The M-Set is connected. But the M-Set viewers I’ve found create cool pictures that don’t really show the connecting filaments.

This mini-Mandel beetle should be connected to a larger min-Mandel by a black filament going into its “butt crack”, but you can’t see it here:


Monochrome pictures that show just the M-Set itself (not the colorful divergence contours) end up showing seemingly disconnected pieces where the filaments are still so thin they disappear between samples:


Denser sampling helps, but it’s very expensive, and pictures end up showing a lot of grey “fuzz” where the filaments become sub-pixel thin, so we still can't see how the mini-Mandels are connected.


I thought of using Mathematica to draw contour plots of $|z_n|==2$. For any given zoom level, there should be some high value of n for which the contour would be visually indistinguishable from the $z_\infty$ contour, right? I posted questions on the Mathematica SE site to get help on this approach, but using ContourPlot[] as a “zoom” function is tricky, though I haven't given up on this approach. My MMa-SE post on this approach is HERE


I also thought of calculating huge numbers of Misiurewicz Points which lie on the boundary of the Mandelbrot Set. Surely enough of them will make a picture, right? But this plot of 17,723 points was as far as this clever approach went before numerically solving $2^{16}$-order irreducible polynomials proved slightly impractical:


(If you like that picture, the MMa code to make it is on my question HERE)

So, does anyone have any other ideas for showing the filaments? Or fixes to my various failed ideas?

  • $\begingroup$ I would search them out. The first problem is identifying the mini-Mandelbrots at all and where the filament should enter. Try finding an area (probably $3 \times 3$ pixels is enough) of black, but then you have to figure out if it is connected to something else or not. Maybe looking for a non-circular region is a good approach. Once you have the main lobe of the mini, as you say you know where the filament connects. Now search with high resolution in that area and trace the filament. Each time you extend the filament you know better where to look for the next bit. Plot it in black. $\endgroup$ Jan 18, 2016 at 0:40
  • $\begingroup$ I don't see what you're talking about. There appear to be zillions of filaments in that first image to me. $\endgroup$ Jan 18, 2016 at 3:10
  • $\begingroup$ @MarkMcClure Most of those pretty swirlies you see in the first pics aren't part of the M-Set at all; the colors encode how many iterations points outside the M-Set take to diverge. But as I described in the Question, there is an actual M-Set filament connecting every mini-Mandel to the bigger structure, but that's not visible in the pic. $\endgroup$ Jan 18, 2016 at 3:38
  • $\begingroup$ @JerryGuern I think your mistaken. Those swirls do, in fact envelope filaments. The fact that colors highlight filaments is exactly the reason that Hubbard suggested coloring these pictures in the first place. $\endgroup$ Jan 18, 2016 at 3:45
  • $\begingroup$ @MarkMcClure What I want to create is a picture similar to the 2nd picture, but with black filaments connecting those disconnected pieces. $\endgroup$ Jan 18, 2016 at 8:03

1 Answer 1


You want to colour the complement of the Mandelbrot set using the exterior distance estimate. For each pixel you calculate the running derivative (w.r.t. $c$) as well as the $z$ iterate, then at the end you combine them to get a distance estimate. Comparing this to the pixel spacing allows you to colour pixels close to the set black and pixels far from the set white. In practice I use $\tanh(d)$ (where $d$ is relative to pixel spacing) to give a smoother transition between white and black.

Mandelbrot set coloured with exterior distance estimate

Pseudo-code (use a larger escape radius for finer appearance):

foreach pixel c
  while not escaped and iteration limit not reached
    dz := 2 * z * dz + 1
    z := z^2 + c
  de := 2 * |z| * log(|z|) / |dz|
  d := de / pixel_spacing

Your second image will look something more like this:



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