**Pure & Poetic: Butterfly in the Quantum World**

The Hofstadter butterfly is a fascinating two-dimensional spectral landscape – a graph of the allowed energies of an electron in a two-dimensional crystal in a magnetic field. It is a quantum fractal made up of integers, describing topological states of matter known as the integer quantum Hall states. My butterfly story tells the tale of its discovery by a graduate student named Douglas Hofstadter and discusses its number theoretical, geometrical and topological aspects [1]. I will describe how the integers of the butterfly are convoluted in the Pythagorean triplets and the integer curvature of Apollonian gaskets, thus providing a marvelous example of a physical incarnation of abstract mathematics. The story will touch the topological roots of integers, hidden in the Berry phase – a quantum cousin of the physics behind a Focault pendulum that reveals the secret behind astonishingly precise quantization of Hall conductivity. As a coda, I will present some novel topological states obtained by simple variations of the butterfly Hamiltonian [2, 3].

[1] *Butterfly in the Quantum World, *Indubala Satija with contributions by Douglas Hofstadter, IOP Concise, Morgan and Claypool publication (2016).

[2] M. Lababidi, I. Satija, and E. Zhao, PRL 112, 026805 (2014).

[3] I. Satija and E. Zhao, PRB 94, 245128 (2016).