Generation of symmetrical patterns using discrete dynamical system

Document Type : Promotional Paper

Author

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, P.O.Box 8731753153, Kashan, Iran

Abstract

Nowadays symmetrical patterns are widely used in various industries, such as jewelry design, carpet design, patterns on wallpaper, and textile design. During the design stage, designers perform most of the work manually. Therefore, the development of methods for symmetrical pattern generation is beneficial. In this paper, we are going to present some methods for generating symmetrical patterns using the discrete dynamical system. For this, the discrete dynamical system is considered as a standard iterative method, and then the general algorithm applied for Polynomiography is used for this to generate patterns. Through phase portrait, we analyze the conditions of the existence of some conventional symmetries. Several non-standard iterative methods can be employed to create a variety of visually appealing patterns. These methods include Mann iteration, Ishikawa iteration, and S-iteration which we use them. Through numerous examples, it is demonstrated that by manipulating the parameters and coefficients, it is possible to generate beautiful symmetrical patterns that have potential artistic applications.

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References

[1] M. Bisheh-Niasar, The effect of the caputo fractional derivative on polynomiography, Math. Interdisc. Res., 8 no. 4 (2023) 347–358.
[2] M. Bisheh-Niasar and K. Gdawiec, Bisheh-Niasar–Saadatmandi root finding method via the S-iteration with periodic parameters and its polynomiography, Math. Comput. Simulation, 160 (2019) 1–12.
[3] M. Bisheh-Niasar, polynomiography, a combination of art and mathematic, Journal of Culture and Mathematical Thinking, 38 (2019) 159–166. [In Persian]
[4] K. W. Chung and H. S. Y. Chan, Symmetrical patterns from dynamics, Computer Graphics Forum, 12 no. 1 (1993) 33–40.
[5] K. Gdawiec, K. Wies law and L. Agnieszka, Polynomiography based on the nonstandard Newton-like root finding methods, Abstr. Appl. Anal., 2015 no. 4 (2015) 19 pp.
[6] K. Gdawiec, Polynomiography and various convergence tests, 21st International conference on computer graphics, in Proceeding sof the WSCG Communication, (2013) 15–20.
[7] K. Gdawiec, Procedural generation of aesthetic patterns from dynamics and iteration processes, Int. J. Appl. Math. Comput. Sci., 27 no. 4 (2017) 827–837.
[8] K. Gdawiec, W. Kotarski and A. Lisowska, Visual analysis of the Newton’s method with fractional order derivatives, Symmetry, 11 no. 9 (2019) pp. 27.
[9] B. Kalantari, Polynomiography and applications in art, education and science, Comput. Graph., 28 no. 3 (2004) 417–430
[10] B. Kalantari, Polynomial root-rinding and rolynomiography, World Scientific, Singapore, 2009.

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History

  • Receive Date: 07 April 2024
  • Revise Date: 30 July 2024
  • Accept Date: 03 August 2024
  • Publish Date: 17 April 2025

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