The lacy web in Figure 7.1 is called the Apollonian gasket. Usually, it is constructed by a simple geometric procedure, dating back to those most famous of geometers, the ancient Greeks. We shall start by explaining the traditional construction, but as we shall disclose shortly, the gasket also represents another remarkable way in which the Schottky dust can congeal. The pictures you see here were actually all drawn using a refinement of the DFS algorithm for tangent Schottky circles.

The starting point of the traditional construction is a chain of three non-overlapping disks, each tangent to both of the others. A region between three tangent disks is a ‘triangle’ with circular arcs for sides. This shape is often called an ideal triangle: the sides are tangent at each of the three vertices so the angle between them is zero degrees. The gasket is activated by the fact that in the middle of each ideal triangle there is always a unique ‘inscribed disk’ or incircle, tangent to the three outer circles.