Thermodynamic properties of Ne, Ar, Kr, and Xe adsorbed on an Al–Ni–Co quasicrystalline (QC) surface are studied with the grand canonical Monte Carlo technique by employing Lennard-Jones interactions with parameter values derived from experiments and traditional combining rules. In all the gas/QC systems, a layer-by-layer film growth is observed at low temperature. The monolayers have regular epitaxial fivefold arrangements which evolve toward sixfold close-packed structures as the pressure is increased. The final states can contain either considerable or negligible amounts of defects. In the latter case, there occurs a structural transition from fivefold to sixfold symmetry which can be described by introducing an order parameter, whose evolution characterizes the transition to be continuous or discontinuous as in the case of Xe/QC (first-order transition with associated latent heat). By simulating fictitious noble gases, we find that the existence of the transition is correlated with the size mismatch between adsorbate and substrate characteristic lengths. A simple rule is proposed to predict the phenomenon.