In the course of the last few chapters, we have been making increasingly frequent reference to two as yet unexplained topics: hyperbolic, otherwise called non-Euclidean geometry, and Teichmüller theory. To conclude the development of our story we want to offer some brief explanations, as it is on these two mathematical pillars which all the deeper developments of our subject rests.

Before we come to that though, let's return briefly to the work of Felix Klein. In the preceding chapters, we have done no more than scratch the surface of Klein and Fricke's epic books. As we have already mentioned, the work which led to the full understanding of Kleinian groups was the topic of an intense rivalry between Klein and Poincaré.

The background to Klein and Fricke's volumes is a subject called at that time Funktionentheorie, the study of differentiable functions of a complex variable. We first meet the familiar trigonometric functions sine and cosine as functions of a real variable, that is, if x is a real number, then so are sin x and cos x. The great significance of these functions is their periodicity, thus sin x = sin(2π + x), cos x = cos(2π + x) and so on.