Scientist Hofstadter dreams of butterflies and other odd things

This spring, there was excitement in the world of physics as a long-predicted butterfly was proved to exist. But, being a creature of physics, this butterfly wasn't an insect, nor anything that would even occur without human minds to construct it.

This was Hofstadter's butterfly, a remarkable spectrum of electron energy levels. It was first described in 1976 by Douglas Hofstadter, who was then in the physics department at the University of Oregon. He was looking at the allowed energy levels of electrons restricted to a two- dimensional plane, with a periodic potential energy and a changing magnetic field. As Hofstadter put it in a summary of his work: "The resultant Schrödinger equation becomes a finite-difference equation whose eigenvalues can be computed by a matrix method."

To which you might respond, "Aha, but of course it does!"

Or even, "Huh?" — in which case, you might simply appreciate that when he plotted a graph of the spectrum, Hofstadter made a remarkable pattern that looked somewhat like a butterfly. And this pattern was recursive, so if you look at a small part of the pattern you see the same butterfly shape, which is repeated at larger and larger scales. The paper was published just one year after the term "fractal" was coined, and Hofstadter had discovered one of the very few fractals known in physics.

Physicists have since searched for proof of the butterfly, yet until recently it proved elusive. This is largely as it results from quantum effects, and when atoms in the two-dimensional plane are close together, observing the butterfly would need unfeasibly strong magnetic fields, while, if they are widely spaced, disorder ruins the pattern.