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).

中文翻译：

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关于斐波那契数列在分形可视化中的演变和重要性


斐波那契数列是一个引人入胜的数学概念，在各个学科中具有深远的意义。除了理论上的阴谋之外，它还在艺术、建筑、自然和金融市场中找到了实际应用。由每个数字定义的斐波那契数列是前两个数字（0、1、1、2、3、5、8、13 等）的总和。这个数字序列通常被认为是 12 世纪意大利数学家比萨的莱昂纳多 （Leonardo of Pisa），被称为斐波那契，其早期起源于印度数学。Acharya Pingala 在几个世纪前的著作中引用了这个序列。本文探讨了斐波那契数列的演变及其现代应用，特别是在分形几何中。我们研究了曼德布洛特集和朱莉娅集的各种函数，并研究了使用斐波那契-曼轨道获得的曼德布洛特集和朱莉娅集的对称性。此外，我们还研究了参数 a 对 Mandelbrot 和 Julia 集的影响。为了量化这些影响，我们采用了三种措施：平均逃逸时间 （AET）、非逃逸区域指数 （NAI） 和平均迭代次数 （ANI）。