3 - Double spirals and Möbius maps

Summary

“First accumulate a mass of Facts: and then construct a Theory. That, I believe, is the true Scientific Method.”

I sat up, rubbed my eyes, and began to accumulate Facts.

Sylvie and Bruno, Lewis Carroll

We now come to a key ingredient of our fractal constructions: the maps we use to make them. As we have seen, one of Klein's fundamental ideas was that any group of transformations can be used to create symmetry. Classically, we think about symmetry in terms of the Euclidean motions of translation, rotation and reflection. But symmetry can also be created from maps which distort, stretch and twist. In this chapter, we shall learn about a beautiful class of maps called Möbius maps, which stretch and twist in just the right controlled way. These are the maps which generate all our fractal pictures, and under which, as in Klein's vision, all our pictures are symmetric.

So what exactly are Möbius maps? The fox picture Figure 2.3 illustrated how maps like T(z) = az + b represent a spiralling expansion or contraction from a fixed point or source. On the other hand, Figure 2.6 showed how the map T(z) = 1/z turns the unit circle inside out and distorts the fox's shape in quite startling ways. Klein taught us that the logic of symmetry demands that when you have two maps, you should always try to compose them.