Theoretically, the semicircle law originates from a representation of the modular group Γ 0 (2), which describes a symmetry between different Hall phases. (Note that this is not a symmetry in the conventional sense; there is no conserved current.) That group's strong connections to number theory also appear: Hall phase transitions (in a single layer) exhibit a selection rule | p q ′ − − p ′ q | = 1 that also governs the Farey sequence. Indeed, plots of the semicircle law are also Farey diagrams.