A tale of two fractals: The Hofstadter butterfly and the integral Apollonian gaskets
- First Online:
- 52 Downloads
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 D3 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.
Unable to display preview. Download preview PDF.
- 1.M.Ya. Azbel’, JETP 19, 634 (1964)Google Scholar
- 15.D. Mackenzie, Am. Scientist 98, 10 (2010)Google Scholar
- 17.The theorem is named after Rene Descartes, who stated it in 1643. See R. Descartes. Oeuvres de Descartes, Correspondence IV, edited by C. Adam and P. Tannery (Leopold Cerf, Paris, 1901)Google Scholar
- 18.I.I. Satija, with contributions by Douglas Hofstadter, Butterfly in the Quantum World: the Story of the Most Fascinating Quantum Fractal (IOP Concise, Morgan and Claypool, San Rafael, 2016)Google Scholar
- 21.A. Avila, S. Jitomikskaya, C.A. Marx [arXiv:http://arxiv.org/abs/1602.05111] (unpublished)