The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations

Abstract

To formulate the intrinsic metrics by using the code representations of the points on the classical fractals is an important research area since these formulas help to prove many geometrical and structural properties of these fractals. In various studies, the intrinsic metrics on the code set of the Sierpinski gasket, the Sierpinski tetrahedron, and the Vicsek (box) fractal are explicitly formulated. However, in the literature, there are not many works on the intrinsic metric that is obtained by the code representations of the points on fractals. Moreover, as seen in the studies on this subject, the contraction coefficients of the associated iterated function systems (IFSs) are the same for each fractal. In this paper, we define the intrinsic metric formula on the added Sierpinski triangle, whose IFS has different contraction factors, by using the code representations of the points of it. Finally, we give several geometrical properties of this fractal by using the intrinsic metric formula.