What makes hyperbolic geometry particularly hard?
As I have already commented in the page on general difficulties with the course , the lack of a natural coordinate system for the hyperbolic plane forces one to think geometrically about it. There is another difficulty which makes hyperbolic geometry seem confusing when it is first introduced, which is that the question “what exactly is the hyperbolic plane?” doesn’t seem to have a very clear answer. In the course you meet so-called models of the hyperbolic plane, all in their different ways somewhat complicated, but in what sense are they models? What are they models of? To understand the answer to this question, one is forced to think not just geometrically, but abstractly , in a way that one is not with the sphere. (In fact, the set of points of norm one in R^3 is just a model of the sphere, but it such a natural model that there is not much harm in saying that it actually is the sphere.) In the abstract, hyperbolic geometry is what you get when you replace the parallel postulate wi
