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This one was found by magnifying 280,000,000 times.

(This one was found by magnifying 280,000,000 times.)

In popular "zoom movies" of the Mandelbrot set the last image is often what appears to be an exact copy of the original set. This is always very moving as it indicates that after a long an fantastic journey or transformation we are in many ways still in the same place. Some writers have compared this self similarity to the way that history can repeat itself. So, you see, the concept is very compelling even to non-mathematicians.

I would like to know if such "Mini Mandelbrots" as they have been called by a few mathematics bloggers are ever perfectly isomorphic to the whole. The quote from the 2nd webpage indicates that at least some of them are not, but not all "Mini Mandelbrots" are found in the location described.

  • Why are there mini sets in the Mandelbrot Set?
  • ANATOMY OF THE MANDELBROT SET At the top and bottom of the main body are smaller bulbs with curved lines including a mini Mandelbrot set. More small bulbs of various descending sizes line the outer edge of the main body, curving inward toward the center and getting ever smaller. Between each bulb are smaller ones, and so on. On each bulb are more smaller bulbs, with more in between, to infinity. That means no matter how much you zoom in and magnify a bulb, you will always find more, smaller ones, each slightly different than the others. The same applies to all of the mini Mandelbrot sets.

If we can't find a perfect copy are some copies more perfect than others? How would one measure such a thing?

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  • $\begingroup$ Wikipedia seems to provide an answer. $\endgroup$
    – Samuel
    Commented Jul 16, 2011 at 4:09
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    $\begingroup$ There's an article by Michael Lyubich: smf4.emath.fr/Publications/Gazette/2007/113/… : "Baby Mandelbrot sets, Renormalization and MLC" $\endgroup$
    – graveolensa
    Commented Jul 16, 2011 at 8:22
  • $\begingroup$ It is easy to see that that cannot be perfect copies. The entire Mandelbrot set exists within the disc of radius 2 in $\mathbb{C}$. Whereas if you scale a "baby Mandelbrot" up, there will be a point in its "parent" more than two units distance away. $\endgroup$
    – Willie Wong
    Commented Jul 16, 2011 at 10:29
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    $\begingroup$ Of course, here I am assuming by "perfectly isomorphic" you mean under linear transformations of the two dimensional plane. You should probably state what notions of isomorphic you are willing to consider. $\endgroup$
    – Willie Wong
    Commented Jul 16, 2011 at 10:31
  • $\begingroup$ This is relevant I think. As well as this. $\endgroup$
    – Raskolnikov
    Commented Jul 16, 2011 at 11:12

2 Answers 2

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I found a distorted "mini mondelbrot"

.

,

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    $\begingroup$ I don't understand the votes to delete this. I think this shows vividly that at least one of the mini-Mandelbrots is not a copy. It is a useful contribution. $\endgroup$
    – Ross Millikan
    Commented Dec 6, 2013 at 21:19
  • $\begingroup$ @Ross Millikan: Yes, and this is the easiest answer. It is also "natural" to "expect" that other copies like the one in the OQ won't be "exact" either, just that the distortions are so subtle as to be invisible to the eye. $\endgroup$
    – The_Sympathizer
    Commented Jan 29, 2017 at 7:11
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    $\begingroup$ It would be best though if this answer provided the value of $c$ and size of the image in units of the complex plane so the readers could confirm it. $\endgroup$
    – Ruslan
    Commented Mar 3, 2017 at 11:44
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    $\begingroup$ @Ruslan : Real +0.272,195,0 Imaginary +0.005,404,7 zoom 15 $\endgroup$
    – mateuszica
    Commented Mar 4, 2017 at 16:00
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    $\begingroup$ OK, found it. Had to spend quite a while to figure out that +0.272,195,0 is a single number of $0.2721950$, and imaginary is likewise $0.0054047$. The size is about $2\times10^{-4}$. $\endgroup$
    – Ruslan
    Commented Mar 4, 2017 at 16:14
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They are not exact copies, as some of the comments have pointed out. They each have a point where the n-fold iteration of $z \to z^2+c$ produces zero. The reason they are approximate copies is that the iteration is differentiable, so points near the n-fold zero behave "like" points near 0 in the main bulb. There are a number of measures of geometric similarity you could use, for example comparing the ratios of radii of the satellites to the main cardioid, their angular positions, etc. You might be able to express these in terms of the derivatives of the iteration at the preimage of 0.

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