A Tale of Two Fractals: The Hofstadter Butterfly and The Integral Apollonian Gaskets

TL;DR

This paper reveals a surprising connection between the quantum Hofstadter butterfly fractal and the geometric Apollonian gasket, uncovering shared integer structures and symmetries that deepen understanding of both phenomena.

Contribution

It introduces a novel mapping between a quantum fractal and a geometric fractal, highlighting their shared integer-based structures and revealing hidden symmetries.

Findings

Identifies a mapping linking the Hofstadter butterfly and Apollonian gasket.

Uncovers a threefold symmetry in the butterfly's scaling properties.

Provides a mini review emphasizing hierarchical and Farey fraction aspects.

Abstract

This paper unveils a mapping between a quantum fractal that describes a physical phenomena, and an abstract geometrical fractal. The quantum fractal is the Hofstadter butterfly discovered in 1976 in an iconic condensed matter problem of electrons moving in a two-dimensional lattice in a transverse magnetic field. The geometric fractal is the integer Apollonian gasket characterized in terms of a 300 BC problem of mutually tangent circles. Both of these fractals are made up of integers. In the Hofstadter butterfly, these integers encode the topological quantum numbers of quantum Hall conductivity. In the Apollonian gaskets an infinite number of mutually tangent circles are nested inside each other, where each circle has integer curvature. The mapping between these two fractals reveals a hidden threefold symmetry embedded in the kaleidoscopic images that describe the asymptotic scaling properties of the butterfly. This paper also serves as a mini review of these fractals, emphasizing their hierarchical aspects in terms of Farey fractions.