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On the evolution and importance of the Fibonacci sequence in visualization of fractals

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  • Sharma, Saurabh
  • Tomar, Anita
  • Padaliya, Sanjay Kumar

Abstract

The Fibonacci sequence is a fascinating mathematical concept with profound significance across various disciplines. Beyond theoretical intrigue, it finds practical applications in art, architecture, nature, and financial markets. The Fibonacci sequence, defined by each number is the sum of the two preceding ones (0, 1, 1, 2, 3, 5, 8, 13, …). This sequence of numbers, often attributed to the 12th century Italian mathematician Leonardo of Pisa, known as Fibonacci, has earlier roots in Indian mathematics. Acharya Pingala referenced this sequence in his work centuries before. This paper explores the evolution of the Fibonacci sequence and its modern applications, particularly in fractal geometry. We examine Mandelbrot and Julia sets for various functions and study the symmetries of the Mandelbrot and Julia sets obtained using the Fibonacci–Mann orbit. Additionally, we investigate the impact of parameter a on the Mandelbrot and Julia sets. To quantify these effects, we employ three measures: Average Escape Time (AET), Non-Escaping Area Index (NAI), and Average Number of Iterations (ANI).

Suggested Citation

  • Sharma, Saurabh & Tomar, Anita & Padaliya, Sanjay Kumar, 2025. "On the evolution and importance of the Fibonacci sequence in visualization of fractals," Chaos, Solitons & Fractals, Elsevier, vol. 191(C).
    • Handle: RePEc:eee:chsofr:v:191:y:2025:i:c:s0960077924014036
    DOI: 10.1016/j.chaos.2024.115851
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    References listed on IDEAS

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    1. William Goetzmann, 2003. "Fibonacci and the Financial Revolution," Yale School of Management Working Papers ysm432, Yale School of Management, revised 01 Mar 2004.
    2. Rawat, Shivam & Prajapati, Darshana J. & Tomar, Anita & Gdawiec, Krzysztof, 2024. "Generation of Mandelbrot and Julia sets for generalized rational maps using SP-iteration process equipped with s-convexity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 148-169.
    3. William Goetzmann, 2003. "Fibonacci and the Financial Revolution," Yale School of Management Working Papers ysm432, Yale School of Management, revised 01 Mar 2004.
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