on it, in order to keep the ant at the highest point. If the ant is making for a point x on the boundary of the disc, then tiny (in the Euclidean sense) triangles will move towards the ant from near x, getting bigger and bigger until the ant is crawling over them. Then the ant leaves them behind and they get smaller again as they approach the boundary point opposite x. As far as the ant is concerned, all the triangles it meets are of the same size, and the view from the centre of one of them is the same as the view from the centre of any other.