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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 quantumarXiv:1802.04585v3 fatcat:c7hsvqc36jb27od72hfy5ljbyi