The differences between unidirectional and bidirectional polymerization are considered. The unidirectional case is discussed in the framework of the RNA world. Similar to earlier models of this type, where polymerization was assumed to proceed in a bidirectional fashion (presumed to be relevant to peptide nucleic acids), left- and right-handed monomers are produced via an autocatalysis from an achiral substrate. The details of the bifurcation from a racemic solution to a homochiral state of either handedness is shown to be remarkably independent of whether the polymerization is unidirectional or bidirectional. Slightly larger differences are seen when dissociation is allowed and the dissociation fragments are recycled into the achiral substrate.