Foundations

In Klein's view a geometry consists of a set S and a subgroup G of the group Bij(S) of all bijections from S to itself.

The prototype example is the case S = R² and then, for example, the group associated with plane Euclidean geometry is the group of rigid motions of the plane. (See example 1. below.)

The elements of S are called points and we think of the elements of G as acting on S by mapping points to points.

Two subsets of S are regarded as equivalent or "the same" if there is an element of G taking one set into the other.

In general, we endow our set S with extra structure and look at the subgroup of Bij(S) which preserves this structure.

Examples

1. In the Euclidean plane the extra structure is the usual metric d which measures the distance between points in the plane.
   We define d((x₁, y₁), (x₂, y₂)) = [(x₁ - x₂)² + (y₁ - y₂)²].
   The group which preserves this is the set of transformations which map pairs of points to points the same distance apart. Such elements g satisfy
   
   d(g(x), g(y)) = d(x, y)
   
   and such transformations are called isometries or rigid motions.
   For plane Euclidean geometry equivalence is called congruence. For example, two triangles are congruent if their corresponding sides have equal length (or equivalently, two sides and the "included angle", ... ).

   We may take a metric on S and then consider the group of bijections from S to itself which preserve this structure by being continuous (and having continuous inverse maps). The geometry associated with this group is then called a topology. For those who have met this idea before, the transformations are called homeomorphisms or topological isomorphisms.

   If we take the set S to be the plane R² and we take the group of bijections on S which preserve angles, we get a group whose associated geometry is called similarity geometry. The concept of equivalence is then called similarity. For example, two triangles which are similar have their angles the same (and their sides are then "in proportion").