Tiles and pretzels

Thus far, our symmetry groups have always gone hand in hand with a tiling; either the bathroom floor tiles in the picture on p. 17 in Chapter 1, or the ring-shaped tiles associated to the group generated by a single loxodromic transformation in Chapter 3. As we already hinted, there is still a set of symmetrical tiles hidden amid the circles in the Schottky array. It is a bit harder to spot them, because their shape is a rather more exotic than those we have met thus far. Perhaps a volunteer could help us. Dr. Stickler, would you do us the favour of guiding us around?

Figure 4.7 shows a time elapsed photograph of Dr. Stickler's journey. Starting boldly at the centre of the picture, you see him shrinking and turning as he progresses ever more deeply into the Schottky array. A new Dr. Stickler appears for each transformation in the group. The most striking feature of the picture is that the arrows exactly reflect the pattern in the word tree. Each Dr. Stickler is at a node, and if you copied over the labelling from the word tree you would find each copy labelled by the exact transformation which gets him from his central position to the given location. Of course, we only show a finite number of levels, but it is not hard to imagine the whole infinite tree of words extending outwards, the edges getting ever shorter until we eventually reach a limit point at the infinite end of the branch.