  Plotting spectra of a range of almost Mathieu operators reveals a beautiful
fractal-like image that contains multiple copies of a butterfly image. We
demonstrate that plotting the butterflies using a gap-labelling scheme based on
K-theory or Chern numbers reveals systematic discontinuities in the gap
positioning. A proper image is produced only when we take into account these
discontinuities, and close the butterfly wingtips at the points of
discontinuity. A conjecture is presented showing a simple formula for locating
the discontinuities, and numerical evidence is given to support the conjecture.
We also present new renderings of this butterfly.
