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A selfsimilar tiling generated by the minimal Pisot number Shigeki Akiyama Taizo Sadahiro
 

Summary: A self­similar tiling generated by the minimal Pisot number
Shigeki Akiyama Taizo Sadahiro
Abstract
Let fi 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 fi. (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 fi ? 1 be a real number. A representation in base fi (or a fi­representation) of a real number
x – 0 is an infinite sequence (x i ) k–i?01 , x i – 0, such that
x = x k fi k + x k01 fi k01 + 1 1 1 + x 1 fi + x 0 + x01 fi 01 + x02 fi 02 + 1 1 1
for a certain integer k – 0. It is denoted by
x = x k x k01 1 1 1 x 1 x 0 :x 01 x02 1 1 1 :
A particular fi­representation -- called the fi­expansion -- can be computed by the 'greedy algorithm':
Denote by [y] and fyg the integer part and the fractional part of y. There exists k 2 Z such that

  

Source: Akiyama, Shigeki - Department of Mathematics, Niigata University

 

Collections: Mathematics