# What makes this course interesting?

An obvious answer is that geometry is one of the oldest branches of mathematics and that not to learn it is to miss out on a huge part of mathematical culture. Regrettably, geometry has fallen out of favour in schools (regrettably, because it used to be a very good way of exposing young mathematicians to the idea of rigorous proofs) with the result that you may not realise just how central it is at the research level. One can get some indication of this by looking at the courses for Part IIB: Riemann Surfaces, Algebraic Curves and Differentiable Manifolds are all strongly geometrical, and they lead to some of the most important and active areas of mathematics, with fascinating connections to theoretical physics that are still being uncovered.

On the other hand, this geometry is very different from what was (and to a small extent still is) taught at schools. Most people associate the word geometry with things like Pythagoras's theorem, the fact that the angles of a triangle add to 180 degrees, how to bisect a line segment using a ruler and compass, and so on. This sort of geometry goes back to Euclid, and is called Euclidean. It rests on five fundamental axioms, which are as follows.

1. Any two distinct points can be joined by a unique line segment.

2. Every line segment can be extended infinitely far in both directions.

3. There is exactly one circle of any given centre and radius.

4. Any two rightangles are congruent to one another.

5. Given two lines L and M and a third line N meeting them both, if the internal angles on one side of N, formed with L and M, add to less than 180, then the lines L and M meet on that side of N.

Assuming axioms 1 to 4, the fifth one can easily be shown to be equivalent to the famous parallel postulate:

5'. Given a line L and a point x not lying on L, there is exactly one line M containing the point x and not intersecting L.

One of the most famous and long-standing open problems in all of the history of mathematics was to determine whether the parallel postulate was a consequence of axioms 1 to 4. (Why did people care so much about it? A brief answer can be found here .) It is now known not to be, thanks to the efforts of Gauss, Bolyai, Lobatchevsky and others, who realised that they could build a perfectly consistent geometry without the parallel postulate. Their geometry is now known as non-Euclidean and will be the main focus of the course.

For various reasons (outlined here ), non-Euclidean geometry is less easy to understand than Euclidean geometry. If you look at the course dependencies given in the lecture schedules, you will find that this course is not an essential prerequisite for any other one. However, even if this course does not provide you with theorems that are directly applied in later courses, it is a good way to get used to the non-Euclidean way of thinking. You will have to do this at some point if you want to understand modern geometry or theoretical physics. If you do so now, you will make your life easier later.

Even if you have no intention of studying geometry or physics at Part II or beyond, it is worth understanding non-Euclidean geometry because its development was of fundamental importance not just to mathematics, but also to philosophy and the history of ideas. A brief indication of why can be found here