The picture in Figure 6.1 shows our first truly impressive fractal construction created by the simultaneous symmetry of two Möbius maps. We made it by introducing what seems like only a small change in the set-up of the previous two chapters: we just let the four circles come together until they were tangent, forming, as we shall say, a kissing chain. In a beautiful way, this causes the limit set to coalesce from dust into a ‘necklace’. If you look carefully, you will see how the nested Schottky circles form smaller and smaller chains of tangent circles. Instead of nesting down on fractal dust, these chains shrink down onto a glowing and rather curiously crinkled loop. The nesting circles seem to condense near the tangency points of the circles, highlighted by our colour coding like brilliant blue jewels. Except for the fact that the circles are tangent, this picture is entirely similar to the one in Figure 5.1.

To see better what is going on, in Figure 6.2 we zoomed in to the small black rectangle marked in the large scale picture. Now you can see more clearly the intricate complexity with which this marvellous necklace has been fashioned. Near the bottom you get a fairly clear view of some of the inner levels of chains of kissing circles, while near the main tangency point, nested tangent circles pile up creating a wonderful jewelled effect only hinted at in Figure 6.1.