I modified Wolfram Demonstration code by Enrique Zeleny to produce a short video of the emergence of the Hofstadter’s Quantum-Mechanical Butterfly
I found an interesting pattern. By modifying the integer used in the solution i=12, the Golden Ratio Ф=(1+Sqrt)/2=1.618 … emerges within the butterfly! Can you find it (or 1/Ф=.618…)?
Hint: Use the interactive version and mouse-over the red or green dots in the white space of the wings of butterfly.
If you have the free Mathematica CDF plugin on a non-Chrome browser, you can interactively analyze this with n<11 being presented with symbolic solutions in a mouse-over Tooltip (below).