Computational Exploration of Mandelbrot Sets for Complex Monic Quadratic Polynomials

The Mandelbrot set is a central object in the study of complex fractals. This set consists of all complex numbers $c$ for which the orbit of point $0$ under the complex polynomial $f_c:\mathbb{C} \rightarrow \mathbb{C}, f_c(z)=z^2+c$ $(c\in \mathbb{C})$ remains bounded. In this paper, we investigate the dynamical behavior of a more general family of complex monic quadratic polynomials of the form $g_c:\mathbb{C} \rightarrow \mathbb{C}, g_c(z)=z^2+az+c$ where $a,c\in \mathbb{C}$. We present a general formula for constructing the Mandelbrot sets corresponding to these functions by analyzing the regions determined by periodic points of period $p$ and their associated centers. Furthermore, we propose algorithms to identify specific parameter values for which $\pi$ emerges within the dynamical context of the system. All computations and visualizations are implemented in Python.