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Dergiler / 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
Sayfa
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Yayın Tarihi
2026-02-23
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

All computations and visualizations are implemented in Python.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:The Mandelbrot set is a central object in the study of complex fractals.
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Temel Bulgular & Veri: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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Sonuç & Katkı:All computations and visualizations are implemented in Python.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler: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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Bu Makaleye Atıf Yap

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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Kaynakça (10)

  1. Barnsley, M., Fractals Everywhere, Dover Publication, New York, USA, 2012.
  2. Danca, M.F., Mandelbrot set as a particular Julia set of fractional order, equipotential lines and external rays of Mandelbrot and Julia sets of fractional order, Fractal Fract., 8(1)(2024), 69.
  3. Devaney R.L., An Introduction to Chaotic Dynamical Systems, CRC Press, New York, USA, 2022.
  4. Gulick, D., Encounters with Chaos and Fractals, CRC Press, Boca Raton, FL, USA, 2012.
  5. Klebanoff, A., π in the Mandelbrot set, Fractals, 9(4)(2001), 393–402.
  6. Lindsey, K., Tiozzo, G., Wu, C., Master teapots and entropy algorithms for the Mandelbrot set, Trans. Amer. Math. Soc., 378 (2025), 3297- 3348.
  7. Liu S., Xu, X., Srivastava, G., Srivastava, H.M., Fractal properties of the generalized Mandelbrot set with complex exponent, Fractals, 32 (04)(2024), 2340121.
  8. Mandelbrot, B.B., The Fractal Geometry of Nature, W.H. Freeman and Company, New York, 1983.
  9. Peitgen, H.O., J¨urgens, H., Saupe, D., Chaos and Fractals, Springer, Berlin/Heidelberg, Germany, 2004.
  10. Roy, S., Gdawiec, K., Saha, P., Choudhury, B.S., Exploring fractal geometry through Das–Debata iteration: A new perspective on Mandelbrot and Julia sets, Chaos, Solitons, Fractals, 199(1) (2025), 116617.