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

Ali Şen ,Mustafa Saltan

doi:10.3906/mat-1909-68

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.

Highlights

The Cantor set, Sierpinski gasket, Koch curve, Sierpinski carpet, Vicsek fractal, Sierpinski tetrahedron, and Menger sponge are some of the fundamental examples of the classical fractals [2, 3, 8, 11, 14]
There have been different ways to define the intrinsic metrics on the classical fractals such as the classical Sierpinski Gasket, the discrete Sierpinski Gasket, the Sierpinski Carpet, and the Vicsek fractals in the last two decades
The intrinsic metric formulas are reformulated on the code set of the equilateral Sierpinski propeller, which is selfsimilar but not strong self-similar, in [12]

Summary

Introduction

The Cantor set, Sierpinski gasket, Koch curve, Sierpinski carpet, Vicsek fractal, Sierpinski tetrahedron, and Menger sponge are some of the fundamental examples of the classical fractals [2, 3, 8, 11, 14]. To compute the length of the shortest paths between points A and B passing through the line segment (Sσak ∩ Sσ0)(Sσbk ∩ Sσ0) , which is an edge of Sσ0 , we must take into account the following cases: i) Let ak+1 ̸= ak and ak+1 ̸= 0. Note that we use the appropriate value A′′ given in Case 3 to compute the length of the shortest paths between points A and Sσak ∩ Sσ0. Note that if min{i | ai= 0, ai= ak, i ≥ k + 2} = ∅, for the computation the length of the shortest paths between A and (Sσak ∩ Sσ0) we add in each step the edge lengths of the related subtriangles where the shortest paths pass through In this case, we get φi = 1 for i = k + 2, k + 3, k + 4,. Proof The code representation of A is σakakak . . . where ak ∈ {1, 2, 3} since A is any vertex point of Sσ

We first get

Conclusion