Relationship between Penrose vertices and decagonal covering configurations

Two-dimensional quasicrystals are generally described by the Penrose tiling model. An alternative is the cluster covering approach developed recently. There exists a correspondence between the two schemes. The distributions of rhombus vertices in Penrose tilings and the nearest neighbor configurations in the decagonal covering structures are studied. Their occurrence probabilities are expressed in terms of the powers of the golden mean τ and their relationship is obtained.