On a numerical approximation of the boundary structure and of the area of the Mandelbrot set

Abstract

Based on the boundary scanning method, a partition of the boundary of the Mandelbrot set is defined. The various classes of points of the boundary according to their divergence from the interior and the exterior of the Mandelbrot set are discussed. Then, numerical invariants of its structure under increase in the lattice resolution and the number of maximum iteration counts used to plot it are provided. Furthermore using Pick's theorem, an alternative numerical approximation of the area of the Mandelbrot set based on the number of points of the boundary and of the interior of the plottedMandelbrot set is provided. An analytic support of the numerical results obtained in approximating the area of the Mandelbrot set is also provided.