Constructing the penrose tiling by means of the five-pointed stars fractal

A. A. Polyakov

ABSTRACT:

The fractal i0w(1b) (in author’s notation) of five-pointed stars is considered. The initiator  of the fractal is a five-pointed star.  When building the fractal the prefractals of previous step  are situated so as its centers coincide with the  vertices  of  a  five-pointed  star.  This  star  is named ‘generalized star’. The size of the generalized star increases in τ times at each step, where τ ≈ 1.618 is the golden mean. The orientation of generalized star reverses at each step. Comparison of the internal parts of the fractal with the layers of Wieringa roof (layers of Pen-  rose tiling) showed their similarity. Computer simulation confirmed that the two-dimensional Penrose tiling can be constructed by a set of four types of the fractal i0w(1b).

KEYWORDS:

 Penrose tiling, fractals, Wieringa roof, quasiperiodic lattice,  quasicrystals.