Hofstadter's ButterflyPut electrons in a crystal and turn on a magnetic field. As the field grows, the allowed energy levels don't just shift — they shatter into bands, then into bands-within-bands, endlessly. Plotting energy against field strength draws this fractal moth. Drag to pan, scroll to zoom into any wing.When the magnetic flux per cell is a fraction p/q, the quantum problem collapses to a tiny q×q matrix (Harper's equation, 1955). Douglas Hofstadter swept every fraction in 1976 and found the self-similar spectrum — later seen for real in graphene superlattices.