Essential cell division protein FtsZ is an assembling GTPase which directs the cytokinetic ring formation in dividing bacterial cells. FtsZ shares the structural fold of eukaryotic tubulin and assembles forming tubulin-like protofilaments, but does not form microtubules. Two puzzling problems in FtsZ assembly are the nature of protofilament association and a possible mechanism for nucleated self-assembly of single-stranded protofilaments above a critical FtsZ concentration. We assembled two-dimensional arrays of FtsZ on carbon supports, studied linear polymers of FtsZ with cryo-electron microscopy of vitrified unsupported solutions, and formulated possible polymerization models. Nucleated self-assembly of FtsZ from Escherichia coli with GTP and magnesium produces flexible filaments 4-6 nm-wide, only compatible with a single protofilament. This agrees with previous scanning transmission electron microscopy results and is supported by recent cryo-electron tomography studies of two bacterial cells. Observations of double-stranded FtsZ filaments in negative stain may come from protofilament accretion on the carbon support. Preferential protofilament cyclization does not apply to FtsZ assembly. The apparently cooperative polymerization of a single protofilament with identical intermonomer contacts is explained by the switching of one inactive monomer into the active structure preceding association of the next, creating a dimer nucleus. FtsZ behaves as a cooperative linear assembly machine.