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

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.

Cited by in corpus (7)

• Kaleidoscopes of Hofstadter Butterflies and Aharonov-Bohm caging from $2^n$-root topology in decorated square lattices
• Ground state fractal crystals
• Lattice random walks and quantum A-period conjecture
• Geometry, Number Theory and the Butterfly Spectrum of Two-Dimensional Bloch Electrons
• Tessellations and Descartes disk configurations
• Asymptotic scaling and universality for skew products with factors in SL(2,R)
• Kaleidoscopic Symmetries and Self-Similarity of Integral Apollonian Gaskets