Introduction

Geometry is one of the earliest well-developed parts of mathematics. It was studied extensively in several ancient civilisations and in particular, the Greeks used it to lay the foundations of our modern "axiomatic" treatment of mathematics.

In particular, in the most influencial mathematical work of all time, the Elements of Euclid (born about 325 BC) described what we can think of as a model for plane geometry.
This started with a set of undefined notions including a point, a straight line, a circle.
In addition Euclid described five postulates which allowed him to develop the theory from these elementary ideas. These were supposed to be properties of the plane which were in some sense self-evident.
Here are the five he defined.

1. A straight line can be drawn from any point to any other.
   
2. A finite straight line can be extended continuously in a straight line.
   
3. A circle may be described with any centre and any radius.
   
4. A right angles are equal.
   
5. If a straight line meets two others so that the sum of the interior angles is less than two right angles, then one may extend the lines to meet on this side.
   

Things which are equal to the same thing are equal to each other.

Euclid left various other ideas like length, distance or the magnitude of an angle undefined and also assumed various other postulates without stating them explicitly. For example, he did not define what it means for a line to be straight or what it means for a point on a straight line to lie between two others. In fact to define an adequate model for plane geometry (or plane Euclidean geometry as it became called) requires a much more complicated set of postulates and this was not all cleared up until much later at the end of the 19th Century.

In the above set of postulates, it is clear that the fifth postulate has a rather different character to the others and right from the beginning mathematicians made strenuous efforts to deduce it from the others. By the 19th Century several different mathematicians including Gauss, Bolyai and Lobachevsky realised that one could define consistent geometries by denying the fifth postulate. Their constructions came to be called non-Euclidean geometries. When they were first introduced they were regarded with great suspicion by other mathematicians.

You can see an article about the history of non-Euclidean geometry

It was in the context of these new and controversial geometries that the German mathematician Felix Klein described a different way of thinking about geometry. In 1872 (at the age of 23) in his Erlangen Program he outlined the principle that a geometry is determined by its group of allowable transformations.

In fact without really realising it, mathematicians had been considering geometries different from Euclidean geometry much earlier. This arose from the efforts of Renaissance artists like Piero della Francesca and others to understand how to represent three dimensional scenes on a two-dimensional canvas in a realistic way. In fact they were studying what we now call Projective Geometry and this too can be fitted into Klein's framework.

Later in the 19th Century Henri Poincaré (among others) was led to other forms of "geometry" by the necessity of understanding what happens in problems involving the stability of the solutions of partial differential equations. This topic, initially called analysis situs is what we now recognise as Topology and can in fact trace it roots back even earlier.

In this course we will concentrate on the link between these various geometries and group theory.