The Hofstadter butterfly graphically represents the eigenvalues of Harper's equation,
In the physical world, eigenvalues are the observable energies of a system. Physics dictates that the measured energy of a system is one of its eigenvalues.
The points in the figure above represent the energy levels of charged, massless fermions confined to a plane permeated by a perpendicular magnetic field. Each horizontal slice holds the eigenvalues of the system at a particular magnetic field strength.
To learn more, read my college thesis