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.

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    $\begingroup$ It's not clear to me how this can be done 'visually' - another catch is that IIRC the finite approximations to the Mandelbrot set are much more easily shown to be connected, so the question of the set's connectivity is really a 'purely' mathematical one rather than a (pseudo)physical one. $\endgroup$ Apr 10, 2014 at 22:27

1 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 $\mathbb{D}$

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

 External rays and equipotential lines of Mandelbrot set as an images of external rays and circles of unit circle under Jungreis function $\Psi_M$

Image with src code and description

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    $\begingroup$ I believe the fact that the image of this covers everything outside the Cantor set $M$ is equivalent to local connectivity of $M$, which is unknown. $\endgroup$
    – GEdgar
    Apr 30, 2014 at 15:33
  • $\begingroup$ " It is conjectured that the Mandelbrot set is locally connected. This famous conjecture is known as MLC (for Mandelbrot Locally Connected)" See : mathoverflow.net/questions/95701/… $\endgroup$
    – Adam
    Apr 30, 2014 at 18:09

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