Applications of Cantor Set to Fractal Geometry

Year 2024, Volume: 17 Issue: 2, 712 - 726, 27.10.2024

İpek Ebru Karaçay , Salim Yüce

https://doi.org/10.36890/iejg.1536179

Abstract

Fractal geometry is a subfield of mathematics that allows us to explain many of the complexities in nature. Considering this remarkable feature of fractal geometry, this study examines the Cantor set, which is one of the most basic examples of fractal geometry. First of all for the Cantor set, which is one of the basic example and important structure of it. Firstly, generalization of Cantor set in on ${\mathbb{R}}$, ${\mathbb{R}}^2$ and ${\mathbb{R}^3}$ are taken into consideration. Then the given structures are examined over curve and surface theory. This approach enables to given a relationship between fractal geometry and differential geometry. Finally, some examples are established.

Keywords

Fractal geometry, Cantor set, iterated function system

References

• [1] Abbena, E., Salamon, S., Gray, A.: Modern differential geometry of curves and surfaces with Mathematica. Chapman and Hall/CRC, (2017).
• [2] Barnsley, M.F., Demko, S.: Iterated function systems and the global construction of fractals. Proc.R.Soc. London, A 399, (243-275), (1985) .
• [3] Brockett, R. W.: Robotic manipulators and the product of exponentials formula. In Mathematical Theory of Networks and Systems: Proceedings of the MTNS-83 International Symposium Beer Sheva, Israel, June 20–24, 1983 (pp. 120-129). Berlin, Heidelberg: Springer Berlin Heidelberg (2005, November).
• [4] Cantor, G.:Uber unendliche, lineare Punktmannigfaltigkeiten. V. Mathematische Annalen, 21, (1883).
• [5] Edgar, G.A.: Measure, Topology and Fractal Geometry. Springer-Verlag, Newyork (1990).
• [6] Islam, J., Islam S.: Generalized Cantor Set and its Fractal Dimension. Bangladesh J. Sci.Ind. Res., 46(4), 499-506, (2011).
• [7] Islam J., Islam S.: Invariant measures for Iterated Function System of Generalized Cantor Sets. German J. Ad. Math. Sci., 1(2) 41-47, (2016).
• [8] Islam J., Islam S.: Lebesgue Measure of Generalized Cantor Set. Annals of Pure and App. Math., 10(1), 75-86, (2015).
• [9] Mandelbrot B.B.: Fractal Geometry of Nature. W. H. Freeman and Company ISBN 0-7167-1186-9, (1983).
• [10] Murray, R. M., Li, Z., Sastry, S. S.: A mathematical introduction to robotic manipulation. CRC press, (2017).
• [11] Yüce, S.: Analytical geometry (in Turkish). Pegem Academy Publication, (2023).
• [12] Yüce, S.: Differential geometry in Euclidean space (in Turkish). Pegem Academy Publication, (2022).