A self-similar tiling generated by the minimal Pisot number

Let be a Pisot unit of degree 3 with a certain niteness condition. A large family of self similar plane tilings can be constructed, by the digit expansion in base . (cf. [7], [5], [8]) In this paper, we prove that the origin is an inner point of the central tile K. Further, in the case corresponds to the minimal Pisot number, we shall give a detailed study on the fractal boundary of each tile. Namely, a su cient condition of "adjacency" of tiles is given and the "vertex" of a tile is determined. Finally, we prove that the boundary of each tile is a union of 5 self similar sets of Hausdor dimension 1:10026 : : : . 1991 Mathematics Classi cation. Primary 11A68, 11R06