Curved crease origami and topological singularities at a cellular scale enable hyper-extensibility of Lacrymaria olor .

Here we present the discovery of curved crease origami at the scale of a single cell. We show how topological singularities in the origami (d-cones) and twist walls in microtubule ribbons control deployment of a hyper-extensile neck in a single-celled protist. Our work establishes a direct link between geometry and cell behavior, connecting form and function of cellular morphodynamics.