Topological Defects in Non-Haüyan Crystallography: The Two-Dimensional Case

A general analysis of dislocations in a 2 d Penrose pattern is given in terms of an inhomogeneous order parameter space, which provides for their topological invariants. The corresponding Volterra processes are described. Dislocations of smallest Burgers' vector in usual Penrose patterns are characterized by stacking faults, or by the appearance of specific tilings belonging to generalized Penrose patterns. In that sense they are partial or imperfect dislocations. They are probably of great physical importance. Our order parameter space provides us also with a characterization of the canonical modes of deformation, which involve 4 hydrodynamic modes and one mode which corresponds to localized structural transitions.