Curved crease origami and topological singularities enable hyperextensibility of L. olor .

Fundamental limits of cellular deformations, such as hyperextension of a living cell, remain poorly understood. Here, we describe how the single-celled protist Lacrymaria olor , a 40-micrometer cell, is capable of reversibly and repeatably extending its necklike protrusion up to 1200 micrometers in 30 seconds. We discovered a layered cortical cytoskeleton and membrane architecture that enables hyperextensions through the folding and unfolding of cellular-scale origami. Physical models of this curved crease origami display topological singularities, including traveling developable cones and cytoskeletal twisted domain walls, which provide geometric control of hyperextension. Our work unravels how cell geometry encodes behavior in single cells and provides inspiration for geometric control in microrobotics and deployable architectures.