Fractal dimensionalities, , for charged polymers are calculated over a continuous range of Bjerrum length for chains with various numbers of beads, N. Drastically decreasing values of , characteristic of a phase transition, are found as increases from zero in the range Å. The fractal dimensionality approaches unity as predicted by de Gennes et al (de Gennes P-G, Pincus P, Valesco R M and Brochard F 1976 J. Physique 37 1461) close to the onset of order. A newly developed smoothing algorithm yields a substantial improvement in the MC results and reveals an interesting fan-shaped behaviour of exponents and prefactor components at slightly above zero. It also enables discernment of a non-monotonic behaviour of versus N at and close to zero. Differences found in bending between short and long chains may provide an additional explanation for the stability of benzene and the low stability of conjugated rings of N>6. Based on fractal concepts, monomer densities are derived at various and it is suggested that the drastic density changes at Å are evidence of the first-order nature of this phase transition.