Geodesics of Higher-Dimensional Sierpinski Gasket

Abstract

It is of great interest to analyze geodesics in fractals. We investigate the structure of geodesics in [Formula: see text]-dimensional Sierpinski gasket [Formula: see text] for [Formula: see text], and prove that there are at most eight geodesics between any pair of points in [Formula: see text]. Moreover, we obtain that there exists a unique geodesic for almost every pair of points in [Formula: see text].

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

“…In the classical Sierpinski gasket ( SG(2)) case, it is shown that the number of geodesics between two points can be 1, 2, 3, 4, or at most 5 (see [16]). In [9], the authors generalized the previous result to the higher dimensional case and proved that the number of geodesics between two points can be 1, 2, 3, 4, 5, 6, or at most 8 on the ( n -dimensional) Sierpinski gasket (on the Sierpinski Tetrahedron for example). However, on the Sierpinski gasket SG(3) , we prove that the number of geodesics between two points can be infinitely many (see between a and t σ ̸ = a is either one of the numbers 2 n , 3 • 2 n for some n ∈ N * or ∞ .…”

“…In several works, the intrinsic metric on the self-similar sets such as classical Sierpinski gasket, Vicsek fractal, and Sierpinski carpet was constructed and defined by using different techniques (see [7][8][9][10][11]). Strichartz defines the intrinsic metric via barycentric coordinates (for details see [17]).…”

“…In [9], the authors investigate the geodesics on the m-dimensional (classical) Sierpinski gasket (for m > 2 ) and prove that there exist at most 8 geodesics between two points.…”

The Intrinsic Metric on the Scale Irregular Sierpinski Triangle SG(2,3)

“…[6] ve [7] çalışmalarında detayları bulunabilecek olan bu açık ifade yardımı ile bu küme üzerindeki farklı iki nokta arasında en fazla 5 farklı jeodezik olduğu da kanıtlanmıştır. [17] [8] çalışmasında açıkça ifade edilmiştir. İlgili çalışmada bu küme üzerindeki içsel metriğin daha rahat ifade edilebilmesi için uygun (daha doğrusu kümeye has) bir kodlama (1/3 benzerlik oranına sahip 6 fonksiyon uygun bir şekilde indislenerek) kullanılmış olup (Şekil 7b-Şekil 7c), bu küme üzerindeki farklı iki nokta arasında,…”

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

“…For example, the code representations of the points on the Sierpinski gasket according to the number of geodesics is also classified in [23]. Then the intrinsic metric formula is given for the n -dimensional Sierpinski gasket and thus the number of geodesics on them are investigated in [10]. 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].…”