It is well known that a Lorenz curve, derived from the distribution function of a random variable, can itself be viewed as a probability distribution function of a new random variable (Arnold, 2015). We prove the surprising result that a sequence of consecutive iterations of this map leads to a non-corner case convergence, independent of the initial random variable. In the primal case, both the limiting distribution and its parent follow a power-law distribution with exponent equal to the golden section. In the reflected case, the limiting distribution is the Kumaraswamy distribution with a conjugate value of the exponent, while the parent distribution is the classical Pareto distribution. Potential applications are also discussed.