The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations
Yıl: 2020 Cilt: 44 Sayı: 2 Sayfa Aralığı: 356 - 377 Metin Dili: İngilizce DOI: 10.3906/mat-1909-68 İndeks Tarihi: 05-05-2020
The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations
Öz: To formulate the intrinsic metrics by using the code representations of the points on the classical fractalsis an important research area since these formulas help to prove many geometrical and structural properties of thesefractals. In various studies, the intrinsic metrics on the code set of the Sierpinski gasket, the Sierpinski tetrahedron, andthe Vicsek (box) fractal are explicitly formulated. However, in the literature, there are not many works on the intrinsicmetric that is obtained by the code representations of the points on fractals. Moreover, as seen in the studies on thissubject, the contraction coefficients of the associated iterated function systems (IFSs) are the same for each fractal. Inthis paper, we define the intrinsic metric formula on the added Sierpinski triangle, whose IFS has different contractionfactors, by using the code representations of the points of it. Finally, we give several geometrical properties of this fractalby using the intrinsic metric formula.
Anahtar Kelime: Konular:
Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık
- [1] Aslan N, Saltan M, Demir B. The intrinsic metric formula and a chaotic dynamical system on the code set of the Sierpinski tetrahedron. Chaos, Solitons and Fractals 2019; 123: 422-428.
- [2] Barnsley M. Fractals Everywhere. San Diego, CA, USA: Academic Press, 1988.
- [3] Barnsley MF. Superfractals. New York, NY, USA: Cambridge University Press, 2006.
- [4] Burago D, Burago Y, Ivanov S. A Course in Metric Geometry. Providence, RI, USA: AMS, 2001.
- [5] Cristea LL. A Geometric property of the Sierpinski carpet. Quaestiones Mathematicae 2005; 28: 251-262.
- [6] Cristea LL, Steinsky B. Distances in Sierpinski graphs and on the Sierpinski gasket. Aequationes Mathematicae 2013; 85: 201-219.
- [7] Denker M, Sato H. Sierpinski gasket as a Martin boundary II (the intrinsic metric). Publications of the Research Institute for Mathematical Sciences 1999; 35: 769-794.
- [8] Falconer K. Fractal Geometry: Mathematical Foundations and Applications. Hoboken, NJ, USA: John Wiley, 2004.
- [9] Grabner P, Tichy RF. Equidistribution and Brownian motion on the Sierpinski gasket. Monatshefte für Mathematik 1998; 125: 147-164.
- [10] Gu J, Ye Q, Xi L. Geodesics of higher-dimensional Sierpinski gasket. Fractals 2019; 27 (4): 1950049. doi: 10.1142/S0218348X1950049X
- [11] Gulick D. Encounters with Chaos and Fractals. Boston, MA, USA: Academic Press, 1988.
- [12] Güneri M, Saltan M. Intrinsic metric formulas on some self-similar sets via the code representation. Fractal and Fractional 2019; 3 (1): 1-13.
- [13] Hinz AM, Schief A. The average distance on the Sierpinski gasket. Probability Theory and Related Fields 1990; 87: 129-138.
- [14] Mandelbrot BB. The Fractal Geometry of Nature. San Francisco, CA, USA: W.H. Freeman and Company, 1982.
- [15] Özdemir Y. The intrinsic metric and geodesics on the Sierpinski gasket SG(3). Turkish Journal of Mathematics 2019; 43: 2741-2754. doi: 10.3906/mat-1907-18
- [16] Özdemir Y, Saltan M, Demir B. The intrinsic metric on the box fractal. Bulletin of the Iranian Mathematical Society 2018, 45 (5): 1269-1281. doi: 10.1007/s41980-018-00197-w
- [17] Romik D. Shortest paths in the Tower of Hanoi graph and finite automata. SIAM Journal on Discrete Mathematics 2006; 20: 610-622.
- [18] Saltan M. Some interesting code sets of the Sierpinski triangle equipped with the intrinsic metric. International Journal of Applied Mathematics and Statistics 2018; 57 (4): 152-160.
