Demonstrating that the Mandelbrot Set is connected

Question

I know that demonstrating the Mandelbrot Set is connected requires a non-trivial proof, and that Mandelbrot himself was fooled at first. But can it be demonstrated visually that the set is connected? If the Set were known to be path-connected, I might hope for a computer program that would let me select two points in the set and demonstrate the path between them. Showing connectedness visually seems harder to me. Is there an intuitive way to do it? Perhaps allow the user to draw a region R and then identify at least one point on the boundary of R is in the Mandelbrot Set? Unfortunately that seems less satisfying than demonstrating a path.

Answer 1

Showing connectedness visually seems harder to me. Is there an intuitive way to do it?

Maybe showing an explicit conformal isomorphism between the complement of the Mandelbrot set $M$ and the complement of the closed unit disk $\mathbb{D}$

$\Psi_M:\mathbb{\hat{C}}\setminus \overline{\mathbb{D}}\to\mathbb{\hat{C}}\setminus M$

Image with src code and description