Geodesics of the Sierpinski Gasket

Abstract

In this paper, we examine the number of geodesics between two points of the Sierpinski Gasket ([Formula: see text]) via code representations of the points and as a main result we show that the maximum number of geodesics between different two points with respect to the intrinsic metric on [Formula: see text] is five.

The Relation Between Binary Tree and the Sierpinski Triangle Which Is Equipped With the Intrinsic Metric

“…Kendine benzer yapılar üzerindeki içsel metrikler, farklı şekillerde ifade edilebilmektedir. Örneğin, Sierpinski üçgenini tanımlamanın farklı yolları olduğundan dolayı bu metrik farklı yollardan ifade edilebilir [5][6][7][8][9][10][11][12]. Bu kümenin noktalarının kod temsilleri kullanılarak açık olarak formüle edilen içsel metrik, [8] de tanımlanmaktadır.…”

“…Bu metrik formülü sayesinde Sierpinski üçgeninin birçok önemli özelliği ispatlanmaktadır. Ayrıca, [12] de Sierpinski üçgeninin jeodezileri sınıflandırılmaktadır. [13] de ise Sierpinski üçgeninin topolojik özellikleri incelenmekte, [14] de Sierpinski üçgeninin çeşitkenar ve ikizkenar üçgen üzerinde inşa edildiğinde bu formülün genelleştirilmesi verilmekte ve bu üçgenlerin geometrik özellikleri karşılaştırılmaktadır.…”

The intrinsic metric and geodesics on the Sierpinski gasketSG(3)

“…In [6], Cristea gives a formula of the intrinsic metric on the Sierpinski carpet by using carpet coordinates and show the equivalence of the Euclidean metric and the intrinsic metric on the Sierpinski carpet. In [13,15,16], the authors use code representations of the points of the classical Sierpinski gasket and Vicsek fractal to express the intrinsic metric and classified the geodesics on the related set. They prove that there exist at most 5 geodesics between two points in the Sierpinski gasket SG (2) .…”

Intrinsic Metric Formulas on Some Self-Similar Sets via the Code Representation

“…In [12], the intrinsic metric on the code set of the Sierpinski gasket, which is one of the instructive examples of strictly self-similar sets, is formulated by the code representations of its points. Due to this metric formula, important geometrical and topological properties of the Sierpinski gasket are expressed by the code sets, the number of geodesics are determined and the code representations of points are classified according to the number of geodesics (for details, see [13][14][15]). As seen in these studies, defining the intrinsic metrics by using the code representations of the points on fractals provides some facilities for different works.…”