Hofstadter's butterfly
In physics, Hofstadter's butterfly is a mathematical object describing the theorised behaviour of electrons in a magnetic field. It was discovered in 1976 by Douglas Hofstadter, who later abandoned physics research and became notable as a cognitive scientist and author. It takes its name from its visual resemblance to a butterfly. It is a fractal structure and as such it shows self-similarity, meaning that small fragments of the structure contain a (distorted) copy of the entire structure. It is one of the rare non-random fractal structures discovered in physics, along with KAM theorem. A central concern in Hofstadter's paper is how the two distinct behaviours predicted for 'rational' and 'irrational' magnetic fields must be unified in the continuous physical world.
To test whether Hofstadter's butterfly describes real electron behaviour requires accurate measurements. Such were not possible when he wrote his paper. However, more recent experimental research has confirmed the characteristic butterfly shape.
Hofstadter described the structure 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 at different frequencies. The intricate mathematical structure of this spectrum was discovered by Soviet physicist Mark Ya. Azbel' in 1964. However, Azbel' did not plot the structure as a geometrical object.
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