- [19] Saltan M. Intrinsic metrics on Sierpinski-like triangles and their geometric properties. Symmetry 2018; 10 (6): 204.
- [20] Saltan M. The relation between binary tree and the Sierpinski triangle which is equipped with the intrinsic metric. Eskişehir Technical University Journal of Science and Technology B - Theoretical Sciences 2019; 7 (1): 1-12.
- [21] Saltan M, Aslan N, Demir B. A discrete chaotic dynamical system on the Sierpinski gasket. Turkish Journal of Mathematics 2019; 43 (1): 361-372.
- [22] Saltan M, Özdemir Y, Demir B. An explicit formula of the intrinsic metric on the Sierpinski gasket via code representation. Turkish Journal of Mathematics 2018; 42: 716-725.
- [23] Saltan M, Özdemir Y, Demir B. Geodesics of the Sierpinski gasket. Fractals 2018; 26 (3): 1850024.
- [24] Strichartz RS. Isoperimetric estimates on Sierpinski gasket type fractals. Transactions of the American Mathematical Society 1999; 351: 1705-1752.
- [25] Zhang Z, Zhou S, Chen L, Yin M, Guan J. The exact solution of the mean geodesic distance for Vicsek fractals. Journal of Physics A: Mathematical and Theoretical 2008; 41: 485102.
APA | İKLİM ŞEN A, SALTAN M (2020). The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations. , 356 - 377. 10.3906/mat-1909-68 |
Chicago | İKLİM ŞEN Aslıhan,SALTAN MUSTAFA The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations. (2020): 356 - 377. 10.3906/mat-1909-68 |
MLA | İKLİM ŞEN Aslıhan,SALTAN MUSTAFA The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations. , 2020, ss.356 - 377. 10.3906/mat-1909-68 |
AMA | İKLİM ŞEN A,SALTAN M The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations. . 2020; 356 - 377. 10.3906/mat-1909-68 |
Vancouver | İKLİM ŞEN A,SALTAN M The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations. . 2020; 356 - 377. 10.3906/mat-1909-68 |
IEEE | İKLİM ŞEN A,SALTAN M "The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations." , ss.356 - 377, 2020. 10.3906/mat-1909-68 |
ISNAD | İKLİM ŞEN, Aslıhan - SALTAN, MUSTAFA. "The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations". (2020), 356-377. https://doi.org/10.3906/mat-1909-68 |
APA | İKLİM ŞEN A, SALTAN M (2020). The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations. Turkish Journal of Mathematics, 44(2), 356 - 377. 10.3906/mat-1909-68 |
Chicago | İKLİM ŞEN Aslıhan,SALTAN MUSTAFA The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations. Turkish Journal of Mathematics 44, no.2 (2020): 356 - 377. 10.3906/mat-1909-68 |
MLA | İKLİM ŞEN Aslıhan,SALTAN MUSTAFA The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations. Turkish Journal of Mathematics, vol.44, no.2, 2020, ss.356 - 377. 10.3906/mat-1909-68 |
AMA | İKLİM ŞEN A,SALTAN M The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations. Turkish Journal of Mathematics. 2020; 44(2): 356 - 377. 10.3906/mat-1909-68 |
Vancouver | İKLİM ŞEN A,SALTAN M The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations. Turkish Journal of Mathematics. 2020; 44(2): 356 - 377. 10.3906/mat-1909-68 |
IEEE | İKLİM ŞEN A,SALTAN M "The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations." Turkish Journal of Mathematics, 44, ss.356 - 377, 2020. 10.3906/mat-1909-68 |
ISNAD | İKLİM ŞEN, Aslıhan - SALTAN, MUSTAFA. "The formulization of the intrinsic metric on the added Sierpinski triangle by using the code representations". Turkish Journal of Mathematics 44/2 (2020), 356-377. https://doi.org/10.3906/mat-1909-68 |