Computational Exploration of Mandelbrot Sets for Complex Monic Quadratic Polynomials
Abstract
The Mandelbrot set is a central object in the study of complex fractals. This set consists of all complex numbers $c$ for which the orbit of point $0$ under the complex polynomial $f_c:\mathbb{C} \rightarrow \mathbb{C}, f_c(z)=z^2+c$ $(c\in \mathbb{C})$ remains bounded. In this paper, we investigate the dynamical behavior of a more general family of complex monic quadratic polynomials of the form $g_c:\mathbb{C} \rightarrow \mathbb{C}, g_c(z)=z^2+az+c$ where $a,c\in \mathbb{C}$. We present a general formula for constructing the Mandelbrot sets corresponding to these functions by analyzing the regions determined by periodic points of period $p$ and their associated centers. Furthermore, we propose algorithms to identify specific parameter values for which $\pi$ emerges within the dynamical context of the system. All computations and visualizations are implemented in Python.
Keywords
Project Number
1919B012203250
References
- Barnsley, M., Fractals Everywhere, Dover Publication, New York, USA, 2012.
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Details
Primary Language
English
Subjects
Numerical Computation and Mathematical Software, Real and Complex Functions (Incl. Several Variables)
Journal Section
Research Article
Publication Date
February 23, 2026
Submission Date
August 29, 2025
Acceptance Date
December 9, 2025
Published in Issue
Year 2026 Volume: 18 Number: 1
APA
Demir, İ., & Saltan, M. (2026). Computational Exploration of Mandelbrot Sets for Complex Monic Quadratic Polynomials. Turkish Journal of Mathematics and Computer Science, 18(1), 267-280. https://doi.org/10.47000/tjmcs.1772953
AMA
1.Demir İ, Saltan M. Computational Exploration of Mandelbrot Sets for Complex Monic Quadratic Polynomials. TJMCS. 2026;18(1):267-280. doi:10.47000/tjmcs.1772953
Chicago
Demir, İbrahim, and Mustafa Saltan. 2026. “Computational Exploration of Mandelbrot Sets for Complex Monic Quadratic Polynomials”. Turkish Journal of Mathematics and Computer Science 18 (1): 267-80. https://doi.org/10.47000/tjmcs.1772953.
EndNote
Demir İ, Saltan M (February 1, 2026) Computational Exploration of Mandelbrot Sets for Complex Monic Quadratic Polynomials. Turkish Journal of Mathematics and Computer Science 18 1 267–280.
IEEE
[1]İ. Demir and M. Saltan, “Computational Exploration of Mandelbrot Sets for Complex Monic Quadratic Polynomials”, TJMCS, vol. 18, no. 1, pp. 267–280, Feb. 2026, doi: 10.47000/tjmcs.1772953.
ISNAD
Demir, İbrahim - Saltan, Mustafa. “Computational Exploration of Mandelbrot Sets for Complex Monic Quadratic Polynomials”. Turkish Journal of Mathematics and Computer Science 18/1 (February 1, 2026): 267-280. https://doi.org/10.47000/tjmcs.1772953.
JAMA
1.Demir İ, Saltan M. Computational Exploration of Mandelbrot Sets for Complex Monic Quadratic Polynomials. TJMCS. 2026;18:267–280.
MLA
Demir, İbrahim, and Mustafa Saltan. “Computational Exploration of Mandelbrot Sets for Complex Monic Quadratic Polynomials”. Turkish Journal of Mathematics and Computer Science, vol. 18, no. 1, Feb. 2026, pp. 267-80, doi:10.47000/tjmcs.1772953.
Vancouver
1.İbrahim Demir, Mustafa Saltan. Computational Exploration of Mandelbrot Sets for Complex Monic Quadratic Polynomials. TJMCS. 2026 Feb. 1;18(1):267-80. doi:10.47000/tjmcs.1772953