Repetitive substitution tilings with respect to rigid motions

A tiling T is repetitive for every r \u3e 0 there exists R = R (r) \u3e 0 such that every R-patch of T contains an equivalent copy of every r-patch of T. in this paper, we describe a construction of a substitution that gives rise to a repetitive tiling T* with respect to rigid motions. The technique applied in the construction­ involves defining dissection rules on inflated edges of tiles and assigning orientations on edges tiles. Other properties pertaining to T* will be presented. One property is the occurrence of dense tile orientation T*. By dense tile orientations (DTO), we mean the orientations of tiles in the tiling are dense in a unit circle. To date, examples of this class of tilings are rarely found in the literatur