We investigated the randomness and uniqueness of human iris patterns by mathematically comparing 2.3 million different pairs of eye images. The phase structure of each iris pattern was extracted by demodulation with quadrature wavelets spanning several scales of analysis. The resulting distribution of phase sequence variation among different eyes was precisely binomial, revealing 244 independent degrees of freedom. This amount of statistical variability corresponds to an entropy (information density) of about 3.2 bits mm−2 over the iris. It implies that the probability of two different irides agreeing by chance in more than 70% of their phase sequence is about one in 7 billion. We also compared images of genetically identical irides, from the left and right eyes of 324 persons, and from monozygotic twins. Their relative phase sequence variation generated the same statistical distribution as did unrelated eyes. This indicates that apart from overall form and colour, iris patterns are determined epigenetically by random events in the morphogenesis of this tissue. The resulting diversity, and the combinatorial complexity created by so many dimensions of random variation, mean that the failure of a simple test of statistical independence performed on iris patterns can serve as a reliable rapid basis for automatic personal identification.