10 - Between the cracks pages 320 to 330

Summary

A spirally degenerate group

The large grey pointed peak just at the left side of the top frame of Figure 10.3 is the cusp corresponding to 0/1. High on its slopes are the smaller cusps μ(1/11) and μ(1/10). Let's put on our crampons and climb up there, which is another way of saying we shall adjust the parameters of our boundary-drawing program to plot just that stretch (or slightly more). The result is the first frame of Figure 10.5.

We shall need technical climbing gear if we wish to scale right up to the rocky grey underside of the 1/11 cusp. The result is shown in the second frame, where we have blown up the nearby stretch from 11/120 to 12/31. Along this stretch, the grey rocks above us are nearly pointing upside down! We are also shrinking ourselves dramatically as we climb into these overhangs. We shall need Batman's suction-cups to climb even further, aiming for the very roof underneath the 12/131 cusp as we do in the third frame. The grey stalagmites in the first frame have now become stalactites, or vice versa, since we always have trouble remembering which is which. Let's carry on and shrink ourselves even smaller. By the time we reach the fourth frame, we have travelled in a very tight spiral so that the small piece marked by arrows towards the bottom right is practically level ground.