View from the Pennines: Ten Martinis and Fractal Carpets

Some insects are living in my children’s sand pit. I don’t know what they are and I don’t know where they are, but without question they are in there. If the sand pit has not been used for a few days the surface of the sand is covered by small tracks. The movement of the insects smooths the larger hills and castles, and the visual effect of the trails, give or take considerations of colour, is not unlike the darker patina of the image on the cover of this issue of Mathematics Today. This image, which I call a fractal carpet, shows how sets of Cantor sets vary with a parameter. The horizontal axis represents the parameter and the vertical axis represents the values of the spectrum of a linear operator for the corresponding parameter. Colours are assigned via the distance of a point to the nearest spectral point, with dark blue being closest and red farthest away. Further examples of spectral sets which vary with a parameter are shown in Figure 1, where the roles of the axes has been exchanged to respect standard practice rather than aesthetic considerations. These have a history which goes back to Douglas Hofstadter [4] and are called Hofstadter butterflies. I can’t seem to get away from insects this month. The fractal sets shown in Figure 1 are derived from the eigenvalue equation of the discrete Schrödinger equation with a quasiperiodic potential