Pythagorean Triplets, Integral Apollonians and The Hofstadter Butterfly [article]

Indubala Satija
2018 arXiv   pre-print
Hierarchical sets such as the Pythagorean triplets (PT) and the integral Apollonian gaskets (IAG) are iconic mathematical sets made up of integers that resonate with a wide spectrum of inquisitive minds. Here we show that these abstract objects are related with a quantum fractal made up of integers, known as the Hofstadter Butterfly. The "butterfly fractal" describes a physical system of electrons in a crystal in a magnetic field, representing exotic states of matter known as integer quantum
more » ... l states. Integers of the butterfly are the quanta of Hall conductivity that appear in a highly convoluted form in the integers of the PT and the IAG. Scaling properties of these integers, as we zoom into the self-similar butterfly fractal are given by a class of quadratic irrationals that lace the butterfly in a highly intricate and orderly pattern, some describing a mathematical kaleidoscope. The number theoretical aspects are all concealed in Lorentz transformations along the light cone in abstract Minkowski space where subset of these are related to the celebrated Pell's equation.
arXiv:1802.04585v3 fatcat:c7hsvqc36jb27od72hfy5ljbyi