Computational Study of Classical Fractals and Newton Basins of Attraction in The Complex Plane

Siti Sahrani; Salmawaty Salmawaty; Said  Munzir; Saiful Amri

doi:10.22373/ekw.v11i2.25438

Authors

Siti Sahrani Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Syiah Kuala, Banda Aceh, Indonesia
Salmawaty Salmawaty Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Syiah Kuala, Banda Aceh, Indonesia
Said Munzir Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Syiah Kuala, Banda Aceh, Indonesia
Saiful Amri Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Syiah Kuala, Banda Aceh, Indonesia

DOI:

https://doi.org/10.22373/ekw.v11i2.25438

Keywords:

Basin of attraction, Fractal, Julia Set, Mandelbrot Set, Newton Method, Sierpinski Gasket

Abstract

Abstract: Fractal sets such as the Sierpinski Gasket, the Julia set, the Mandelbrot, and the Newton basin attraction provide rich examples of self-similar structures arising from simple iterative rules. In this work, we present a computational study of these fractals with emphasis on parameter sensitivity and basin dynamics. Using C++, we generated classical fractals (Sierpinski, Julia, and Mandelbrot) and verified their self-similar properties through coordinate transformations. For the Newton basin attraction of the polynomial, implemented in MATLAB, we analyzed the effect of the exponent, complex constant, and iteration depth on convergence regions. The results show that the number of basin attractions corresponds to the polynomial degree, with clear rotational symmetries, while the parameter controls the orientation and distribution of convergence regions. Furthermore, the boundary of the Newton basin attraction was observed to align with Julia sets, highlighting the intrinsic connection between these two fractal structures. This study demonstrates how computational visualization provides analytical insight into the structure and parameter dependence of fractal systems, establishing a systematic framework for parameter-based fractal analysis that bridges classical visualization with numerical dynamics.

Abstrak: Himpunan fraktal seperti segitiga Sierpinski, Julia, Mandelbrot, dan Newton basin atraksi merupakan representasi khas dari struktur self-similarity yang terbentuk melalui aturan iteratif sederhana. Penelitian ini menyajikan kajian komputasional terhadap fraktal-fraktal tersebut dengan fokus pada sensitivitas parameter serta dinamika basin atraksi. Algoritma fraktal klasik diimplementasikan menggunakan C++ untuk menghasilkan pola Sierpinski, Julia, dan Mandelbrot sekaligus memverifikasi sifat swasimilaritasnya melalui transformasi koordinat. Sementara itu, Newton basin atraksi dari polinomial dianalisis menggunakan MATLAB untuk mengkaji pengaruh eksponen, konstanta kompleks, dan kedalaman iterasi terhadap daerah konvergensi. Hasil penelitian memperlihatkan bahwa jumlah basin atraksi sebanding dengan derajat polinomial dan menampilkan simetri rotasional yang khas, sedangkan parameter mempengaruhi orientasi serta distribusi wilayah konvergensi. Selain itu, batas Newton basin atraksi ditemukan berimpit dengan himpunan Julia, menegaskan keterkaitan mendasar antara kedua struktur fraktal tersebut. Penelitian ini menunjukkan bahwa visualisasi komputasional mampu memberikan pemahaman analitis terhadap struktur dan pengaruh parameter pada sistem fraktal. Selain itu, penelitian ini juga membangun kerangka kerja sistematis untuk analisis fraktal berbasis parameter, yang menghubungkan visualisasi klasik dengan dinamika numerik.

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