Hofstadter's butterfly

Hofstadter's butterfly is the name of a fractal structure discovered by Douglas Hofstadter, which he described in 1976 in an article on the energy levels of Bloch electrons in magnetic fields. It gives a graphical representation of the spectrum of the almost Mathieu operator for λ = 1 at different frequencies, which turned out to display an obvious self-similarity. As such, it is one of the rare fractal structures discovered in physics, along with KAM tori.

Written while Hofstadter was at the University of Oregon, his paper was influential in directing further research. Hofstadter predicted on theoretical grounds that the allowed energy level values of an electron in a two-dimensional square lattice, as a function of a magnetic field applied to the system, formed what is now known as a fractal set. That is, the distribution of energy levels for small scale changes in the applied magnetic field recursively repeat patterns seen in the large-scale structure. This fractal structure is generally known as "Hofstadter's butterfly"; it has recently been experimentally confirmed in transport measurements in two-dimensional electron systems with a superimposed nano-fabricated lattice.

Rendering of the butterfly by Hofstadter

Hofstadter's butterfly "Gplot", as Hofstadter called the figure, was described as a recursive structure in his 1976 article in Physical Review B, written before Benoit Mandelbrot's newly coined word "fractal" was introduced in an English text.