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 Hofstadter Butterfly. The "butterfly fractal" describes a physical system of electrons in a crystal in a magnetic field, representing an exotic state of matter known as integer quantum Hall state. 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 quadratic irrationals that lace the butterfly in a highly intricate and orderly pattern, some describing a mathematical kaleidoscope, all concealed in Lorentz transformations in an abstract Minkowski space in $(2+1)$ dimension and in the solutions of the celebrated Pell's equation.