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

Abstract

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 their centers coincide with the vertices of a five-pointed star. This star is called a “generalized star.” The size of the generalized star increases τ times at each step, where τ ≈ 1.618 is the golden mean. The orientation of the generalized star reverses at each step. A comparison of the internal parts of the fractal with the layers of the Wieringa roof (layers of Penrose tiling) shows their similarity. Computer simulation confirms that the two-dimensional Penrose tiling can be constructed from a set of four types of fractal ⁱ0w(1b).