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Journals / Turkish Journal of Mathematics and Computer Science / 2026 / Volume: 18 Issue: 1

Computational Exploration of Mandelbrot Sets for Complex Monic Quadratic Polynomials

İbrahim Demir(ESKİŞEHİR TEKNİK ÜNİVERSİTESİ, FEN BİLİMLERİ ENSTİTÜSÜ)0009-0005-5444-9035
Mustafa Saltan(ESKİŞEHİR TEKNİK ÜNİVERSİTESİ, FEN FAKÜLTESİ, MATEMATİK BÖLÜMÜ)0000-0002-3252-3012
Pages
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Publication Date
2026-02-23
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

All computations and visualizations are implemented in Python.

📌 3 Key Takeaways

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Objective & Hypothesis:The Mandelbrot set is a central object in the study of complex fractals.
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Key Findings & Data: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.
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Conclusion & Impact:All computations and visualizations are implemented in Python.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Mandelbrot set · complex fractals · periodic points

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.

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Cite This Article

Demir, İ., Saltan, M. (2026). Computational Exploration of Mandelbrot Sets for Complex Monic Quadratic Polynomials. Turkish Journal of Mathematics and Computer Science, Volume: 18 Issue: 1. https://doi.org/10.47000/tjmcs.1772953

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