Our progression through the book has been the investigation of more and more remarkable ways in which two Möbius maps a and b can dance together. Figure 9.1 shows another level of complexity, an array of interlocking spirals which literally took our breath away when we first drew it. It results from creating a double cusp group in which the generator b and the word a15B are both parabolic. Surely it cannot be coincidence that there are exactly 16 coloured circles forming a chain across the centre of the picture? Let's pick apart the dynamics of a and b, using the diagrammatic version Figure 9.2 for notation. In particular, let's try to see from the picture why a15B is parabolic and where its fixed point is located.

The action of b is quite easy. It is parabolic with fixed point at the bottom of the picture at —i. It pushes points out from its fixed point along clockwise circular trajectories. (You may like to compare with Figure 8.4 on p. 233 to help follow this.) One trajectory lies along the boundary of the outer unit circle framing the picture (note b(—1) = +1), and another is the boundary of the white circle tangent to the unit circle at —i.