The Hofstadter butterfly graphically represents the eigenvalues of Harper's equation,

Warning:
Generating this image takes O(n^3) time

Lattice Size: 00

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