  The \lq Hofstadter butterfly', a plot of the spectrum of an electron in a
two-dimensional periodic potential with a uniform magnetic field, contains
subsets which resemble small, distorted images of the entire plot. We show how
the sizes of these sub-images are determined, and calculate scaling factors
describing their self-similar nesting, revealing an un-expected simplicity in
the fractal structure of the spectrum. We also characterise semi-infinite
chains of sub-images, showing one end of the chain is a result of gap closure,
and the other end is at an accumulation point.
