One of the outstanding insights obtained by physicists working on "Quantum Chaos" is a conjectural description of local statistics of the energy levels of simple quantum systems according to crude properties of the dynamics of classical limit, such as integrability, where one expects Poisson statistics, versus chaotic dynamics, where one expects the same statistics as those of a suitable class of random matrices (GOE). I will describe in general terms for a lay audience what these conjectures say and discuss recent joint work with Valentin Blomer, Jean Bourgain and Maksym Radziwill, in which we study the size of the minimal gap between the first N eigenvalues for one such simple integrable system, a rectangular billiard having irrational squared aspect ratio. For certain quadratic irrationalities, such as the golden ratio, we show that the minimal gap is about 1/N, consistent with Poisson statistics. In the case of the golden ratio, the problem involves some curious properties of the Fibonacci sequence.