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

Saurabh Sharma, Anita Tomar and Sanjay Kumar Padaliya

Chaos, Solitons & Fractals, 2025, vol. 191, issue C

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).

Keywords: Average Escape Time; Non-Escaping Area Index; Average Number of Iterations; Fibonacci sequence; Fractal (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:191:y:2025:i:c:s0960077924014036

DOI: 10.1016/j.chaos.2024.115851

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