A Geometric Exploration of Generalized Mandelbrot Sets

Abstract

The exploration of generalized Mandelbrot sets offers insight into the deep and elegant connections between complex numbers and geometry. By plotting generalized Mandelbrot sets of the form f(z) = z^j + c, where j is an arbitrary number, and running simulations using Python tools such as matplotlib and numpy, along with Google Colab, I was able to derive several conjectures and test them empirically. The problem-solving process throughout this project proved essential in guiding both the formation and proof of these conjectures. Key properties analyzed include graphical relationships between polynomial degree and area, rotational and reflectional symmetries, and petal-like structures. This research contributes to the field of complex dynamics by expanding the classical Mandelbrot set framework and addressing open questions regarding symmetry and convergence behavior in higher-degree systems.