Throughout the natural world – in corals, cactuses, sea-slugs and lettuce leaves – we see swooping, curving and crenelated forms. All these are biological manifestations of hyperbolic geometry an alternative to the Euclidean geometry we learn about in school. While nature has been playing with permutations of hyperbolic space for hundreds of millions of years, human mathematicians spent centuries trying to prove that such forms were impossible. The discovery of hyperbolic geometry in the nineteenth century helped to usher in a mathematical revolution, giving rise to new ways of mapping and analyzing curved surfaces. Such “non-Euclidean geometry,” now underlies the general theory of relativity and thus our understanding of the universe as a whole.