Heading for the boundary

In Chapter 8, we played (rather irresponsibly) with traces around 2. One especially spiralliferous spot was near trace 1.91+0.05i, exactly the complex conjugate of the trace corresponding to the μ-value 0.05+1.91i, the red dot in Figure 9.12. (Complex conjugate traces, remember, give mirror image groups.) Looking back at Figure 8.17, perhaps you can see how we might have predicted that the μ-value for this group should be near the 1/9 and 1/10 cusps. The spiral head is largest around words with prefix ab10 and ab9, and in fact the most extreme point seems to be about ab9ab10. The roles of a and b are reversed because of the mirror symmetry, which more or less explains why in our present set-up it is a good idea to declare a10B a special word.

Figure 9.12 is like a road map, delineating the boundary between order and chaos. How about using it to drive right up to one of these two nearby cusps? With any luck, it should exhibit both the beautiful spirals of Figure 8.17 and the delicate lacework of the Apollonian gasket. Imagine how those two effects will be combined. The results are – wait one moment, let's not get ahead of ourselves. Perhaps we should look very closely at the boundary near μ = 0.05 + 1.91i. Zoom in to the red dot in Figure 9.12 to get the very small (0.03 × 0.03) frame in Figure 9.14.