Pythagorean Triplets, Integral Apollonians and The Hofstadter Butterfly

Hierarchical sets such as the Pythagorean triplets ($\cal{PT}$) and the integral Apollonian gaskets ($\cal{IAG}$) 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 {\it Hofstadter Butterfly}. The "butterfly fractal" describes a {\it physical system} of electrons in a crystal in a magnetic field, representing exotic states of matter known as {\it integer quantum Hall} states. Integers of the butterfly are the quanta of Hall conductivity that appear in a highly convoluted form in the integers of the $\cal{PT}$ and the $\cal{IAG}$. 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 {\it 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 {\it Pell's equation}.