Phase Defects and Order Parameter Space for Penrose Tilings

(Received 22 June 1994, revised 8 November 1994, accepted 30 November 1994)

Abstract
A new invariant classifying phase defects of Penrose tilings is constructed. This invariant takes values in the group closely related to the fundamental group of a certain topological space, which is the image of the tiling itself under identifications dictated by matching rules. This space plays the role of the order parameter space for pentagonal quasicrystals. The invariant enables us to discriminate between mismatches of different directions.