Sparsity over the Mandelbrot set

Some setup

import matplotlib.pyplot as plt
import chaoseverywhere as chaos

Sparsity

The use of projecting non-zeros values over an area to determine its area is very well known (it’s even how most of us learn the Monte-Carlo algorithm to calculate an approximation of pi). Let’s say someone needs to do the same process with the Mandelbrot set. Then, a simple way to graphically overset the two objets is like below.

fig = plt.figure()
mandel = chaos.Mandelbrot_disp(-.5,0,1.5).mandel_loop(go_up=True)
plt.imshow(mandel, cmap='Spectral')
chaos.sparse_matrix(400,400,.02)
plt.show()

Some values

It can be estimated that the Mandelbrot set has an area between \(1.50\) and \(1.51\). It was proved by Mitsuhiro Shishikura that the Haussdorf dimension of the boundary of the Mandelbrot set equals \(2\).