5 - Fractal dust and infinite words

Summary

Even in a single pore are inconceivably many lands, countless as particles of dust… In every particle of dust in these lands, one also differentiates countless lands, some small, others large…

Avatamsaka Sutra

Allowing our circle program to run for a long time, we find ourselves drawing great numbers of circles, circles within circles within circles. The disks of higher and higher levels seem to recede ‘into the distance’ as their diameters rapidly shrink to zero. As we learnt in the last chapter, there is a pot of gold at the end of each infinite sequence of nested circles, a very special point called a limit point. You can see these limit points glowing yellow in Figure 5.1. They are a smattering of points in the plane, what Mandelbrot calls ‘fractal dust’, known to the experts rather less colourfully as the limit set of the Schottky group generated by the transformations a and b. We shall be explaining more about what is meant by fractal dust as we go along.

In the last chapter we described a way of drawing all the Schottky circles. We can easily change it to draw only the limit set: run the BFS circle drawing program without plotting any circles until the words we are examining are extremely long. Eventually, all the circles will become extremely small, and the plot we get will be a very good approximation to the fractal dust.