The butterfly and the remarkable Professor Hofstadter

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 with the Physics Department, University of Oregon, USA. 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” had been coined, and Hofstadter had discovered one of the very few fractals known in physics.

Quest for the elusive butterfly

Physicists have since searched for experimental 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 very close together observing the butterfly would require unfeasibly strong magnetic fields, while if they are widely spaced disorder ruins the pattern.

Graphene, a quirky form of carbon, has been the key to finding the butterfly. It is a one-atom thick layer of carbon atoms arranged in hexagonal patterns – somewhat like chicken wire. A layer of this was placed on atomically flat boron nitride substrate, which likewise has a honeycomb atomic lattice structure, but with slightly longer bonds between atoms. This combination resulted in the electrons experiencing a periodic potential, akin to a marble rolling over a surface shaped like the tray of an egg carton.

City College of New York Assistant Professor of Physics Cory Dean developed the material. He was a member of an international group that published its findings in May. Separate groups at the University of Manchester (UK) and Massachusetts Institute of Technology simultaneously reported similar results.

According to a City College press release, the light and dark sections of the butterfly pattern correspond to “gaps” in energy levels that electrons cannot cross and dark areas where they can move freely. While efficient conductors like copper have no gaps, and there are very large gaps in insulators, Dean believes the very complicated structure of the Hofstadter spectrum suggests as yet unknown electrical properties.

“We are now standing at the edge of an entirely new frontier in terms of exploring properties of a system that have never before been realized,” he said. “The ability to generate this effect could possibly be exploited to design new electronic and optoelectronic devices.”

Graphene the wonder material, and father n son physicists

Graphene planes had already shown promise as a new wonder material. They were first isolated in 2004, and have a thickness almost a millionth of a human hair. Graphene is stronger than steel and more conductive than copper, and can help make ultrafast optical switches for applications including communications, as well as lead to more efficient solar cells, enhanced printed circuits, unbreakable touchscreens and microscale Lithium-ion batteries. It may even prove to be the ideal material for 3D printing.

Rather as graphene may have multiple uses, the man who described the butterfly spectrum has proven multi-talented. Douglas Hofstadter was the son of Stanford University physicist Robert Hofstadter, who in 1961 was the joint winner of the Nobel Prize for Physics, “for his pioneering studies of electron scattering in atomic nuclei and for his consequent discoveries concerning the structure of nucleons."

Like father, like son, you might think, as Douglas also became a physicist. Yet he did not remain so for long. The year after his paper on the spectrum was published, Hofstadter joined Indiana University's Computer Science Department faculty, and launched a research program in computer modeling of mental processes, which he then called "artificial intelligence research", though he now prefers "cognitive science research".

Miracles, mirages and butterfly dreaming

Hofstadter pondered the question of what is a self, and how can one come out of stuff that is as selfless as a stone or a puddle? In an attempt to provide an answer, he wrote a book, Gödel, Escher, Bach: an Eternal Golden Braid. This interwove several narratives, and featured word play, puzzles, and recursion and self-reference, with objects and ideas referring to themselves.

The book was a success, winning the Pulitzer Prize for general non-fiction. Yet in an interview with Wired, Hofstadter later expressed disappointment that most people found its point was simply to have fun, albeit noting that hundreds of people had written to him, saying it launched them on a path of studying computer science or cognitive science or philosophy.

Some of these people might have been startled when Hofstadter, by then professor of cognitive science at Indiana University, USA, later told the New York Times, “I have no interest in computers,” adding, “People who claim that computer programs can understand short stories, or compose great pieces of music — I find that stuff ridiculously overblown.”

The NY Times interview accompanied the publication of a straighter book on questions of consciousness and soul, I Am a Strange Loop. Within this, Hofstadter wrote, “In the end, we are self-perceiving, self-inventing, locked-in mirages that are little miracles of self-reference.”

Here, Hofstadter seems to echo the butterfly pattern he discovered, with its multitude of versions of itself. But there’s far more to consciousness, which he believes derives from a self-model.

Over 2300 years ago, a butterfly featured in an anecdote by another thinker. The Chinese philosopher Zhuangzi wrote of dreaming he was a butterfly, and awaking to wonder if he was a man who dreamt of being a butterfly, or a butterfly dreaming of being a man. After Hofstadter, you might wonder if either of these is but a dream within a dream. Which may remind you of a movie, which you may find within another column, which is currently a hazy looping form within the mirage that’s this writer … but will not feature butterflies.

Martin Williams