Gerrymandering, the structuring of voting districts to favour certain politicians and political groups, undermines fair elections and presents a serious challenge to democracy. We introduce a solution to gerrymandering inspired by the biological process of cell division in sexually reproducing organisms, meiosis, in which the boundaries of electorates are frequently redrawn by randomizing algorithms. By demonstrating the deep parallels between meiosis and John Rawls's concept of a 'veil of ignorance', we also show how one of the biggest threats to the integrity of meiosis-selfish genetic elements, genes that promote their own transmission at the expense of organismal fitness-can inspire another potential advantage to frequent random redistricting.