Higher Order Mandelbrot Sets and their Varying Shapes

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 circle of unit radius. It is also shown that the arguments of the points where secondary lobes are attached to the primary lobes are given by the roots of unity. These results provide an easy way to predict shapes of the main lobes of Mandelbrot sets and the locations of the points where secondary lobes are attached to the main lobe and may be helpful in understanding features of the Mandelbrot set.