With Felix Klein's historic Erlangen address of 1872, the crucial role of group theory as a unifying and classifying force in geometry assumed centre stage. Klein was able to show how projective, affine, Euclidean and hyperbolic geometries were all different aspects of the same phenomenon of a group (typically a Lie group) acting on a homogeneous space.
With the rational or algebraic approach to planar geometry, a remarkable three-fold symmetry emerges which brings together Euclidean and two relativistic geometries. We need to go beyond Klein's program, and consider a broader landscape of interacting groups and associated complex number structures. Remarkable new phenomenon appear, valid also over finite fields, where explicit combinatorial and computational issues naturally arise. This talk will have lots of pictures --in colour!