Hofstadter's butterfly is the name of a fractal

Fractal

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

 structure discovered by Douglas Hofstadter

Douglas Hofstadter

Douglas Richard Hofstadter is an American academic whose research focuses on consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics...

, which he described in 1976 in an article on the energy level

Energy level

A quantum mechanical system or particle that is bound -- that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy. These discrete values are called energy levels...

s of Bloch electrons in magnetic field

Magnetic field

A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

s. It gives a graphical representation of the spectrum of the almost Mathieu operator for at different frequencies, which turned out to display an obvious self-similarity

Self-similarity

In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...

. As such, it was the first fractal structure ever discovered in physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

.

Written while Hofstadter was at the University of Oregon

University of Oregon

-Colleges and schools:The University of Oregon is organized into eight schools and colleges—six professional schools and colleges, an Arts and Sciences College and an Honors College.- School of Architecture and Allied Arts :...

, 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

Square lattice

In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice. It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group is known symbolically as p4m.Two...

, 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 pattern

Pattern

A pattern, from the French patron, is a type of theme of recurring events or objects, sometimes referred to as elements of a set of objects.These elements repeat in a predictable manner...

s 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

2DEG

A two-dimensional electron gas is a gas of electrons free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantized energy levels for motion in that direction, which can then be ignored for most problems. Thus the electrons appear to be a 2D sheet...

 with a superimposed nano-fabricated lattice.

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

Benoît Mandelbrot

Benoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child...

's newly coined word "fractal" was introduced in an English text.