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Hi,

I'm curious about the distribution of "minibrots" within the Mandelbrot set.  I don't recall ever seeing a dense clustering of them.  So I wonder if that's even possible?

To make the question more specific, what's the greatest density of minibrots that you've actually seen in an image?  It's probably true that any image containing part of the actual set contains an infinite number of minibrots...  but they're always small enough that they're effectively invisible.

Have you ever seen an area with more than two or three minibrots actually large enough to be recognizable?  Or, is it perhaps a property of the M-set that this is impossible?

Pat

I am not sure as to what you would refer to as a "dense clustering".  But there are several places that you can see numerous minibrots within the same area of an image.


Yes, I have seen many such images.  Just look at the "stem" (or spike) as an example.  You can find several along it which are quite recognizable and large enough to view.

Here are some links for further reading:
    http://66.39.71.195/Derbyshire/manguide.html#mini (http://66.39.71.195/Derbyshire/manguide.html#mini)
    http://www.Nahee.com/spanky/www/fractint/dz.html (http://www.Nahee.com/spanky/www/fractint/dz.html)
    http://www.mrob.com/pub/muency/minibrot.html (http://www.mrob.com/pub/muency/minibrot.html)
 

Many years ago, I looked at the distribution of midgets (minibrots) along the spike out to -2.  If I remember correctly, the spike is dense in them; the only points along the spike not inside a midget are isolated Misiurewicz points (the tip at -2 is one, another is at approximately -1.543689).  However, the midgets get very small very quickly.  So, off of the spike, they become relatively more isolated--the smaller the midget, the more space (relative to its size) between them.

By "stem," do you mean the same area that lkmitch mentions here?


I should have specified "away from the x axis" in my original post.  The minibrots (or "midgets" as lkmitch says) there are no less minibrots than anywhere else, but I was thinking more along the lines of the rotated variations found elsewhere.

The fact that the rotated versions exist at all has always been one of the most amazing aspects of the M-set to me.

i have not thought about the density yet, i was just recalling what makes a minibrot,
isnt the center of each minibulb a strange attractor with getting longer and longer periods ?
e.g. point 0.0 +0.0i start and 0.0+0.0i seed has a single period ( 0.0+0.0i )
thus as a test should be quite demanding, because pixel center seldom resemble
exact center points ...

the scheme i can report of is that each "knob" you find on the spiral has its center a minibulb, and the bulbs
can be spread in many different ways, i think i have a picture with 3 clearly visible minibrots that i can remember :D

 :angel1:

Go to the UFVP website (http://www.fractal-animation.net/ufvp.html) and look for a viddie called "Slider_x".

I find the skewed minibrots to be much more incredible... I have seen some, occasionally (and no, the screen itself was not skewed), where the minibrot itself is not only rotated, but is taken with a skew. Rare, but existent.

http://www.fractal-animation.net/ufvp.html (http://www.fractal-animation.net/ufvp.html)

    http://www.fractal-animation.net/vid/slider_x.zip (http://www.fractal-animation.net/vid/slider_x.zip)  (30.7-MB)

Here is a link to an interesting image that was passed along from another group:

   http://www.abm-enterprises.net/fractals/mandelbrotgalaxywallpaper.html (http://www.abm-enterprises.net/fractals/mandelbrotgalaxywallpaper.html)
 

You see those patterns zooming into seahorse valley, midpoint, a little above the trench but not too close to the "top". It is clear that many minibrots are there, but often too small to see clearly. But in each frame you realize you are looking at zillions of them. And ponder that they are ALL connected to each other, even though you can't see the connecting "lines". My Universe #5 viddie (an early effort) goes there also.

Wow, great site.  The "Descent" animation is spectacular.

The animation called "where no man has gone before" is also interesting.  It mentions that the iteration count gets up around 2.1 billion!  I bet the authors of Fractint didn't foresee that they'd need arbitrary precision math just for the iteration count!

This recalls me 2 animations I did a few months ago, where 2 minibrots intersect each other. But the fractals are not the Mandelbrot set.

http://www.youtube.com/watch?v=efLkHgDMFOY

http://www.youtube.com/watch?v=3ew2RNbxaLc

Here are a few more links that might be of interest for this topic:

   http://www.Nahee.com/FOTD/FotD_10-01-16.html (http://www.Nahee.com/FOTD/FotD_10-01-16.html)
   http://www.Nahee.com/FOTD/FotD_10-01-17.html (http://www.Nahee.com/FOTD/FotD_10-01-17.html)
   http://www.Nahee.com/FOTD/FotD_10-01-18.html (http://www.Nahee.com/FOTD/FotD_10-01-18.html)
   http://www.Nahee.com/FOTD/FotD_10-01-19.html (http://www.Nahee.com/FOTD/FotD_10-01-19.html)
 

A gathering of "minibrots" in the Perpendicular Mandelbrot
http://www.fractalforums.com/videos/the-other-dimension-mandelbrot/msg11601/#msg11601
(http://www.fractalforums.com/gallery/1/492_17_01_10_11_08_35.jpeg)

Not quite. It is true that points on the spike all belong to midgets except for a set of measure zero, but a) they aren't all Misiurewicz points (there are also points on the spike with irrational external angles like the "band-merging point") and b) they aren't, strictly speaking, isolated; they are all points of accumulation of more such points. For example, consider the set of tips of spike minibrot spikes; the tip of the main spike is a Misiurewicz point and is one accumulation point of that set.

I have done a lot of work on this question. Because this topic is old, I started in a new topic:

http://www.fractalforums.com/mandelbrot-and-julia-set/clusters-of-minibrots/ (http://www.fractalforums.com/mandelbrot-and-julia-set/clusters-of-minibrots/)