 
Summary: A selfsimilar tiling generated by the minimal Pisot number
Shigeki Akiyama Taizo Sadahiro
Abstract
Let be a Pisot unit of degree 3 with a certain finiteness 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 sufficient 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 Hausdorff dimension 1.10026 . . ..
1991 Mathematics Classification. Primary 11A68, 11R06
Key words and phrases. Fractal, Plane Tiling, Pisot number.
1 Plane tiling and Pisot numeration system
Let > 1 be a real number. A representation in base (or a representation) of a real number
x 0 is an infinite sequence (xi)ki>, xi 0, such that
x = xkk
+ xk1k1
+ · · · + x1 + x0 + x11
+ x22
+ · · ·
