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 fc:C → C, fc(z)=z²+c (c∈ C) remains bounded. In this paper, we investigate the dynamical behavior of a more general family of complex monic quadratic polynomials of the form gc:C → C, gc(z)=z²+az+c where a,c∈ 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 π emerges within the dynamical context of the system. All computations and visualizations are implemented in Python.