Higher Order Mandelbrot Sets and their Varying Shapes
Arshdeep Singh Pareek

TL;DR
This paper explores how the shape and structure of Mandelbrot sets change with higher degree polynomials, revealing predictable patterns in their lobes and shapes as the degree increases.
Contribution
It extends the analysis of Mandelbrot sets to higher degree polynomials, establishing relationships between polynomial degree and the set's shape and lobes.
Findings
Number of primary lobes is n-1 for degree n polynomials.
As degree n tends to infinity, the primary lobe approaches a circle of unit radius.
Arguments of secondary lobes are roots of unity.
Abstract
Mandelbrot set arose from the pioneering work of French mathematician Gaston Julia in the field of complex dynamics at the beginning of the 20th century. French-American mathematician Benoit Mandelbrot used computers to calculate iterations of complex polynomials of second order and displayed intricate images of fractal geometry. While studying fundamental properties of the Mandelbrot set, little attention has been paid to study the relationship between the degree of the generating polynomials and the shape of the main body of the Mandelbrot set. This paper extends the work from generating polynomials of second degree to polynomials of higher degrees using basic principles of complex numbers and calculus and shows that the number of primary lobes in the Mandelbrot set corresponding to a polynomial of nth degree is n-1 and as n tends to infinity the shape of the primary lobe tends to a…
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Taxonomy
TopicsMathematical Dynamics and Fractals
