Living systems propagate by undergoing rounds of cell growth and division. Cell division is at heart a physical process that requires mechanical forces, usually exerted by assemblies of cytoskeletal polymers. Here we developed a physical model for the ESCRT-III-mediated division of archaeal cells, which despite their structural simplicity share machinery and evolutionary origins with eukaryotes. By comparing the dynamics of simulations with data collected from live cell imaging experiments, we propose that this branch of life uses a previously unidentified division mechanism. Active changes in the curvature of elastic cytoskeletal filaments can lead to filament perversions and supercoiling, to drive ring constriction and deform the overlying membrane. Abscission is then completed following filament disassembly. The model was also used to explore how different adenosine triphosphate (ATP)-driven processes that govern the way the structure of the filament is changed likely impact the robustness and symmetry of the resulting division. Comparisons between midcell constriction dynamics in simulations and experiments reveal a good agreement with the process when changes in curvature are implemented at random positions along the filament, supporting this as a possible mechanism of ESCRT-III-dependent division in this system. Beyond archaea, this study pinpoints a general mechanism of cytokinesis based on dynamic coupling between a coiling filament and the membrane.