The effect of this policy of appeasement is that the domain associated to each circle is no longer an annulus, but an annulus with certain pieces snipped away. The question arises, whether two of the snipped-away pieces can overlap, as shown in Figure 14. This will depend on how large the annuli are that we originally tried to associate to the hemispheres.
This question of the possible overlap of snipped-away parts is important, because now that we don't have a complete annulus to work in, we can no longer simply spread our tubes out radially away from the center of the hemisphere. If we do this, the tubes near the middle of the snipped-away part will not be growing. If these tubes are to grow, we must allow them to expand laterally, which will squeeze tubes out away from the middle of the snipped-away part. (Peek ahead at Figure 24.) But if two snipped-away parts overlap, there will be nowhere for the tubes to spread out into. We want to arrange things so that snipped-away parts can never overlap.
