Inversions and Fractal Patterns in Alpha Plane

Year 2023, Volume: 16 Issue: 1, 398 - 411, 30.04.2023

Özcan Gelişgen , Temel Ermiş

https://doi.org/10.36890/iejg.1244520

Cited By: 3

Abstract

In this paper, we introduce the alpha circle inversion by using alpha distance function instead of Euclidean distance in definition of classical inversion. We give some proporties of alpha circle inversion. Also this new transformation is applied to well known fractals. Then new fractal patterns are obtained. Moreover we generalize the method called circle inversion fractal be means of the alpha circle inversion. In alpha plane, we give a generalization of alpha circle inversion fractal by using the concept of star-shaped set inversion which is a generalization of circle inversion fractal.

Keywords

Alpha circle inverion, Alpha distance, Limit set, Fractal, Star-shaped set, Star-shaped set inversion

References

• [1] Barnsley, M.: Fractals Everywhere. Academic Press, Boston (1988).
• [2] Bayar, A., Ekmekci, S.: On circular inversions in taxicab plane. J. Adv. Res. Pure Math. 6 (4), 33-39 (2014).
• [3] Blair, D.: Inversion Theory and Conformal Mapping. Student Mathematical Library, American Mathematical Society, 9 (2000).
• [4] Boreland, B., Kunze, H.: Circle Inversion Fractals. In: Belair, J., Frigaard, I., Kunze, H., Makarov, R., Melnik, R., Spiteri, R. (eds) Mathematical and Computational Approaches in Advancing Modern Science and Engineering, Springer, Cham. 609-619 (2016).
• [5] Brown, C. T.,Witschey,W. R. T. and Liebovitch, L. S.: The Broken past: Fractals in Archaeology. Journal of Archaeological Method and Theory. 12 (1), 37-78 (2005).
• [6] Childress, N.: Inversion with respect to the central conics. Mathematics Magazine. 38 (3), 147-149 (1965).
• [7] Clancy, C. and Frame M.: Fractal geometry of restricted sets of circle inversions. Fractals. 3 (4), 689-699 (1995).
• [8] Colakoglu, H. B.: Concerning the alpha distance. Algebras Groups Geom. 8, 1-14 (2011).
• [9] Frame, M., Cogevina, T.: An infinite circle inversion limit set fractal. Comput. Graph. 24(5), 797-804 (2000).
• [10] Fitzsimmons, M., Kunze, H.: Circle Inversion IFS. Springer Proceedings in Mathematics & Statistics. 259, 81-91 (2018).
• [11] Gdawiec, K.: Star-shaped set inversion fractals. Fractals. 22 (4), 1450009-1-1450009-7 (2014).
• [12] Gdawiec, K.: Pseudoinversion Fractals. Lecture Notes in Computer Science. 9972, 29-36 (2016).
• [13] Gdawiec, K.: Inversion Fractals and Iteration Processes in the Generation of Aesthetic Patterns. Computer Graphics Forum. 36 (1), 35-45 (2017).
• [14] Gelişgen, O., & Kaya, R.: On αi−distance in Three Dimensional Space. Applied Sciences (APPS). 8, 65-69 (2006).
• [15] Gelişgen, O., & Kaya, R.: Generalization of α−distance to ndimensional Space. Scientific-Professional Information Journal of Croatian Society for Constructive Geometry and Computer Graphics (KoG). 10, 33-35 (2006).
• [16] Helt, G.: Inversive Diversions and Diversive Inversions. Bridges Conference Proceedings, 467-470 (2017).
• [17] Kozai, K., Libeskind, S.: Circle Inversions and Applications to Euclidean Geometry. online supplement to Euclidean and Transformational Geometry: A Deductive Inquiry.
• [18] Leys, J.: Sphere inversion fractals. Comput. Graph. 29 (3), 463-466 (2005).
• [19] Losa, G. A., Merlini, D., Nonnenmacher, T. F., Weibel E. R.: Fractals in Biology and Medicine. Birkhäuser Basel (1998).
• [20] Mandelbrot, B.: The Fractal Geometry of Nature. W. H. Freeman and Company, New York (1983).
• [21] Nickel, J. A.: A Budget of inversion. Math. Comput. Modelling. 21 (6), 87-93 (1995).
• [22] Ostwald, M.J.: Fractal Architecture: Late Twentieth Century Connections Between Architecture and Fractal Geometry. Nexus Netw J. 3, 73-84 (2001).
• [23] Patterson, B. C.: The origins of the geometric princible of inversion. Isis. 19 (1), 154-180 (1933).
• [24] Ramirez, J. L.: Inversions in an Ellipse. Forum Geometricorum. 14, 107-115 (2014).
• [25] Ramirez, J. L. G., Rubiano N. & Zlobec, B. J.: Generating Fractal Patterns by Using p−Circle Inversion. Fractals. 23 (4), 1550047-1-1550047-13 (2015).
• [26] Smith, R.: Fractal producing iterative mapping systems on circles. M.S. thesis, University of Newcastle, Australia (2010).
• [27] Tian, S.: Alpha Distance-A Generalization of Chinese Checker Distance and Taxicab Distance. Missouri J. of Math. Sci. (MJMS). 17 (1), 35-40 (2005).
• [28] Zhang, Y., He, X.: Fractal geometry derived from geometric inversion. Comput. Graph. 11 (11), 1075-1079 (1990).